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EXCLUDING A FOREST IN (THETA, PRISM)-FREE GRAPHS +TARA ABRISHAMI∗†, BOGDAN ALECU∗∗¶, MARIA CHUDNOVSKY∗∐, SEPEHR HAJEBI §, +AND SOPHIE SPIRKL§∥ +Abstract. Given a graph H, we prove that every (theta, prism)-free graph of sufficiently large +treewidth contains either a large clique or an induced subgraph isomorphic to H, if and only if +H is a forest. +1. Introduction +All graphs in this paper are finite and simple unless specified otherwise. Let G, H be graphs. +We say that G contains H if G has an induced subgraph isomorphic to H, and we say G is +H-free if G does not contain H. For a family H of graphs we say G is H-free if G is H-free +for every H ∈ H. A class of graphs is hereditary if it is closed under isomorphism and taking +induced subgraphs, or equivalently, if it is the class of all H-free graphs for some other family +H of graphs. +For a graph G = (V (G), E(G)), a tree decomposition (T, χ) of G consists of a tree T and a +map χ : V (T) → 2V (G) with the following properties: +(i) For every v ∈ V (G), there exists t ∈ V (T) such that v ∈ χ(t). +(ii) For every v1v2 ∈ E(G), there exists t ∈ V (T) such that v1, v2 ∈ χ(t). +(iii) For every v ∈ V (G), the subgraph of T induced by {t ∈ V (T) | v ∈ χ(t)} is connected. +For each t ∈ V (T), we refer to χ(t) as a bag of (T, χ). The width of a tree decomposition +(T, χ), denoted by width(T, χ), is maxt∈V (T) |χ(t)| − 1. The treewidth of G, denoted by tw(G), +is the minimum width of a tree decomposition of G. +Treewidth was first popularized by Robertson and Seymour in their graph minors project, +and has attracted a great deal of interest over the past three decades. Particularly, graphs of +bounded treewidth have been shown to be well-behaved from structural [19] and algorithmic [6] +viewpoints. +This motivates investigating the structure of graphs with large treewidth, and especially, +the substructures emerging in them. The canonical result in this realm is the Grid Theorem +of Robertson and Seymour [19], the following, which describes the unavoidable subgraphs of +graphs with large treewidth. For a positive integer t, the (t × t)-wall, denoted by Wt×t, is a +planar graph with maximum degree three and treewidth t (see Figure 1; a formal definition can +be found in [3]). +∗Princeton University, Princeton, NJ, USA +∗∗School of Computing, University of Leeds, Leeds, UK +§Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, +Canada +† Supported by NSF-EPSRC Grant DMS-2120644. +∐ Supported by NSF-EPSRC Grant DMS-2120644 and by AFOSR grant FA9550-22-1-0083. +¶ Supported by DMS-EPSRC Grant EP/V002813/1. +∥ We acknowledge the support of the Natural Sciences and Engineering Research Council of +Canada (NSERC), [funding reference number RGPIN-2020-03912]. Cette recherche a été financée +par le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG), [numéro de +référence RGPIN-2020-03912]. This project was funded in part by the Government of Ontario. +Date: January 6, 2023. +1 + +2 +INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII. +Figure 1. W5×5 +Theorem 1.1 (Robertson and Seymour [19]). For every integer t ≥ 1 there exists w = w(t) ≥ 1 +such that every graph of treewidth more than w contains a subdivision of Wt×t as a subgraph. +Theorem 1.1 can also be reformulated into a full characterization of unavoidable minors in +graphs of large treewidth, that every graph of sufficiently large treewidth contains any given +planar graph as a minor (and no non-planar graph has this property). In contrast, unavoidable +induced subgraphs of graphs with large treewidth are far from completely understood. There +are some natural candidates though, which we refer to as the “basic obstructions”: complete +graphs and complete bipartite graphs, subdivided walls mentioned above, and line graphs of +subdivided walls, where the line graph L(F) of a graph F is the graph with vertex set E(F), +such that two vertices of L(F) are adjacent if and only if the corresponding edges of F share an +end. Note that the complete graph Kt+1, the complete bipartite graph Kt,t, and the line graph +of every subdivision of Wt×t all have treewidth t. For a positive integer t, let us say a graph H +is a t-basic obstruction if H is one of the following graphs: Kt, Kt,t, a subdivision of Wt×t, or +the line graph of a subdivision of Wt×t. We say a graph G is t-clean if G does not contain a +t-basic obstruction. +The basic obstructions do not form a comprehensive list of induced subgraph obstructions +for bounded treewidth. Equivalently, there are t-clean graphs of arbitrarily large treewidth for +small values of t. A well-known hereditary class of graphs evidencing this fact is the class of +even-hole-free graphs, where a hole is an induced cycle on at least four vertices, the length of +a hole is its number of edges and an even hole is a hole with even length. In fact, for every +positive integer t ≥ 1, one may observe that an even-hole-free graph is t-clean if and only if it is +Kt-free. It is therefore tempting to ask whether even-hole-free graphs excluding a fixed complete +graph have bounded treewidth. Sintiari and Trotignon [20] answered this with a vehement no, +providing a construction of (even-hole, K4)-free graphs with arbitrarily large treewidth, hence +proving that there are t-clean (even-hole-free) graphs of arbitrarily large treewidth for every +fixed t ≥ 4. In addition, graphs from this construction are rather sparse, in the sense that they +exclude short holes. +Theorem 1.2 (Sintiari and Trotignon [20]). For all integers w, l ≥ 1, there exists an (even-hole, +K4)-free graph Gw,l of treewidth more than w and with no hole of length at most l. +Note that t-clean graphs for t ≤ 2 have empty vertex set or edge set. But one might still +hope for 3-clean graphs to have bounded treewidth. This is in fact supported by a result from +[7] asserting that 3-clean even-hole-free graphs have treewidth at most five. However, another +construction by Sintiari and Trotignon [20] shows that being 3-clean fails to guarantee bounded +treewidth in the more general class of theta-free graphs (see the next section for the definition of +a theta; one may check that the every t-basic obstruction for t ≥ 3 contains either a theta or a +triangle). Indeed, the treewidth of theta-free graphs remains unbounded even when forbidding +short cycles. +Theorem 1.3 (Sintiari and Trotignon [20]). For all integers w, g ≥ 1, there exists a theta-free +graph Gw,g of treewidth more than w and girth more than g. +A natural question to ask then is what further conditions must be imposed to force bounded +treewidth in even-hole-free graphs. For instance, graphs from both Theorems 1.2 and 1.3 have + +INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII. +3 +vertices of arbitrarily large degree, and so it was conjectured in [1] that (theta, triangle)- +free graphs of bounded maximum degree have bounded treewidth and even-hole-free graphs +of bounded maximum degree have bounded treewidth. These were proved in [3] and [4], respec- +tively. In the same paper [1], a stronger conjecture was made, asserting that basic obstructions +are in fact the only obstructions to bounded treewidth in graphs of bounded maximum degree. +This was later proved in [16], which closed the line of inquiry into graph classes of bounded +maximum degree. +Theorem 1.4 (Korhonen [16]). For all integers t, δ ≥ 1, there exists w = w(t, δ) such that every +t-clean graph of maximum degree at most δ has treewidth at most w. +Despite its generality, the proof of Theorem 1.4 is surprisingly short. However, the case of +proper hereditary classes containing graphs of unbounded maximum degree seems to be much +harder. For graph classes G and H, let us say H modulates G if for every positive integer t, +there exists a positive integer w(t) (depending on G and H) such that every t-clean H-free graph +in G has treewidth at most w(t). An induced-subgraph analogue to Theorem 1.1 is therefore +equivalent to a full characterization of graph classes H which modulate the class of all graphs. +This remains out of reach, but the special case where |H| = 1 turns out to be more approachable. +For a graph H and a graph class G, let us say H modulates G if {H} modulates G. Building +on a method from [17], recently we characterized all graphs H which modulate the class of all +graphs: +Theorem 1.5 (Abrishami, Alecu, Chudnovsky, Hajebi and Spirkl [2]). Let H be a graph. Then +H modulates the class of all graphs if and only if H is a subdivided star forest, that is, a forest +in which every component has at most one vertex of degree more than two. +In general, for a hereditary class G containing t-clean graphs of arbitrarily large treewidth for +small t, one may ask for a characterization of graphs H modulating G. Given Theorem 1.2, a +natural class G to consider is the class of even-hole-free graphs. Note that Theorem 1.2 shows +that a graph H modulates even-hole-free graphs only if H is a chordal graph (that is, a graph +with no hole) of clique number at most three. As far as we know, the converse may also be +true, that every chordal graph of clique number at most three modulates even-hole-free graphs. +In fact, in this paper we narrow the gap, showing that every chordal graph of clique number at +most two, that is, every forest, modulates the class of even-hole-free graphs. +Theorem 1.6. For every forest H and every integer t ≥ 1, every even-hole-free graph of suffi- +ciently large treewidth contains either H or a clique of cardinality t. +This aligns with the observation [21] that every forest is contained in some graph Gw,l from +Theorem 1.2. As mentioned above, one way to improve on Theorem 1.6 is to push H towards +being an arbitrary chordal graph of clique number three. Another way to strengthen Theorem 1.6 +is to find a superclass G of even-hole-free graphs for which forests are the only graphs modulating +G. While the former remains open, we provide an appealing answer to the latter: our main result +shows that forests are exactly the graphs which modulate the class of (theta, prism)-free graphs +(see the next section for the definition of a prism; again one may check that in (theta, prism)-free +graphs, being t-clean is equivalent to being Kt-free for every positive integer t). +Theorem 1.7. Let H be a graph. Then H modulates (theta, prism)-free graphs if and only if +H is a forest. In other words, given a graph H, for every integer t ≥ 1, every (theta, prism)-free +graph of sufficiently large treewidth contains either H or a clique of cardinality t, if and only if +H is a forest. +Let C be the class of all (theta, prism)-free graphs. It is easily seen that C contains all even- +hole-free graphs, and so Theorem 1.7 implies Theorem 1.6. Note that the “only if” direction +of Theorem 1.7 follows immediately from Theorem 1.3 as prisms contain triangles. Since every +forest is an induced subgraph of a tree, in order to prove Theorem 1.7, it suffices to prove + +4 +INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII. +Theorem 1.8 below, which we do in Section 7. For a positive integer t and a tree F, we denote +by Ct the class of all graphs in C with no clique of cardinality t (that is, t-clean graph in C), and +by Ct(F) the class of all F-free graphs in Ct. +Theorem 1.8. For every tree F and every integer t ≥ 1, there exists an integer τ(F, t) ≥ 1 +such that every graph in Ct(F) has treewidth at most τ(F, t). +We conclude this introduction by sketching our proofs (the terms we use here are defined in +later sections). The proof of Theorem 1.8 begins with a two-step preparation which culminates +in the proof of Theorem 6.2, a result we will also use in subsequent papers in this series. As the +first step, inspired by a result from [9], we show that for every graph G ∈ C which contains a +pyramid with certain conditions on the apex and its neighbors, G admits a construction which we +call a “(T, a)-strip-structure,” where a is the apex of the pyramid and T is an optimally chosen +tree. Roughly speaking, we show that G\{a} can be partitioned into two induced subgraphs H +and J where H is more or less similar to the line graph of the tree T and every vertex in J with a +neighbor in H attaches at a pyramid lurking in H in a restricted way; we call the latter vertices +“jewels”. The proof of this theorem occupies Sections 3 and 4. The second step is to employ the +previous result to show that if G ∈ Ct admits a (C, a)-strip-structure where C is a caterpillar, +then every vertex in G \ NG[a] can be separated from a by removing a few vertices (our proof +works more generally when C is any tree of bounded maximum degree, but the caterpillar case +suffices for our application). We prove this in Section 6. The central difficulty in the proof is to +deal with the jewels separately. This is surmounted in Section 5 where we prove several results +concerning the properties of jewels. Most notably, we show that jewels only attach at “local +areas of the line-graph-like part” of G, and that only a few jewels attach at each local area. This +concludes the preparation for proving Theorem 1.8. +Next, we embark on the proof of Theorem 1.8. We assume that G ∈ Ct has large treewidth, +which together with results from Section 2 implies that G contains two vertices x, y joined by +many pairwise internally disjoint induced paths P1, . . . , Pm. Now we analyze the structure of +the graph G[P1 ∪ · · · ∪ Pm]. It turns out that, if m is large enough, then either +• there are many paths among Pi’s whose union H admits a (C, x)-strip-structure for some +caterpillar C, or +• for some large value of d, G[P1 ∪ · · · ∪ Pm] contains a tree S isomorphic to the complete +bipartite graph K1,d, such that x is the vertex of degree d in S, and for every leaf l of S, +there are many pairwise internally disjoint induced paths between l and y, such that in +addition, paths corresponding to distinct leaves of S are also pairwise internally disjoint. +The former case implies that y can be separated from x by removing few vertices, which using +a result from Section 6, yields a contradiction with Menger’s theorem. The latter case is the first +step towards building the large tree in G as a subgraph. We now iterate the argument we just +described, applying it to each leaf l of S and y, obtaining larger and larger trees. The process is +stopped once we reach a sufficiently large tree as a subgraph of G. This, combined with the fact +that G ∈ Ct and a result of Kierstead and Penrice [15], yields the desired tree F as an induced +subgraph of G. +This paper is organized as follows. Section 2 covers preliminary definitions as well as some +results from the literature used in our proofs. Section 3 investigates the behavior of pyramids in +graphs from C. Section 4 is devoted to defining strip-structures and jewels, and showing how they +arise from pyramids in graphs in C. Section 5 takes a closer look at jewels for the strip-structures +obtained in Section 4. In Section 6 we show that admitting certain strip-structures weakens the +connectivity of most vertices to the apex. Finally, in Section 7, we prove Theorem 1.8. +2. Preliminaries and results from the literature +Let G = (V (G), E(G)) be a graph. For a set X ⊆ V (G) we denote by G[X] the subgraph of +G induced by X. For X ⊆ V (G)∪E(G), G\X denotes the subgraph of G obtained by removing + +INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII. +5 +X. Note that if X ⊆ V (G), then G \ X denotes the subgraph of G induced by V (G) \ X. In +this paper, we use induced subgraphs and their vertex sets interchangeably. +Let x ∈ G and d be a positive integer. We denote by N d +G(x) the set of all vertices in G at +distance d from some x, and by N d +G[x] the set of all vertices in G at distance at most d from x. +We write NG(x) for N 1 +G(x) and NG[x] for N 1 +G[x]. For an induced subgraph H of G, we define +NH(x) = NG(x) ∩ H, NH[x] = NG[x] ∩ H. Also, for X ⊆ G, we denote by NG(X) the set of all +vertices in G \ X with at least one neighbor in X, and define NG[X] = NG(X) ∪ X. +Let X, Y ⊆ G be disjoint. We say X is complete to Y if all edges with an end in X and an +end in Y are present in G, and X is anticomplete to Y if there are no edges between X and Y . +A path in G is an induced subgraph of G that is a path. +If P is a path in G, we write +P = p1- · · · -pk to mean that V (P) = {p1, . . . , pk} and pi is adjacent to pj if and only if |i−j| = 1. +We call the vertices p1 and pk the ends of P, and say that P is from p1 to pk. The interior of +P, denoted by P ∗, is the set P \ {p1, pk}. The length of a path is its number of edges (so a path +of length at most one has empty interior). Similarly, if C is a cycle, we write C = c1- · · · -ck-c1 +to mean that V (C) = {c1, . . . , ck} and ci is adjacent to cj if and only if |i − j| ∈ {1, k − 1}. The +length of a cycle is its number edges (or equivalently, vertices.) +A theta is a graph Θ consisting of two non-adjacent vertices a, b, called the ends of Θ, and +three pairwise internally disjoint paths P1, P2, P3 from a to b of length at least two, called the +paths of Θ, such that P ∗ +1 , P ∗ +2 , P ∗ +3 are pairwise anticomplete to each other. For a graph G, by a +theta in G we mean an induced subgraph of G which is a theta. +A prism is a graph Π consisting of two disjoint triangles {a1, a2, a3}, {b1, b2, b3} called the +triangles of Π, and three pairwise disjoint paths P1, P2, P3 called the paths of Π, where Pi has +ends ai, bi for each i ∈ {1, 2, 3}, and for distinct i, j ∈ {1, 2, 3}, aiaj and bibj are the only edges +between Pi and Pj. For a graph G, by a prism in G we mean an induced subgraph of G which +is a prism. +A pyramid is a graph Σ consisting of a vertex a, a triangle {b1, b2, b3} and three paths P1, P2, P3 +of length at least one with Pi from a to bi for each i ∈ {1, 2, 3} and otherwise pairwise disjoint, +such that for distinct i, j ∈ {1, 2, 3}, bibj is the only edge between Pi \ {a} and Pj \ {a}, and +at most one of P1, P2, P3 has length exactly one. We say that a is the apex of the pyramid and +b1b2b3 is the base of the pyramid. The pyramid Σ is said to be long if Pi has length more than +one for every i ∈ {1, 2, 3}. For a graph G, by a pyramid in G we mean an induced subgraph of +G which is a pyramid. +Figure 2. Theta, pyramid and prism. The dotted lines represent paths of +length at least one. +Let us now mention a few results from the literature which we will use in this paper. Let +G be a graph. By a separation in G we mean a triple (L, M, R) of pairwise disjoint subsets of +vertices in G with L ∪ M ∪ R = G, such that neither L nor R is empty and L is anticomplete +to R in G. Let x, y ∈ G be distinct. We say a set M ⊆ G \ {x, y} separates x and y if there +exists a separation (L, M, R) in G with x ∈ L and y ∈ R. Also, for disjoint sets X, Y ⊆ G, we +say a set M ⊆ G \ (X ∪ Y ) separates X and Y if there exists a separation (L, M, R) in G with +X ⊆ L and Y ⊆ R. If X = {x}, we say that M separates x and Y to mean M separates X and +Y . Recall the following well-known theorem of Menger [18]: + +6 +INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII. +Theorem 2.1 (Menger [18]). Let k ≥ 1 be an integer, let G be a graph and let x, y ∈ G be +distinct and non-adjacent. Then either there exists a set M ⊆ G \ {x, y} with |M| < k such that +M separates x and y, or there are k pairwise internally disjoint paths in G from x to y. +Let k be a positive integer and let G be a graph. A strong k-block in G is a set B of at least k +vertices in G such that for every 2-subset {x, y} of B, there exists a collection P{x,y} of at least +k distinct and pairwise internally disjoint paths in G from x to y, where for every two distinct +2-subsets {x, y}, {x′, y′} ⊆ B of G, and every choice of paths P ∈ P{x,y} and P ′ ∈ P{x′,y′}, we +have P ∩ P ′ = {x, y} ∩ {x′, y′}. +For a tree T and xy ∈ E(T), we denote by Tx,y the component of T − xy containing x. Let G +be a graph and (T, χ) be a tree decomposition for G. For every S ⊆ T, let χ(S) = � +x∈S χ(x). +By an adhesion of (T, χ) we mean the set χ(x) ∩ χ(y) = χ(Tx,y) ∩ χ(Ty,x) for some xy ∈ E(T). +For every x ∈ V (T), by the torso at x, denoted by ˆχ(x), we mean the graph obtained from +the bag χ(x) by, for each y ∈ NT (x), adding an edge between every two non-adjacent vertices +u, v ∈ χ(x, y). In [2], we used Theorem 1.4 and the following result from [13]: +Theorem 2.2 (Erde and Weißauer [13], see also [14]). Let r be a positive integer, and let G +be a graph containing no subdivision of Kr as a subgraph. Then G admits a tree decomposition +(T, χ) for which the following hold. +• Every adhesion of (T, χ) has cardinality less than r2. +• For every x ∈ V (T), either ˆχ(x) has fewer than r2 vertices of degree at least 2r4, or ˆχ(x) +has no minor isomorphic to K2r2. +to prove the following. +Theorem 2.3 (Abrishami, Alecu, Chudnovsky, Hajebi and Spirkl [2]). Let k, t ≥ 1 be integers. +Then there exists an integer w = w(k, t) ≥ 1 such that every t-clean graph with no strong k-block +has treewidth at most w. +Note that for every t ≥ 3, every subdivision of Wt×t contains a theta and the line graph of +every subdivision of Wt×t contains a prism. It follows that for every t ≥ 1, every graph in Ct is +t-clean, and so the following is immediate from Theorem 2.3: +Corollary 2.4. For all integers k, t ≥ 1, there exists an integer β = β(k, t) such that every +graph in Ct with no strong k-block has treewidth at most β(k, t). +A vertex v in a graph G is said to be a branch vertex if v has degree more than two. By +a caterpillar we mean a tree C with maximum degree three such that there is a path P in +C containing all branch vertices of C (our definition of a caterpillar is non-standard for two +reasons: a caterpillar is often allowed to be of arbitrary maximum degree, and the path P from +the definition often contains all vertices of degree more than one). By a subdivided star we mean +a graph isomorphic to a subdivision of the complete bipartite graph K1,δ for some δ ≥ 3. In +other words, a subdivided star is a tree with exactly one branch vertex, which we call its root. +For every graph H, a vertex v of H is said to be simplicial if NH(v) is a clique. We denote by +Z(H) the set of all simplicial vertices of H. Note that for every tree T, Z(T) is the set of all +leaves of T. An edge e of a tree T is said to be a leaf-edge of T if e is incident with a leaf of +T. It follows that if H is the line graph of a tree T, then Z(H) is the set of all vertices in H +corresponding to the leaf-edges of T. The following is proved in [2] based on (and refining) a +result from [11]. +Theorem 2.5 (Abrishami, Alecu, Chudnovsky, Hajebi and Spirkl [2]). For every integer h ≥ 1, +there exists an integer µ = µ(h) ≥ 1 with the following property. Let G be a connected graph +with no clique of cardinality h and let S ⊆ G such that |S| ≥ µ. Then either some path in G +contains h vertices from S, or there is an induced subgraph H of G with |H ∩ S| = h for which +one of the following holds. +• H is either a caterpillar or the line graph of a caterpillar with H ∩ S = Z(H). +• H is a subdivided star with root r such that Z(H) ⊆ H ∩ S ⊆ Z(H) ∪ {r}. + +INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII. +7 +3. Jumps and jewels on pyramids with trapped apices +For a graph G, an induced subgraph H of G and a vertex a ∈ H, we say a is trapped in H if +• we have N 2 +G[a] ⊆ H, and; +• every vertex in NH(a) = NG(a) has degree two in H (and so in G). +The goal of this section is, for a graph G ∈ C, H ⊆ G and a pyramid Σ in H, to investigate the +adjacency between Σ and a path in G \ H, assuming that the apex of Σ is trapped in H. This +will be of essential use in the next section. +We begin with a few definitions. Let G be a graph and let Σ be a pyramid in G with apex +a, base b1b2b3 and paths P1, P2, P3. A set X ⊆ Σ is said to be local (in Σ) if either X ⊆ Pi for +some i ∈ {1, 2, 3} or X ⊆ {b1, b2, b3}. Let P be a path in G \ Σ with (not necessarily distinct) +ends p1, p2. For i ∈ {1, 2, 3}, we say P is a corner path for Σ at bi if +• p1 has at least one neighbor in Pi \ {bi}; +• p2 is complete to {b1, b2, b3} \ {bi}, and; +• except for the edges between {p1, p2} and Σ described in the above two bullets, there is +no edge with an end in P and an end in Σ \ {bi}. +By a corner path for Σ we mean a corner path for Σ at one of b1, b2 or b3. +Let p ∈ G \ Σ. Then p is said to be narrow for Σ if NΣ(p) is local in Σ. Otherwise, we say +p is wide for Σ. For i ∈ {1, 2, 3}, we say p is a jewel for Σ at bi if p is anticomplete to Pi (in +particular, p is anticomplete to a), and for every j ∈ {1, 2, 3} \ {i}, we have NPj(p) = NPj[bj]. +By a jewel for Σ we mean a jewel for Σ at one of b1, b2 or b3. Note that if p is either a corner +path or a jewel for Σ, then p is wide for Σ. The following lemma establishes a converse to this +fact for graphs in C and pyramids with a trapped apex. +Lemma 3.1. Let G ∈ C be graph, let H ⊆ G and let a ∈ H be trapped in H. Let Σ be a pyramid +in H with apex a, base b1b2b3 and paths P1, P2, P3. Let p ∈ G \ H. Then p is wide for Σ if and +only if p is either a corner path for Σ or a jewel for Σ. +Proof. We only need to prove the “only if” direction. Assume that p ∈ G \ H is wide for Σ and +p is not a corner path for Σ. Since a is trapped in H and p ∈ G \ H, it follows that Σ is long +and p is anticomplete to NΣ[a]. First, we show that: +(1) There exists i ∈ {1, 2, 3} for which p is anticomplete to Pi. +Suppose for a contradiction that p has a neighbor in each of P1, P2, P3. Since p is wide for Σ +and p is not a corner path for Σ, we may assume without loss of generality that p has a neighbor +in P ∗ +1 and a neighbor in P ∗ +2 . For each i ∈ {1, 2, 3}, traversing Pi from a to bi, let xi be the first +neighbor of p in Pi. Since a is trapped, it follows that x1 ∈ P ∗ +1 , x2 ∈ P ∗ +2 and x3 ∈ P3\NΣ[a]. But +then there is a theta in G with ends a, p and paths a-Pi-xi-p for i ∈ {1, 2, 3}, a contradiction. +This proves (1). +By (1) and without loss of generality, we may assume that p is anticomplete to P3. Note that +since p is wide for Σ, it follows that for every j ∈ {1, 2}, p has a neighbor in Pj, and there exists +j ∈ {1, 2} for which p has a neighbor in P ∗ +j . For each j ∈ {1, 2}, traversing Pj from a to bj, let xj +and yj be the first and the last neighbor of p in Pj, respectively. Then we have xj ∈ P ∗ +j \NPj(a) +for some j ∈ {1, 2}. In fact, the following holds. +(2) For every j ∈ {1, 2}, we have xj ∈ P ∗ +j \ NPj(a). +Suppose not. Then since p is wide for Σ, we may assume without loss of generality that p has +a neighbor in P ∗ +1 and we have x2 = y2 = b2. But now there is a theta in G with ends a, b2 and +paths a-P1-x1-p-b2, a-P2-b2 and a-P3-b3-b2, a contradiction. This proves (2). +(3) For every j ∈ {1, 2}, NPj(p) is a clique of cardinal ity two. +Suppose not. Then we may assume without loss of generality that either x1 = y1 or x1 and y1 +are distinct and non-adjacent. By (2), for every j ∈ {1, 2}, we have xj ∈ P ∗ +j \NPj(a). Therefore, + +8 +INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII. +if x1 = y1, then there is a theta in G with ends a, x1 and paths a-P1-x1, a-P2-x2-p-x1 and +a-P3-b3-b1-P1-x1, which is impossible. Thus, x1 and y1 are distinct and non-adjacent. But now +there is a theta in G with ends a, p and paths a-P1-x1-p, a-P2-x2-p and a-P3-b3-b1-P1-y1-p, a +contradiction. This proves (3). +The proof is almost concluded. By (3), for every j ∈ {1, 2}, we have NPj(p) = {xj, yj} and xj +is adjacent to yj. If yj ∈ P ∗ +j for some j ∈ {1, 2}, then there is a prism in G with triangles xjyjp +and b1b2b3 and paths xj-Pj-a-P3-b3, yj-Pj-bj and p-y3−j-P3−j-b3−j, a contradiction. Hence, we +have yj = bj for every j ∈ {1, 2}, and so p is a jewel corner for Σ at bi. This completes the proof +of Lemma 3.1. +■ +We can now prove the main result of this section. +Theorem 3.2. Let G ∈ C be a graph, let H ⊆ G and let a ∈ H be trapped in H. Let Σ be a +pyramid in H with apex a, base b1b2b3 and paths P1, P2, P3. Let P be a path in G \ H. Then +one of the following holds. +• NΣ(P) is local in Σ. +• P contains a corner path for Σ. +• P contains a jewel for Σ. +Proof. Suppose for a contradiction that there exists a path P in G \ H for which none of the +outcomes of Theorem 3.2 hold. We choose such a path P with |P| as small as possible. It follows +that NΣ(P) is not local in Σ, NΣ(X) is local in Σ for every connected set X ⊊ P, P contains +no corner path for Σ and P contains no jewel for Σ. Therefore, by Lemma 3.1, we have |P| > 1. +Since a is trapped in H and P ⊆ G\H, it follows that Σ is long and P is anticomplete to NΣ[a]. +For every i ∈ {1, 2, 3}, let P ′ +i = Pi \ NPi[a]. Since NΣ(P) is not local and P is minimal subject +to this property, we may assume without loss of generality that +• NΣ(p1) ⊆ P ′ +1 and p1 has a neighbor in P ′ +1 \ {b1}, and; +• p2 has a neighbor in P ′ +2, and either NΣ(p2) ⊆ P ′ +2, or NΣ(p2) ⊆ {b1, b2, b3}. +It follows from the choice of P that P ∗ is anticomplete to Σ\{b1}. For each i ∈ {1, 2}, traversing +Pi from a to bi, let xi and yi be the first and the last neighbor of pi in Pi, respectively. So we +have x1 ∈ P ′ +1 \ {b1}, y1 ∈ P ′ +1 and x2, y2 ∈ P ′ +2. In fact, the following holds. +(4) We have x2 ∈ P ′ +2 \ {b2}. +Suppose not. Then we have x2 = y2 = b2, and so b2 ∈ NΣ(p2) ⊆ {b1, b2, b3}. Note that if p2 +is adjacent to b3, then P is a corner path for Σ at b1, which is impossible. So p2 is not adjacent +to b3. But now there is a theta in G with ends a, b2 and paths a-P1-x1-p1-P-p2-b2, a-P2-b2 and +a-P3-b3-b2, a contradiction. This proves (4). +In view of (4) and the choice of P, we conclude that P ∗ is anticomplete to Σ, and for every +i ∈ {1, 2}, we have NΣ(pi) = NP ′ +i (pi), xi ∈ P ′ +i \ {bi} and yi ∈ P ′ +i. +(5) For every i ∈ {1, 2}, xi and yi are distinct and adjacent. +Suppose not. Then we may assume without loss of generality that either x1 = y1 or x1 and +y1 are distinct and non-adjacent. In the former case, there is a theta in G with ends a, x1 and +paths a-P1-x1, a-P2-x2-p2-P-p1-x1 and a-P3-b3-b1-P1-x1, which is impossible. It follows that x1 +and y1 are distinct and non-adjacent. But then there is a theta in G with ends a, p1 and paths +a-P1-x1-p1, a-P2-x2-p2-P-p1 and a-P3-b3-b1-P1-y1-p1, a contradiction. This proves (5). +By (5), for every i ∈ {1, 2}, we have NPi(p) = {xi, yi} and xi is adjacent to yi. But now there is +a prism in G with triangles p1x1y1 and p2x2y2 and paths P, x1-P1-a-P2-x2 and y1-P1-b1-b2-P2-y2, +a contradiction. This completes the proof of Theorem 3.2. +■ + +INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII. +9 +4. Strip structures with an ornament of jewels +The main result of this section, Theorem 4.2, provides a description of the structure of graphs +in C which have an induced subgraph containing a pyramid with a trapped apex. +We first set up a framework that allows us to think of a pyramid with apex a as a special case +of a construction similar to the line graph of a tree T, which we call a “(T, a)-strip-structure.” +We start with an induced subgraph W of G that admits an “optimal” (T, a)-strip-structure in +G in a certain sense, and show that the rest of the graph fits into the same construction, except +for vertices which are jewels for certain canonically positioned pyramids in W. +First, we need to properly define a strip-structure (this is similar to [8], [9] and [10]). A +tree T is said to be smooth if T has at least three vertices and every vertex of T is either +a branch vertex or a leaf. +Let G be a graph, let a ∈ G, let T be a smooth tree, and let +η : V (T) ∪ E(T) ∪ (E(T) × V (T)) → 2G\{a} be a function. For every S ⊆ V (T), we define +η(S) = � +v∈S,e∈E(T[S])(η(v) ∪ η(e)) and η+(S) = η(S) ∪ {a}. For every vertex v ∈ V (T), we +define Bη(v) to be the union of all sets η(e, v) taken over all edges e ∈ E(T) incident with v (we +often omit the subscript η unless there is ambiguity). +The function η is said to be a (T, a)-strip-structure in G if the following conditions are satisfied. +(S1) For all distinct o, o′ ∈ V (T) ∪ E(T), we have η(o) ∩ η(o′) = ∅. +(S2) If l ∈ V (T) is a leaf of T, then η(l) is empty. +(S3) For all e ∈ E(T) and v ∈ V (T), we have η(e, v) ⊆ η(e) and η(e, v) ̸= ∅ if and only if e is +incident with v. +(S4) For all distinct edges e, f ∈ E(T) and every vertex v ∈ V (T), η(e, v) is complete to η(f, v), +and there are no other edges between η(e) and η(f). In particular, if e and f share no end, +the η(e) is anticomplete to η(f). +(S5) For every e ∈ E(T) with ends u, v, define η◦(e) = η(e) \ (η(e, u) ∪ η(e, v)). Then for every +vertex x ∈ η(e), there is a path in η(e) from x to a vertex in η(e, u) with interior contained +in η◦(e), and there is a path in η(e) from x to a vertex in η(e, v) with interior contained in +η◦(e). +(S6) For all v ∈ V (T) and e ∈ E(T), η(v) is anticomplete to η(e) \ η(e, v). In other words, we +have Nη(T)(η(v)) ⊆ Bη(v). +(S7) For every v ∈ V (T) and every connected component D of η(v), we have NBη(v)(D) ̸= ∅. +(S8) For every leaf l ∈ V (T) of T, assuming e ∈ E(T) to be the leaf-edge of T incident with l, +a is complete to η(e, l). Also, a has no other neighbors in η(T). +Let S ⊆ η(T). We say that S is local in η if S ⊆ η(e) for some e ∈ E(T) or S ⊆ Bη(v) ∪ η(v) +for some v ∈ V (T). The following lemma shows that every non-local subset contains a 2-subset +(that is, a subset of cardinality two) which is non-local. +Lemma 4.1. Let G be a graph and a ∈ G. Let T be a smooth tree and η be a (T, a)-strip- +structure in G. Assume also that C ⊆ η(T) is not local in η. Then there is a 2-subset of C +which is not local in η. +Proof. First, suppose there exists a vertex x ∈ C ∩ η◦(e) for some e ∈ E(T). Since C is not +local, there exists y ∈ C \ η(e). Now {x, y} is a 2-subset of C which is not local in η, as desired. +Therefore, we may assume that C ⊆ � +v∈V (T)(B(v) ∪ η(v)). Since the empty set is local in η, +we have C ̸= ∅; thus, we may pick x ∈ C, v ∈ V (T) and e ∈ E(T) such that x ∈ η(e, v) ∪ η(v). +If there exists a vertex y ∈ C \ (η(e) ∪ B(v) ∪ η(v)), then {x, y} is a 2-subset of C which is not +local in η, and so we are done. Consequently, we may assume that C ⊆ η(e) ∪ B(v) ∪ η(v). +Since C is not local, there exist x′ ∈ η(e) \ (B(v) ∪ η(v))) and y′ ∈ (B(v) ∪ η(v)) \ η(e) such that +{x′, y′} ⊆ C. Now {x′, y′} is a 2-subset of C which is not local in η, as required. This completes +the proof of Lemma 4.1. +■ +In order to state and prove the main result of this section, we need to define several notions +related to strip-structures. From here until the statement of Theorem 4.2, let us fix a graph G, +a vertex a ∈ G, a smooth tree T and a (T, a)-strip-structure η in G. + +10 +INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII. +For every edge e ∈ E(T) with ends u, v, by an η(e)-rung, we mean a path P in η(e) ⊆ η(T) +for which either |P| = 1 and P ⊆ η(e, u) ∩ η(e, v), or P has an end in η(e, u) \ η(e, v) and an +end in η(e, v) \ η(e, u) and we have P ∗ ⊆ η◦(e). Equivalently, a path P in η(e) is an η(e)-rung +if P has an end in η(e, u) and an end in η(e, v) and we have |P ∩ η(e, u)| = |P ∩ η(e, v)| = 1. It +follows from (S5) that every vertex in η(e) \ η◦(e) is contained in an η(e)-rung. In particular, +if either η(e, u) ⊆ η(e, v) or η(e, v) ⊆ η(e, u), then η(e, u) = η(e, v). An η(e)-rung is said to be +long if it is of non-zero length. +For every edge e ∈ E(T), let ˜η(e) be the set of vertices in η(e) that are not in any η(e)-rung +(so ˜η(e) ⊆ η◦(e).) We say that η is tame if +• η(v) = ∅ for every v ∈ V (T), and; +• ˜η(e) = ∅ for every e ∈ E(T). +In other words, η is tame if and only if every vertex in η(T) is in an η(e)-rung for some e ∈ E(T). +For a (T, a)-strip-structure η′ in G, we write η ≤ η′ to mean that for every o ∈ V (T)∪E(T)∪ +(E(T)×V (T)), we have η(o) ⊆ η′(o). We say a (T, a)-strip-structure η is substantial if for every +e ∈ E(T), there exists a long η(e)-rung in G. Equivalently, η is substantial if for every edge +e ∈ E(T) with ends u, v, we have η(e, u) ̸= η(e, v), and so η(e, u) \ η(e, v), η(e, v) \ η(e, u) ̸= ∅. +One may observe that since T has at least three vertices, if η is substantial and η ≤ η′, then η′ +is substantial too. +We say η is rich if +• a is trapped in η+(T), and; +• for every leaf l ∈ V (T) of T, assuming e ∈ E(T) to be the leaf-edge of T incident with +l, we have |η(e, l)| = 1. +It follows that if there exists a rich (T, a)-strip-structure η in G, then T has exactly |NG(a)| +leaves, and for every leaf l ∈ V (T) of T, assuming e ∈ E(T) to be the leaf-edge of T incident +with l and v ∈ V (T) to be the unique neighbor of l in T, we have η(e, v) ∩ η(e, l) = ∅. +By a seagull in T we mean a triple (v, e1, e2) where v ∈ V (T) and e1, e2 are two distinct +edges of T incident with v. By a claw in T we mean a 4-tuple (v, e1, e2, e3) where v ∈ V (T) and +e1, e2, e3 are three distinct edges of T incident with v. +Let (v, e1, e2, e3) be a claw in T. By an η-pyramid at (v, e1, e2, e3), we mean a pyramid Σ with +apex a, base b1b2b3 and paths P1, P2, P3, satisfying the following for each i ∈ {1, 2, 3}. +• bi ∈ η(ei, v). +• There exists a leaf li of T with the following properties: +(1) li belongs to the component of T \ {ei} not containing v. +(2) Let Λi be the unique path in T from v to li (so ei ∈ E(Λi)). Then Pi = Γi ∪ {a}, +where Γi is a path in � +e∈E(Λi) η(e) such that Ri = Γi ∩ η(ei) is a long η(ei)-rung +and Γi ∩ η(e) is a η(e)-rung for each e ∈ E(Λi) \ {ei}. +In particular, assuming ui to be the ends of ei distinct from v and ci to be the unique vertex in +NRi(bi) = NPi(bi) for each i ∈ {1, 2, 3}, we have bi ∈ η(ei, v) \ η(ei, ui) and ci ∈ η(ei) \ η(ei, v). +For a branch vertex v ∈ V (T), by an η-pyramid at v we mean an η-pyramid at (v, e1, e2, e3) for +some claw (v, e1, e2, e3) in T. Also, by an η-pyramid we mean an η-pyramid at v for some branch +vertex v ∈ V (T). It follows that every η-pyramid is a long pyramid. Also, if η is substantial, +then for every claw (v, e1, e2, e3) in T there is a η-pyramid at (v, e1, e2, e3). +Let (v, e1, e2) be a seagull in T. A vertex p ∈ G\η+(T) is said to be a jewel for η at (v, e1, e2) +if for some edge e3 ∈ E(T)\{e1, e2} incident with v, there exists an η-pyramid Σ at (v, e1, e2, e3) +with base b1b2b3 where bi ∈ η(ei, v) for each i ∈ {1, 2, 3}, such that p is a jewel for Σ at b3. In +particular, for each i ∈ {1, 2}, p is adjacent to bi and the unique vertex ci in NPi(bi). Therefore, +since Σ is an η-pyramid at (v, e1, e2, e3), assuming ui to be the end of ei distinct from v, it +follows that p has a neighbor bi ∈ η(ei, v) \ η(ei, ui) and a neighbor ci ∈ η(ei) \ η(ei, v). +For a vertex v ∈ V (T), by a jewel for η at v we mean a jewel for η at (v, e1, e2) for some +seagull (v, e1, e2) in T. Also, by a jewel for η we mean a jewel for η at v for some branch vertex +v ∈ V (T). We denote by Jη the set of all jewels for η. It follows that Jη ⊆ G \ η+(T). + +INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII. +11 +We are now in a position to prove the main result of this section: +Theorem 4.2. Let G ∈ C, let a ∈ G and let T be a smooth tree. Suppose that there exists +a tame, substantial and rich (T, a)-strip-structure in G. Then there is a substantial and rich +(T, a)-strip-structure ζ in G for which G \ (ζ+(T) ∪ Jζ) is anticomplete to ζ+(T). +Proof. Let η be a tame, substantial and rich (T, a)-strip-structure in G such that η(T) is maximal +with respect to inclusion. Let M = G \ (η+(T) ∪ Jη). +(6) +Let P be a path in M with ends p1 and p2 such that there exists x1 ∈ Nη(T)(p1) and +x2 ∈ Nη(T)(p2) for which {x1, x2} is not local in η, and such that |P| ≥ 1 is as small as possible +subject to this property. Then there exists {j1, j2} = {1, 2} and f = v1v2 ∈ E(T) such that +xj1 ∈ B(vj1) \ η(f) and xj2 ∈ (B(vj2) ∪ η(f)) \ B(vj1). +Suppose not. For each i ∈ {1, 2}, let ei ∈ E(T) such that xi ∈ η(ei) (hence e1 ̸= e2) and +si be an end of ei such that there exists a path Λ0 (possibly of length zero) from s1 to s2 in +T \ {e1, e2}. We claim that there is a vertex v ∈ Λ0 such that B(v) ∩ {x1, x2} = ∅. Suppose first +that s1 ̸= s2; let v1 be unique neighbor of s1 in Λ0. Then we have x1 /∈ B(v1) and x2 /∈ B(s1). +Also, since f = s1v1 does not satisfy (6), we have either x1 /∈ B(s1) or x2 /∈ B(v1). But then +either v = s1 or v = v1 satisfies the claim. Thus, we may assume that v = s1 = s2. Note +that since neither e1 nor e2 satisfies (6), we have x1 /∈ B(s1) and x2 /∈ B(s2). In other words, +we have B(v) ∩ {x1, x2} = ∅, and the claim follows. Henceforth, let v be as promised by the +above claim. For each i ∈ {1, 2}, let ui be the end of ei distinct from si (hence u1 ̸= u2). Let +Λ = u1-s1-Λ0-s2-u2 and let u′ +1, u′ +2 be the neighbors of v in Λ such that Λ traverses u1, u′ +1, v, u′ +2, u2 +in this order (so either of u1 = u′ +1 and u2 = u′ +2 is possible). Let e′ +i = u′ +iv for each i ∈ {1, 2}. Since +T is smooth, there exists a vertex u′ +3 ∈ NT (v) \ Λ; let e′ +3 = u′ +3v. For each i ∈ {1, 2, 3}, let Ti be +the component of T \(NT (v)\{u′ +i}) containing v (so e′ +i ∈ E(Ti)). Then since B(v)∩{x1, x2} = ∅ +and since η is tame and substantial, there exists an η-pyramid Σ at (v, e′ +1, e′ +2, e′ +3) with apex a, +base b1b2b3 and paths P1, P2, P3 such that we have +• bi ∈ η(e′ +i, v) and Pi \ {a, bi} ⊆ η(Ti) \ B(v) for each i ∈ {1, 2, 3}, and; +• xi ∈ P ∗ +i for each i ∈ {1, 2}. +In particular, the second bullet above implies that NΣ(P) is not local in Σ and P is not a corner +path for Σ. Since P ⊆ M, we have P ∩ Jη = ∅. Thus, Σ being an η-pyramid, it follows that +P contains no jewel for Σ. Also, since η is rich, a is trapped in η+(T). Therefore, applying +Theorem 3.2 to G, H = η+(T), a, Σ and P, we deduce that P contains a corner path for Σ. +On the other hand, note that by the second bullet above, for every vertex x ∈ Σ \ {a}, either +{x, x1} or {x, x2} is not local in η. From this, the minimality of |P| and the fact that η is rich, +it follows that P ∗ is anticomplete to Σ. But then P is a corner path for Σ, a contradiction. This +proves (6). +(7) Let P be a path in M with ends p1 and p2 such that there exists x1 ∈ Nη(T)(p1) and +x2 ∈ Nη(T)(p2) for which {x1, x2} is not local in η, and such that |P| ≥ 1 is as small as possible +subject to this property. Let f = v1v2 ∈ E(T) and {j1, j2} = {1, 2} be as guaranteed by (6) applied +to P, x1 and x2. Then we have Nη(T)(P ∗) ⊆ η(f, vj1) and Nη(T)({p1, p2}) ⊆ η(f)∪B(v1)∪B(v2). +Suppose not. Without loss of generality, we may assume that j1 = 1 and j2 = 2. Note that by +the minimality of |P|, we have Nη(T)(P ∗) ⊆ η(f, v1). Therefore, one of p1 and p2 has a neighbor +in η(T) \ (η(f) ∪ B(v1) ∪ B(v2)); say p1 is adjacent to x′ +1 ∈ η(T) \ (η(f) ∪ B(v1) ∪ B(v2)). For +each i ∈ {1, 2}, let Ti be the component of T \ {f} containing vi. It follows that there exists +j ∈ {1, 2} such that x′ +1 ∈ η(Tj) \ B(vj). Assume that |P| > 1. By the minimality of |P|, we +have j = 1. But then P, x′ +1 and x2 violate (6). We deduce that |P| = 1. But now P, x′ +1 and x3−j +violate (6). This proves (7). + +12 +INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII. +(8) Let P be a path in M with ends p1 and p2 such that there exists x1 ∈ Nη(T)(p1) and +x2 ∈ Nη(T)(p2) for which {x1, x2} is not local in η, and such that |P| ≥ 1 is as small as +possible subject to this property. Suppose that there exist {k1, k2} = {1, 2}, f = v1v2 ∈ E(T) +and e1 ∈ E(T) \ {f} incident with vk1 such that pk1 has a neighbor in η(e1, vk1) and pk2 has a +neighbor in (B(vk2) ∪ η(f)) \ B(vk1). Then pk1 is complete to B(vk1) \ (η(e1, vk1) ∪ η(f)). +Due to symmetry, we may assume that k1 = 1 and k2 = 2. Let e3 ∈ E(T)\{e1, f} be incident +with v1 and let b3 ∈ η(e3, v1) be arbitrary. We need to show that p1 is adjacent to b3. Suppose +for a contradiction that p1 and b3 are non-adjacent. Let b1 ∈ η(e1, v1) be adjacent to p1 and let +x ∈ (B(v2) ∪ η(f)) \ B(v1) be adjacent to p2. Let T2 be the component of T \ (NT (v1) \ {v2}) +containing v1 (so f ∈ E(T2)). Also, for each i ∈ {1, 3}, let ui be the end of ei distinct from v1 +and let Ti be the component of T \ (NT (v1) \ {ui}) containing v1 (so ei ∈ E(Ti)). By (6) and +(7), there exists an edge f ′ = v′ +1v′ +2 ∈ E(T) such that Nη(T)({p1, p2}) ⊆ η(f ′) ∪ B(v′ +1) ∪ B(v′ +2). +This, along with the minimality of |P|, implies that p1 is anticomplete to (η(T1)∪η(T3))\B(v1), +P \ {p1} is anticomplete to η(T1) ∪ η(T3) and P \ {p2} is anticomplete to η(T2) \ B(v1). Since +p2 has a neighbor x ∈ (B(v2) ∪ η(f)) \ B(v1) and since η is tame, there exists a path P2 in G +from a to p2 with P ∗ +2 ⊆ η(T2) \ B(v1). Also, for each i ∈ {1, 3}, there exists a path Pi in G +from a to bi with P ∗ +i ⊆ η(Ti) \ B(v1). Note that since η is rich, P anticomplete to NG[a]; in +particular, P1 has length at least two. But now there is a theta in G with ends a and b1 and +paths P1, a-P2-p2-P-p1-b1 and b1-b3-P3-a, a contradiction. This proves (8). +The following is immediate from (8) and the fact that T is smooth. +(9) Let P be a path in M with ends p1 and p2 such that there exists x1 ∈ Nη(T)(p1) and +x2 ∈ Nη(T)(p2) for which {x1, x2} is not local in η, and such that |P| ≥ 1 is as small as possible +subject to this property. Suppose that there exist {k1, k2} = {1, 2} and f = v1v2 ∈ E(T) such that +xk1 ∈ B(vk1) \ (η(f)) and xk2 ∈ (B(vk2) ∪ η(f)) \ B(vk1). Then pk1 is complete to B(vk1) \ η(f). +We now deduce: +(10) Let D be a component of M. Then Nη(T)(D) is local in η. +Suppose not. By Lemma 4.1, there exist x1, x2 ∈ Nη(T)(D) such that {x1, x2} is not local in η. +For each i ∈ {1, 2}, let pi be a neighbor of xi in D. Since D is connected, there exists a path P +in D ⊆ M from p1 to p2. In other words, there exists a path P in M with ends p1, p2 along with +x1 ∈ Nη(T)(p1) and x2 ∈ Nη(T)(p2) such that {x1, x2} is not local in η. Now, let P be a path in M +with ends p1 and p2 such that there exists x1 ∈ Nη(T)(p1) and x2 ∈ Nη(T)(p2) for which {x1, x2} +is not local in η, and such that |P| ≥ 1 is as small as possible subject to this property. So we can +apply (6) to P, x1 and x2; let {j1, j2} = {1, 2} and f = v1v2 ∈ E(T) be as in (6). We may assume +without loss of generality that j1 = 1 and j2 = 2; thus, v1 is a branch vertex of T. It follows from +(7) that Nη(T)(P ∗) ⊆ η(f, v1) and Nη(T)({p1, p2}) ⊆ η(f) ∪ B(v1) ∪ B(v2). By (9) applied to +k1 = 1 and k2 = 2, p1 is complete to B(v1) \ η(f). Also, from (9) applied to k1 = 2 and k2 = 1, +it follows that either p2 is complete to B(v2) \ η(f) and B(v2) \ η(f) ̸= ∅, or p2 is anticomplete +to B(v2) \ η(f). Note that if |P| > 1, then by the minimality of |P|, we have Nη(T)(p1) ⊆ B(v1) +and Nη(T)(p2) ⊆ (B(v2)∪η(f))\B(v1). Let us define η′ : V (T)∪E(T)∪(E(T)×V (T)) ⊆ 2G\{a} +as follows. Let η′(f) = η(f) ∪ P and let η′(f, v1) = η(f, v1) ∪ {p1}. Let +• η′(f, v2) = η(f, v2) ∪ {p2} if p2 is complete to B(v2) \ η(f) and B(v2) \ η(f) ̸= ∅, and; +• η′(f, v2) = η(f, v2) if p2 is anticomplete to B(v2) \ η(f). +Let η′ = η elsewhere on V (T) ∪ E(T) ∪ (E(T) × V (T)). Then since η is tame, substantial and +rich, and p2 is adjacent to x2 ∈ B(v2) ∪ η(f)) \ B(v1), it is straightforward to check that η′ +is also a tame, substantial and rich (T, a)-strip-structure. But we have η′(T) = η(T) ∪ P, a +contradiction with the maximality of η(T). This proves (10). +The proof is almost concluded. Let X be the union of all the components D of M such that +D is anticomplete to η+(T). Since η is rich, it follows that for every component D of M \ X, a + +INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII. +13 +is anticomplete to X and Nη+(T)(D) = Nη(T)(D) is non-empty. By (10), for every component D +of M \ X, Nη(T)(D) is local in η. Let D be the set of all components D of M \ X for which we +have Nη+(T)(D) ⊆ Bη(v) for some v ∈ V (T). Breaking the ties arbitrarily and by the definition +of X, we may write D = � +v∈V (T) Dv, where +• for all distinct u, v ∈ V (T), we have Du ∩ Dv = ∅, and; +• for all v ∈ V (T) and every D ∈ Dv, we have Nη+(T)(D) ⊆ Bη(v) and Nη+(T)(D) ̸= ∅. +Also, for every e = uv ∈ E(T), let De be the set of all components D of M \ X for which we +have Nη+(T)(D) ⊆ η(e) and +• either Nη(T)(D) ∩ η◦(e) ̸= ∅, or; +• Nη(T)(D) ∩ (η(e, u) \ η(e, v)) ̸= ∅ and Nη(T)(D) ∩ (η(e, v) \ η(e, v)) ̸= ∅. +From the definition of X, it follows that every component of M \ X belongs to exactly one of +the sets {Dv, De : v ∈ V (T), e ∈ E(T)} (note that since η is rich, a is anticomplete to each such +component). +Let ζ : V (T) ∪ E(T) ∪ (E(T) × V (T)) ⊆ 2G\{a} be defined as follows. For all v ∈ V (T) and +e ∈ E(T), let +• ζ(v) = � +D∈Dv D; +• ζ(e) = η(e) ∪ (� +D∈De D), and; +• ζ(e, v) = η(e, v). +It is easily seen that ζ satisfies the conditions (S1-S8) from the definition of a (T, a)-strip- +structure. In particular, since η is rich, ζ satisfies (S2), and from the definitions of X, Dv’s and +De’s, it follows that ζ satisfies (S5) and (S7). Also, we have η ≤ ζ. +Now, since η is substantial and rich, since η ≤ ζ and from the definitions of X and ζ, it follows +that ζ is a substantial and rich (T, a)-strip-structure with Jζ = Jη. Moreover, note that we have +ζ+(T) = η(T)+ ∪ (M \ X), and so G \ (ζ+(T) ∪ Jζ) = G \ (ζ+(T) ∪ Jη) = X is anticomplete to +ζ+(T). This completes the proof of Theorem 4.2. +■ +5. Jewels under the loupe +Here we revisit jewels for strip-structures, establishing several results about their proper- +ties in various settings. This will help attune Theorem 4.2 for its application in the proof of +Theorem 6.1. +First we need to introduce some notation. Let G be a graph and let a ∈ G. Let T be a smooth +tree and let ζ be a (T, a)-strip-structure in G. Let v ∈ V (T) and let e ∈ E(T) be incident with +v. We denote by ζe(v) the set of all components D of ζ(v) for which we have NB(v)(D) ⊆ η(e, v), +or equivalently, Nζ(T)\ζ(e,v)(D) = ∅. +Let (v, e1, e2) be a seagull in T and let ui be the end of ei distinct from v for each i ∈ {1, 2}. +We define +ζ(v, e1, e2) = ζ(e1) ∪ ζ(e2) ∪ ζe1(u1) ∪ ζe2(u2) ∪ ζ(v). +We denote by Jζ,(v,e1,e2) the set of all jewels for ζ at (v, e1, e2), and for every vertex v ∈ V (T), +Jζ,v stands for the set of all jewels for η at v. It follows that Jζ,v = ∅ if v is a leaf of T. +The first result in this section describes, for a (T, a)-strip-structure in a theta-free graph, the +attachments of jewels at a vertex of T. +Theorem 5.1. Let G be a theta-free graph and let a ∈ G. Let T be a smooth tree and let ζ be +a (T, a)-strip-structure in G. Let (v, e1, e2) be a seagull in T and let x ∈ Jζ,(v,e1,e2). Then the +following hold. +• We have Nζ+(T)(x) ⊆ ζ(v, e1, e2), and so Nζ+(T)(Jζ,(v,e1,e2)) ⊆ ζ(v, e1, e2). Consequently, +for every vertex v ∈ V (T), we have Nζ+(T)(Jζ,v) ⊆ ζ(NT [v]), and for every two distinct +vertices v, v′ ∈ V (T), we have Jζ,v ∩ Jζ,v′ = ∅. + +14 +INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII. +• Assume that ζ is rich. Let i ∈ {1, 2} and let R be a long ζ(ei)-rung, let r be the end of +R in ζ(ei, v) and let r′ be the unique neighbor of r in R. Then either x is anticomplete +to R or NR(x) = {r, r′}. +Proof. Note that v is a branch vertex of T. For each i ∈ {1, 2}, let ui be the end of ei distinct +from v and let Ti be the component of T \(NT (v)\{ui}) containing v. Let T ′ be the component +of T \{u1, u2} containing v. Let x ∈ Jζ,(v,e1,e2). Since x ∈ Jζ,(v,e1,e2) is a jewel for ζ, there exists +an edge e3 ∈ E(T)\{e1, e2} incident with v and a ζ-pyramid Σ at (v, e1, e2, e3) with apex a, base +b1b2b3 and paths P1, P2, P3 such that x is a jewel for Σ at b3. In particular, for each j ∈ {1, 2, 3}, +Pj ∩ ζ(ej) is a long ζ(ej)-rung Rj with bj as its end in ζ(ej, v). +Also, x is anticomplete to +P3 (and so x is not adjacent to a), and for each j ∈ {1, 2}, assuming cj to be the unique +vertex in NRj(bj) = NPj(bj), x is adjacent to bj ∈ ζ(ej, v) \ ζ(ej, uj) and cj ∈ ζ(ej) \ ζ(ej, v). +Therefore, there exist paths Qi, Si of length more than one in G from a to x for which we have +bi ∈ Q∗ +i ⊆ (ζ(T ′) \ ζ(v)) ∪ (ζ(ei, v) \ ζ(ei, ui)) and ci ∈ S∗ +i ⊆ ζ(Ti) \ (B(v) ∪ ζ(ui) ∪ ζ(v)). +To prove the first assertion of the Theorem 5.1, assume for a contradiction that x has a +neighbor y ∈ ζ+(T) \ ζ(v, e1, e2). Since x is not adjacent to a, we have y ∈ ζ(T) \ ζ(v, e1, e2). +First, assume that y ∈ ζ(T ′)\ζ(v). Then by (S5) and (S7) from the definition of a strip-structure, +there exists a path Q′ of length more than one in G from a to x with Q′∗ ⊆ ζ(T ′) \ ζ(v). But +now there is a theta in G with ends a, x and paths a-S1-x, a-S2-x and a-Q′-x, a contradiction. +It follows that y ∈ ζ(T1 ∪ T2) \ ζ(v, e1, e2). +In other words, for some i ∈ {1, 2}, we have +y ∈ ζ(Ti) \ (ζ(ei) ∪ ζei(ui) ∪ ζ(v)). As a result, by (S5) and (S7) from the definition a strip- +structure, and by the definition of ζei(ui), there exists a path S′ +i of length more than one in G +from a to x with S′∗ +i ⊆ ζ(Ti)\(ζ(ei)∪ζei(ui)∪ζ(v)). But now assuming i′ ∈ {1, 2} to be distinct +from i, there is a theta in G with ends a, x and paths a-Qi-x, a-S′ +i-x and a-Si′-x, a contradiction. +This proves the the first assertion. +Next we prove the second assertion of Theorem 5.1. By symmetry, we may assume that i = 1. +Assume that x has a neighbor y ∈ R. Let P ′ +1 = (P1 \ R1) ∪ R. Let Σ′ be the pyramid with +apex a, base rb2b3 and paths P ′ +1, P2 and P3. Recall that since ζ is rich, a is trapped in ζ+(T). +Also, Σ′ is a pyramid in ζ+(T), x is adjacent to y ∈ P ′ +1, x is adjacent to b2, c2 ∈ P2 and x is +anticomplete to P3. It follows that x is a wide vertex for Σ′ which is not a corner path for Σ′. +Now applying Lemma 3.1 to G, a, H = ζ+(T), Σ′ and p = x, we deduce that x is a jewel for Σ′ +at b3, and so NR(x) = NP ′ +1(x) = {r, r′}. This completes the proof of Theorem 5.1. +■ +Our next goal is to show that for every rich (T, a)-strip-structure in a graph G ∈ Ct, there +are only a few jewels at each vertex of T. Let us begin with a lemma, asserting that for a rich +(T, a)-strip-structure ζ in a theta-free graph, each set Bζ(v) is almost a clique. +Lemma 5.2. Let G be a theta-free graph and a ∈ G. Let T be a smooth tree and ζ be a rich +(T, a)-strip-structure in G. Then for every v ∈ V (T), there exists at most one edge f ∈ E(T) +such that η(f, v) is not a clique. +Proof. Suppose for a contradiction that there are two distinct edges f1, f2 ∈ E(T) incident with +v, and for each i ∈ {1, 2}, there exist xi, yi ∈ ζ(fi, v) such that xi is not adjacent to yi. Then +v is not a leaf of T and H = x1-x2-y1-y2-x1 is a hole of length four in G. Since ζ is rich, a +is anticomplete to H. Let f1 = u1v. Let l1 be a leaf of T which belongs to the component of +T \ {v} containing u1, and let Λ1 be the unique path in T from v to l1 (so f1 ∈ E(Λ1)). Let +Rx1 be an ζ(f)-rung containing x1 and let Ry1 be an ζ(f)-rung containing y1. Since ζ is rich, +H1 = Rx1 ∪ Rx2 ∪ B(u1) is a connected induced subgraph of G, and so there is a path Q in H1 +from x1 to y1. It follows that Q has length more than one and Q∗ ⊆ (B(u1) ∪ ζ(f1))\B(v). But +now there is a theta in G with ends x1, y1 and paths Q, x1-x2-y1 and x1-y2-y1, a contradiction. +This completes the proof of Lemma 5.2. +■ +Recall the following classical result of Ramsey (see, for instance, [5] for an explicit bound.) + +INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII. +15 +Theorem 5.3 (See [5]). For all integers a, b ≥ 1, there exists an integer R = R(a, b) ≥ 1 such +that every graph G on at least R(a, b) vertices contains either a clique of cardinality a or a stable +set of cardinality b. +We can now prove the second main result of this section. +Theorem 5.4. For all positive integers t, δ, there exists a positive integer j = j(t, δ) with the +following property. Let G ∈ Ct be a graph and let a ∈ G and let T be a smooth tree of maximum +degree δ. Let ζ be a rich (T, a)-strip-structure in G. Then for every vertex v ∈ V (T), we have +|Jζ,v| < j. +Proof. Let j = j(t, δ) = +�δ +2 +�R(t, 3) with R(·, ·) as in Theorem 5.3. +Then in order to prove +|Jζ,v| < j, it is enough to show that |Jζ,(v,e1,e2)| < R(t, 3) for every seagull (v, e1, e2) in T. +Suppose for a contradiction that |Jζ,(v,e1,e2)| ≥ R(t, 3) for some seagull (v, e1, e2) in T. Then v is +a branch vertex of T. For each i ∈ {1, 2}, let ui be the end of ei different from v. Since G ∈ Ct, +it follows from Theorem 5.3 that Jζ,(v,e1,e2) contains a stable set X of cardinality three. For +every x ∈ X, since x is a jewel for ζ at (v, e1, e2), it follows that for every i ∈ {1, 2}, there exists +a long ζ(ei)-rung Rx +i such that Qx +i = Rx +i \ ζ(ei, v) is a path in ζ(ei) \ ζ(ei, v) from a neighbor +of x to a vertex in ζ(ei, ui) \ ζ(ei, v); in particular, Rx +i contains a neighbor of x. Therefore, for +each i ∈ {1, 2}, we may pick a non-empty set Ri of long ζ(ei)-rungs such that every vertex in X +has a neighbor in at least one rung in Ri, and with Ri minimal with respect to inclusion. We +deduce: +(11) There exists i ∈ {1, 2} with |Ri| > 1. +Suppose not. Then for every i ∈ {1, 2}, there exists a long ζ(ei)-rung Si such that every vertex +in X has a neighbor in Si. Let si be the end of Si in ζ(ei, v) and s′ +i be unique neighbor of si in +Si. By the second assertion of Theorem 5.1, X is complete to {s′ +1, s′ +2}. But now X ∪ {s′ +1, s′ +2} is +a theta in G with ends s′ +1, s′ +2, a contradiction. This proves (11). +By (11) and due to symmetry, we may assume that |R1| > 1. +This, together with the +minimality of R1, implies that there exist distinct vertices x, y ∈ X as well as distinct long +ζ(e1)-rungs Rx, Ry ∈ R1 such that x has a neighbor in Rx, y has a neighbor in Ry, x is +anticomplete to Ry, and y anticomplete to Rx. Let rx and ry be the ends of Rx and Ry in +ζ(e1, v), respectively. Let r′ +x be the unique neighbor of rx in Rx and r′ +y be the unique neighbor +of ry in Ry; so we have r′ +x, r′ +y ∈ ζ(e1) \ ζ(e1, v). By the second assertion of Theorem 5.1, we +have NRx∪Ry(x) = {rx, r′ +x} and NRx∪Ry(y) = {ry, r′ +y}. It follows that rx, r′ +x ∈ Rx \ Ry and +ry, r′ +y ∈ Ry \ Rx. Also, rx is anticomplete to Ry \ {ry}, as otherwise (Ry \ {ry}) ∪ {rx} contains +a long ζ(e1)-rung R with NR(x) = {rx}, which violates the second assertion of Theorem 5.1. +Similarly, ry is anticomplete to Rx \ {rx}. +Now, let G1 = G[(B(u1)\ζ(e1, u1))∪((Rx∪Ry)\{rx, ry})] and let G2 = G[(B(u2)\ζ(e2, u2))∪ +Qx +2 ∪ Qy +2]. Since ζ is rich, the second bullet in the definition of a rich strip-structure implies that +G1 and G2 are connected. Consequently, there exists a path Q1 in G1 from r′ +x to r′ +y, and there +exists a path Q2 from x to y with Q∗ +2 ⊆ G2. Also, since v is a branch vertex of T, we may choose +an edge e3 ∈ E(T) \ {e1, e2} incident with v. By the first assertion of Theorem 5.1, {x, y} is +anticomplete to ζ(e3, v). Let Q3 be a path from rx to ry with Q∗ +3 ⊆ ζ(e3, v) (thus |Q3| ∈ {2, 3}). +But now there is a prism with triangles xrxr′ +x and yryr′ +y and paths Q1, Q2, Q3, a contradiction. +This completes the proof of Theorem 5.4. +■ +Our last theorem in this section examines the connectivity within G \ ζ+(T) for a (T, a)- +strip-structure ζ arising from Theorem 4.2. We need the following lemma, the proof of which is +similar to that of Theorem 5.1. +Lemma 5.5. Let G be a theta-free graph and let a ∈ G. Let T be a smooth tree and let ζ be a +(T, a)-strip-structure in G. Let v, v′ ∈ V (T) be distinct and let P be a path in G \ ζ+(T) with + +16 +INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII. +ends x, x′ such that x ∈ Jζ,v, x′ ∈ Jζ,v′ and P ∗ is anticomplete to ζ+(T). Then v and v′ are +adjacent in T. +Proof. Suppose not. Note that by Theorem 5.1, x and x′ are distinct. Let Λ be the path in T +from v to v′. Then Λ has length more than one, and so there are two distinct edges f, f ′ ∈ E(Λ) +such that f is incident with v and f ′ is incident with v′. Let u be the end of f distinct from +v and u′ be the end of f ′ distinct from v′. Let (v, e1, e2) and (v′, e′ +1, e′ +2) be two seagulls in G +such that x ∈ Jζ,(v,e1,e2) and x′ ∈ Jζ,(v′,e′ +1,e′ +2). For each i ∈ {1, 2}, let ui be the end of ei distinct +from v and let u′ +i be the end of e′ +i distinct from v′. Without loss of generality, we may assume +that u2, u′ +2 /∈ Λ. Let T2 be the component of T \ (NT (v) \ {u2}) containing v and let T ′ +2 be the +component of T \(NT (v′)\{u′ +2}) containing v′. Let T ′ be the component of T \{u′, u′ +2} containing +v′. Since x is a jewel for ζ at (v, e1, e2), it follows that x is not adjacent to a, and x has a neighbor +c ∈ ζ(e2) \ ζ(e2, v) ⊆ ζ(T2) \ (B(v) ∪ ζ(u2) ∪ ζ(v)). Therefore, there exists a path Q of length +more than one in G from a to x for which we have c ∈ Q∗ ⊆ ζ(T2) \ (B(v) ∪ ζ(u2) ∪ ζ(v)). Also, +since x′ is a jewel for ζ at (v′, e′ +1, e′ +2), it follows that x′ is not adjacent to a, and x′ has a neighbor +b′ ∈ B(v′)\(ζ(f ′, u′)∪ζ(e′ +2, v′)) and a neighbor c′ ∈ ζ(e′ +2)\ζ(e′ +2, v′) ⊆ ζ(T ′ +2)\(B(v′)∪ζ(u′ +2)∪ζ(v′)). +Therefore, there exist paths P ′, Q′ of length more than one in G from a to x′ for which we have +b′ ∈ P ′∗ ⊆ (ζ(T ′) \ ζ(v′)) ∪ (ζ(f ′, v′) \ ζ(f ′, u′)) and c′ ∈ Q′∗ ⊆ ζ(T ′ +2) \ (B(v′) ∪ ζ(u2) ∪ ζ(v′)). +But now there is a theta in G with ends a, x′ and paths a-P ′-x′, a-Q′-x′ and a-Q-x-P-x′, a +contradiction. This proves Lemma 5.5. +■ +Theorem 5.6. Let t, δ ≥ 1 be integers and let j(t, δ) be as in Theorem 5.4. Let G ∈ Ct be a +graph and let a ∈ G. Let T be a smooth tree of maximum degree δ and let v ∈ V (T). Let ζ +be a rich (T, a)-strip-structure in G such that G \ (ζ+(T) ∪ Jζ) is anticomplete to ζ+(T). Let +x ∈ G \ (ζ+(T) ∪ Jζ). Then there exists Sx ⊆ G \ (ζ+(T) ∪ {x}) such that |Sx| < 2j(t, δ) and Sx +separates x and Jζ \ ({x} ∪ Sx) in G \ ζ+(T). Consequently, Sx separates x and ζ+(T) in G. +Proof. By Theorem 5.1, {Jζ,v : v ∈ V (T)} is a partition of Jζ. Let G′ be the graph obtained +from G\ζ+(T) by contracting the set Jζ,v into a vertex zv for each v ∈ V (T) with Jζ,v ̸= ∅, and +then adding a new vertex z such that NG′(z) = {zv : v ∈ V (T), Jζ,v ̸= ∅}. We claim that there +is a set Y ⊆ G′ \ {x, z} of cardinality at most two which separates x and z in G′. Suppose not. +By Theorem 2.1, there are three pairwise internally disjoint paths in G′ from x to z. Thus, there +exist S ⊆ T with |S| = 3 as well as three paths {Pv : v ∈ S} in G \ ζ+(T) all having x as an end +and otherwise disjoint, such that for each v ∈ S, Pv has an end yv ∈ Jζ,v distinct from x, and +we have P ∗ +v ⊆ G\(ζ+(T) ∪ Jζ). As a result, for all distinct v, v′ ∈ S, Pv,v′ = yv-Pv-x-Pv′-yv′ is a +path in G\ζ+(T) from yv ∈ Jζ,v to yv′ ∈ Jζ,v′ such that P ∗ +v,v′ ⊆ G\(ζ+(T)∪Jζ). In particular, +P ∗ +v,v′ is anticomplete to ζ+(T). But then by Lemma 5.5, S is a clique in T, which is impossible. +The claim follows. +Let Y be as in the above claim. For each y ∈ Y , if y = zv for some v ∈ V (T), then let +Ay = Jζ,v. Otherwise, let Ay = {y}. Let Sx = � +y∈Y Ay. Then Sx ⊆ G\(ζ+(T)∪{x}) separates +x and Jζ \({x}∪Sx) in G\ζ+(T). Also, by Theorem 5.4, we have |Sx| < 2j(t, δ). This completes +the proof of Theorem 5.6. +■ +6. Strip structures and connectivity +In this section, we investigate the connectivity implications of the presence of certain (T, a)- +strip-structures in graphs from Ct. The main result is the following. +Theorem 6.1. For all integers t, δ ≥ 1, there exists an integer σ = σ(t, δ) ≥ 1 with the following +property. Let G ∈ Ct be a graph and let a ∈ G. Let T be a smooth tree of maximum degree δ and +let ζ be a rich (T, a)-strip-structure in G such that G \ (ζ+(T) ∪ Jζ) is anticomplete to ζ+(T). +Then for every vertex x ∈ G \ NG[a], there exists a set Sx ⊆ G \ {a, x} with |Sx| < σ such that +S separates a and x in G. + +INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII. +17 +Proof. Let j(t, δ) be as in Theorem 5.4. We claim that +σ = σ(t, δ) = 2δ(j(t, δ) + t) +satisfies Theorem 6.1. For every vertex v ∈ V (T), we define Cv = B(v) if v is a leaf of T and +Cv = ∅ otherwise. Also, for every vertex v ∈ V (T), let Kv be a maximal clique of G contained +in B(v). Thus, we have |Kv| < t. Moreover, Lemma 5.2 along with the assumption that ζ is +rich implies if v is a leaf of T, then we have Kv = B(v) = Cv (and so |Kv| = 1), and if v is a +branch vertex of T, then Kv contains all but possibly one of the sets η(f, v) for f ∈ E(T). For +every S ⊆ T, we define +MS = +� +w∈NT (S) +Jη,w, +NS = +� +w∈NT (S) +Kw. +Also, we write Mv for M{v} and Nv for N{v}. For every v ∈ V (T), let Ov = Mv ∪ Nv. The +following is immediate from Theorems 5.1 and 5.4 and Lemma 5.5. +(12) For every v ∈ V (T), we have +• Ov ⊆ G \ (Jζ,v ∪ {a}); +• |Ov| < δ(j(t, δ) + t) ≤ σ, and; +• Ov separates a and Jζ,v in G. +Now, for every x ∈ G \ NG[a], we define Sx as follows. First, assume that x ∈ ζ(T) \ NG[a]. +Then either x ∈ ζ(e) for some edge e = uv ∈ E(T), or x ∈ ζ(v) for some branch vertex v ∈ V (T). +In the former case, let +Ex = Mu ∪ Mv, +Ix = N{u,v} ∪ Cu ∪ Cv. +In the latter case, let +Ex = Mv ∪ Jζ,v +Ix = Nv. +Let Sx = Ex ∪ Ix. +Observe that since x ∈ G \ NG[a], we have Sx ⊆ G \ {a, x}. +Also, by +Theorem 5.4, we have |Ex| ≤ 2δj(t, δ) and so |Sx| < 2δ(j(t, δ) + t) = σ. +Moreover, from +Theorem 5.1 and the fact that ζ is rich, it is easy to check that for every path P in G from a +to x, if P ⊆ ζ+(T), then P contains a vertex from Ix, and otherwise P contains a vertex from +either Ix or Ex. Therefore, Sx separates a and x in G. +Next, assume that x ∈ Jζ. Then by Theorem 5.1, there exists a unique vertex v ∈ V (T) such +that x ∈ Jζ,v. Let Sx = Ov. Then by (12), we have Sx ⊆ G \ {a, x}, |Sx| < σ and Sx separates +a and x in G. +Finally, assume that x ∈ G\(ζ+(T)∪Jζ). Then letting Sx to be as in Theorem 5.6, it follows +from Theorem 5.6 that Sx ⊆ G \ {a, x}, |X| < 2j(t, δ) ≤ σ and Sx separates a and x in G. This +completes the proof of Theorem 6.1. +■ +Our application of Theorem 6.1 though is confined to the case where T is a caterpillar. More +precisely, for a graph G and a vertex a ∈ G, an induced subgraph H ⊆ G \ {a} is said to be an +a-seed in G if the following hold. +• There exists a caterpillar C such that H is the line graph of a 1-subdivision of C and +NG(a) = Z(H). +• The vertex a is trapped in H ∪ {a}. +It follows that Z(H) is the set of all degree-one vertices of H. We now combine Theorems 4.2 +and 6.1 to deduce the following. + +18 +INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII. +Theorem 6.2. For every integer t ≥ 1, there exists an integer s = s(t) ≥ 1 with the following +property. Let G ∈ Ct be a graph and a ∈ G. Assume that there is an a-seed in G. Then for +every vertex x ∈ G \ NG[a], there exists Sx ⊆ G \ {a, x} with |Sx| < s such that Sx separates a +and x in G. +Proof. Let σ(·, ·) be as in Theorem 6.1. We show that s = s(t) = σ(t, 3) satisfies Theorem 6.2. +Pick an a-seed H in G. Let T be the unique smooth caterpillar with |NG(a)| leaves. Then T has +maximum degree three. Also, one may immediately observe that there is a tame, substantial +and rich (T, a)-strip-structure η in G with η(T) = H. Now we can apply Theorem 4.2 to G, a +and T, deducing that there exists a substantial and rich (T, a)-strip-structure ζ in G such that +G \ (ζ+(T) ∪ Jζ) is anticomplete to ζ+(T). Hence, by Theorem 6.1 applied to G, a, T and ζ, +for every vertex x ∈ G \ NG[a], there exists Sx ⊆ G \ {a, x} with |Sx| < s such that Sx separates +a and x in G. This completes the proof of Theorem 6.2. +■ +7. From blocks to trees +In this section, we prove Theorem 1.8. We begin with a result which captures the use of +Theorem 6.2 in the proof of Theorem 1.8. For a positive integer n, we write [n] = {1, . . . , n}. +Theorem 7.1. For all integers ν, t ≥ 1, there exists an integer ψ = ψ(t, ν) ≥ 1 with the +following property. Let G ∈ Ct, let a, b ∈ G be distinct and non-adjacent and let {Pi : i ∈ [ψ]} +be a collection of ψ pairwise internally disjoint paths in G from a to b. For each i ∈ [ψ], let +ai be the neighbor of a in Pi (so ai ̸= b). Then there exists I ⊆ [ψ] with |I| = ν for which the +following holds. +• {ai : i ∈ I} ∪ {b} is a stable set in G. +• For all i, j ∈ I with i < j, ai has a neighbor in P ∗ +j \ {aj}. +Proof. Let s = s(t) be as in Theorem 6.2 and let µ = µ(max{2s + 1, t}), where µ(·) is as in +Theorem 2.5. Let R(·, ·) be as in Theorem 5.3. For every integer p ≥ 1, let Rtourn(p) be the +smallest positive integer n such that every tournament on at least n vertices contains a transitive +tournament on p vertices; the existence of Rtourn(p) follows easily from Theorem 5.3 (in fact, +one may observe that Rtourn(p) ≤ R(p, p)). Let γ = R(Rtourn(ν + 1), µ). We prove that +ψ = ψ(t, ν) = R(γ, t) +satisfies Theorem 7.1. Let P1, . . . , Pψ be ψ pairwise internally disjoint paths in G from a to +b. Since G is Kt-free, it follows from Theorem 5.3 and the definition of ψ that there exists a +stable set N ⊆ {ai : i ∈ [ψ]} in G with |N| = γ; we may assume without loss of generality that +N = {ai : i ∈ [γ]}. +Let D be a directed graph with V (D) = N such that for distinct i, j ∈ [γ], there is an arc +from ai to aj in D if and only if xi has a neighbor in P ∗ +j \ {aj}. Note that D may contain both +arcs (ai, aj) and (aj, ai), and so the undirected underlying graph of D might not be simple. Let +D− be the simple graph obtained from the undirected underlying graph of D by removing one +of every two parallel edges. +(13) D− contains no stable set of cardinality µ. +Suppose for a contradiction that D− contains a stable set S of cardinality µ. We may assume +without loss of generality that S = {a1, . . . , aµ}. +Let G1 = G[(�µ +j=1 Pj) \ {a}]. +Note that +by the definition of D, for every i ∈ [µ], we have NG1(ai) = NPi(ai) \ {a}, and in particular +|NG1(ai)| = 1. Since G1 is connected and Kt-free, and since and |S| = µ = µ(max{2s + 1, t}), +we can apply Theorem 2.5 to G1 and S. Note that every vertex in S has a unique neighbor in +G1, and so no path in G1 contains max{2s + 1, t} ≥ 3 vertices from S. Consequently, there is +an induced subgraph H1 of G1 with |H1 ∩ S| = 2s + 1 for which one of the following holds. +• H1 is either a caterpillar or the line graph of a caterpillar with H1 ∩ S = Z(H1). +• H1 is a subdivided star with root r1 such that Z(H1) ⊆ H1 ∩ S ⊆ Z(H1) ∪ {r1}. + +INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII. +19 +If H1 is a caterpillar, then G[H1∪{a}] contains a theta with ends a and a′ for every vertex a′ ∈ H1 +of degree more than two, a contradiction. Also, if the second bullet above holds, then since every +vertex in S is of degree one in G1, we have H1 ∩ S = Z(H1), and so r1 is not adjacent to a. But +then G[H1 ∪ {a}] contains a theta with ends x and r1, a contradiction. It follows that H1 is the +line graph of a caterpillar with |H1 ∩ S| = 2s + 1 and H1 ∩ S = Z(H1). This, together with the +fact that every vertex in H1 ∩ S ⊆ S has a unique neighbor in H1 ⊆ G, implies that H1 contains +the line graph H2 of a 1-subdivision of a caterpillar with |H2 ∩ S| = s and H2 ∩ S = Z(H2). +Let S2 = H2 ∩ S = Z(H2); then S2 is the set of all vertices of degree one in H2, and we may +assume without loss of generality that S2 = {a1, . . . , as}. Let G2 = G[H2 ∪(�s +j=1 Pj)]. It follows +that G2 ∈ Ct, NG2(a) = S2 = Z(H2) and a is trapped in H2 ∪ {a}. Therefore, H2 is an a-seed +in G2. Since b ∈ G2 \ NG2[a], applying Theorem 6.2 to G2 and a, we deduce that there exists +Sb ⊆ G2 \{a, b} such that |Sb| < s and Sb separates a and b in G2. But P1, . . . , Ps are s pairwise +internally disjoint paths in G2 from a to b, a contradiction with Theorem 2.1. This proves (13). +By (13), Theorem 5.3 and the definition γ, D− contains a clique of cardinality Rtourn(ν + 1). +This, along with the definition of Rtourn(·), implies that D contains (as a subdigraph) a transitive +tournament K on ν + 1 vertices. +We may assume without loss of generality that V (K) = +{a1, . . . , aν+1} such that for distinct i, j ∈ [ν + 1], (ai, aj) is an arc in K if i < j. From the +definition of D, it follows that {a2, . . . , aν+1, b} is a stable set in G, and for all i, j ∈ {2, . . . , ν+1} +with i < j, ai has a neighbor in P ∗ +j \{aj}. Hence, I = {2, . . . , ν + 1} satisfies Theorem 7.1. This +completes the proof. +■ +For positive integers d and r, let T r +d denote the rooted tree in which every leaf is at distance +r from the root, the root has degree d, and every vertex that is neither a leaf nor the root has +degree d + 1. We need a result from [15]: +Theorem 7.2 (Kierstead and Penrice [15]). For all integers d, r, s, t ≥ 1, there exists an integer +f = f(d, r, s, t) ≥ 1 such that if G contains T f +f as a subgraph, then G contains one of Ks,s, Kt +and T r +d as an induced subgraph. +The following lemma is the penultimate step in the proof of Theorem 1.8. +Lemma 7.3. For all integers d, r, t ≥ 1, there exists an integer m = m(d, r, t) with the following +property. Let G ∈ Ct be a graph, let a, b ∈ G be non-adjacent and let {Pi : i ∈ [m]} be a collection +of m pairwise internally disjoint paths in G from a to b. Then G[�m +j=1 Pj] contains a subgraph +J isomorphic to T r +d such that a ∈ J and a has degree d in J (that is, a is the root of J), and we +have b /∈ J. +Proof. Let d, t ≥ 1 be fixed. Let m1 = d. For every integer r > 1, let mr = ψ(t, (mr−1 + 1)d) +where ψ(·, ·) is as in Theorem 7.1. We prove by induction on r ≥ 1 that m(d, r, t) = mr satisfies +Lemma 7.3. Let P1, . . . , Pmr be mr pairwise internally disjoint paths in G from a to b. Since a +and b are not adjacent, it follows that for each i ∈ [mr], we have P ∗ +i ̸= ∅; let ai be the neighbor +of a in Pi. In particular, we have b /∈ {ai : i ∈ [mr]}. Suppose first that r = 1. Then we have +|{ai : i ∈ [m1]}| = m1 = d, and so G[{ai : i ∈ [mr]} ∪ {a}] contains a (spanning) subgraph +J isomorphic to T 1 +d such that a ∈ J and a has degree d in J, and we have b /∈ J, as desired. +Therefore, we may assume that r ≥ 2. Since mr = ψ(t, (mr−1 + 1)d), we can apply Theorem 7.1 +to a, b and {Pi : i ∈ [mr]}, obtaining I ⊆ [mr] with |I| = (mr−1 + 1)d which satisfies the two +outcomes of Theorem 7.1. Without loss of generality, we may assume that I = [(mr−1 + 1)d]. +It follows that {a1, · · · , a(mr−1+1)d, b} is a stable set in G, and for all i, j ∈ [(mr−1 + 1)d] with +i < j, ai has a neighbor in P ∗ +j \ {aj}. For every i ∈ [d], let a′ +i = a(i−1)mr−1+i and let +Ai = {(i − 1)mr−1 + i + 1, . . . , (i − 1)mr−1 + i + mr−1}. +In particular, we have |Ai| = mr−1. Then for each i ∈ [d] and each j ∈ Ai, a′ +i has a neighbor in +P ∗ +j \ {aj}, and so there exists a path Qj in G from a′ +i to b with Q∗ +j ⊆ P ∗ +j . Now, for every i ∈ [d], +a′ +i and b are non-adjacent, and {Qj : j ∈ Ai} is a collection of mr−1 pairwise internally disjoint + +20 +INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII. +paths in G from a′ +i to b. It follows from the induction hypothesis that G[� +j∈Ai Qj] contains a +subgraph Ji isomorphic to T r−1 +d +such that a′ +i ∈ Ji and a′ +i has degree d in Ji, and we have b /∈ Ji. +But now G[(�d +i=1 V (Ji)) ∪ {a}] ⊆ G[�mr +j=1 Pj] contains a (spanning) subgraph J isomorphic to +T r +d such that a ∈ J and a has degree d in J, and we have b /∈ J. This completes the proof of +Lemma 7.3. +■ +Finally, we prove Theorem 1.8, which we restate: +Theorem 1.8. For every tree F and every integer t ≥ 1, there exists an integer τ(F, t) ≥ 1 +such that every graph in Ct(F) has treewidth at most τ(F, t). +Proof. Let d and r be the maximum degree and the radius of F, respectively. It follows that +T r +d contains F as an induced subgraph. +Let f = f(d, r, 3, t) be as in Theorem 7.2 and let +m = m(f, f, t) be as in Lemma 7.3. Let β(·, ·) be as in Corollary 2.4. We claim that τ(F, t) = +β(max{m, t + 1}, t) satisfies Theorem 1.8. Suppose for a contradiction that tw(G) > τ for some +G ∈ Ct(F). By Corollary 2.4, G contains a max{m, t + 1}-block B. Consequently, since G is +Kt-free, there are two distinct and non-adjacent vertices a, b ∈ B, and m pairwise internally +disjoint paths P1, . . . , Pm in G from a to b. It follows from Lemma 7.3 that G contains T f +f as a +subgraph. Also, since G ∈ Ct(F) ⊆ Ct, G is (K3,3, Kt)-free. But now by Theorem 7.2, G contains +T r +d , and so F, as an induced subgraph, a contradiction. This completes the proof. +■ +References +[1] P. Aboulker, I. Adler, E. J. Kim, N. L. D. Sintiari, and N. Trotignon. “On the treewidth of even-hole-free +graphs.” European Journal of Combinatorics 98, (2021), 103394. +[2] T. Abrishami, B. Alecu, M. Chudnovsky, S. Hajebi, and S. Spirkl, “Induced subgraphs and tree decomposi- +tions VII. Basic obstructions in H-free graphs.” arXiv:2212.02737, (2022). +[3] T. Abrishami, M. Chudnovsky, C. Dibek, S. Hajebi, P. Rzążewski, S. Spirkl, and K. Vušković, “Induced +subgraphs and tree decompositions II. Toward walls and their line graphs in graphs of bounded degree.” +arXiv:2108.01162, (2021). +[4] T. Abrishami, M. Chudnovsky and K. Vušković, “Induced subgraphs and tree decompositions I. Even-hole- +free graphs of bounded degree.” J. Combin. Theory Ser. B, 157 (2022), 144-175. +[5] M. Ajtai, J. Komlós and E. Szemerédi. “A note on Ramsey numbers.” J. Combinatorial Theory, Ser. A 29 +(1980), 354–360. +[6] H. L. Bodlaender. “Dynamic programming on graphs with bounded treewidth.” Springer, Berlin, Heidelberg, +(1988), pp. 105–118. +[7] K. Cameron, M.V. da Silva, S. Huang, and K. Vušković, “Structure and algorithms for (cap, even hole)-free +graphs.” Discrete Mathematics 341, 2 (2018), 463-473. +[8] M. Chudnovsky, N. Robertson, P. Seymour, and R. Thomas, “The strong perfect graph theorem.” Annals of +Math 164 (2006), 51-229. +[9] M. Chudnovsky and P. Seymour, “Even-hole-free graphs still have bisimplicial vertices.” arXiv:1909.10967, +(2019). +[10] M. Chudnovsky and P. Seymour, “The three-in-a-tree problem.” Combinatorica 30, 4 (2010): 387-417. +[11] J. Davies, “Vertex-minor-closed classes are χ-bounded.” arXiv:2008.05069, (2020). +[12] J. Davies, appeared in an Oberwolfach technical report DOI:10.4171/OWR/2022/1. +[13] J. Erde and D. Weißauer. “A short derivation of the structure theorem for graphs with excluded topological +minors.” SIAM Journal of Discrete Mathematics 33, 3 (2019), 1654–1661. +[14] M. Grohe and D. Marx. “Structure theorem and isomorphism test for graphs with excluded topological +subgraphs,” SIAM Journal on Computing 44, 1 (2015), 114–159. +[15] H.A. Kierstead and S. G. Penrice, “Radius two trees specify χ-bounded classes.” J. Graph Theory 18, 2 +(1994): 119–129. +[16] T. Korhonen, “Grid Induced Minor Theorem for Graphs of Small degree.” arXiv:2203.13233, (2022). +[17] V. Lozin and I. Razgon. “Tree-width dichotomy.” European J. Combinatorics 103 (2022): 103517. +[18] K. Menger, “Zur allgemeinen Kurventheorie.” Fund. Math. 10, 1927, 96–115. +[19] N. Robertson and P. Seymour. “Graph minors. V. Excluding a planar graph.” J. Combin. Theory Ser. B, 41 +(1) (1996), 92–114. +[20] N.L.D. Sintiari and N. Trotignon. “(Theta, triangle)-free and (even-hole, K4)-free graphs. Part 1: Layered +wheels.” J. Graph Theory 97 (4) (2021), 475-509. +[21] N. Trotignon, private communication, 2021. + diff --git a/19A0T4oBgHgl3EQfMv_l/content/tmp_files/load_file.txt b/19A0T4oBgHgl3EQfMv_l/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..274785991a40bab5f36e738b1556b3e5b98d151b --- /dev/null +++ b/19A0T4oBgHgl3EQfMv_l/content/tmp_files/load_file.txt @@ -0,0 +1,1145 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf,len=1144 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='02138v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='CO] 5 Jan 2023 INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' EXCLUDING A FOREST IN (THETA, PRISM)-FREE GRAPHS TARA ABRISHAMI∗†, BOGDAN ALECU∗∗¶, MARIA CHUDNOVSKY∗∐, SEPEHR HAJEBI §, AND SOPHIE SPIRKL§∥ Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Given a graph H, we prove that every (theta, prism)-free graph of sufficiently large treewidth contains either a large clique or an induced subgraph isomorphic to H, if and only if H is a forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Introduction All graphs in this paper are finite and simple unless specified otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let G, H be graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We say that G contains H if G has an induced subgraph isomorphic to H, and we say G is H-free if G does not contain H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For a family H of graphs we say G is H-free if G is H-free for every H ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' A class of graphs is hereditary if it is closed under isomorphism and taking induced subgraphs, or equivalently, if it is the class of all H-free graphs for some other family H of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For a graph G = (V (G), E(G)), a tree decomposition (T, χ) of G consists of a tree T and a map χ : V (T) → 2V (G) with the following properties: (i) For every v ∈ V (G), there exists t ∈ V (T) such that v ∈ χ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' (ii) For every v1v2 ∈ E(G), there exists t ∈ V (T) such that v1, v2 ∈ χ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' (iii) For every v ∈ V (G), the subgraph of T induced by {t ∈ V (T) | v ∈ χ(t)} is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For each t ∈ V (T), we refer to χ(t) as a bag of (T, χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' The width of a tree decomposition (T, χ), denoted by width(T, χ), is maxt∈V (T) |χ(t)| − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' The treewidth of G, denoted by tw(G), is the minimum width of a tree decomposition of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Treewidth was first popularized by Robertson and Seymour in their graph minors project, and has attracted a great deal of interest over the past three decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Particularly, graphs of bounded treewidth have been shown to be well-behaved from structural [19] and algorithmic [6] viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This motivates investigating the structure of graphs with large treewidth, and especially, the substructures emerging in them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' The canonical result in this realm is the Grid Theorem of Robertson and Seymour [19], the following, which describes the unavoidable subgraphs of graphs with large treewidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For a positive integer t, the (t × t)-wall, denoted by Wt×t, is a planar graph with maximum degree three and treewidth t (see Figure 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' a formal definition can be found in [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' ∗Princeton University, Princeton, NJ, USA ∗∗School of Computing, University of Leeds, Leeds, UK §Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada † Supported by NSF-EPSRC Grant DMS-2120644.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' ∐ Supported by NSF-EPSRC Grant DMS-2120644 and by AFOSR grant FA9550-22-1-0083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' ¶ Supported by DMS-EPSRC Grant EP/V002813/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' ∥ We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), [funding reference number RGPIN-2020-03912].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Cette recherche a été financée par le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG), [numéro de référence RGPIN-2020-03912].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This project was funded in part by the Government of Ontario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Date: January 6, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' 1 2 INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' W5×5 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1 (Robertson and Seymour [19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For every integer t ≥ 1 there exists w = w(t) ≥ 1 such that every graph of treewidth more than w contains a subdivision of Wt×t as a subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1 can also be reformulated into a full characterization of unavoidable minors in graphs of large treewidth, that every graph of sufficiently large treewidth contains any given planar graph as a minor (and no non-planar graph has this property).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' In contrast, unavoidable induced subgraphs of graphs with large treewidth are far from completely understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' There are some natural candidates though, which we refer to as the “basic obstructions”: complete graphs and complete bipartite graphs, subdivided walls mentioned above, and line graphs of subdivided walls, where the line graph L(F) of a graph F is the graph with vertex set E(F), such that two vertices of L(F) are adjacent if and only if the corresponding edges of F share an end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Note that the complete graph Kt+1, the complete bipartite graph Kt,t, and the line graph of every subdivision of Wt×t all have treewidth t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For a positive integer t, let us say a graph H is a t-basic obstruction if H is one of the following graphs: Kt, Kt,t, a subdivision of Wt×t, or the line graph of a subdivision of Wt×t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We say a graph G is t-clean if G does not contain a t-basic obstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' The basic obstructions do not form a comprehensive list of induced subgraph obstructions for bounded treewidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Equivalently, there are t-clean graphs of arbitrarily large treewidth for small values of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' A well-known hereditary class of graphs evidencing this fact is the class of even-hole-free graphs, where a hole is an induced cycle on at least four vertices, the length of a hole is its number of edges and an even hole is a hole with even length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' In fact, for every positive integer t ≥ 1, one may observe that an even-hole-free graph is t-clean if and only if it is Kt-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' It is therefore tempting to ask whether even-hole-free graphs excluding a fixed complete graph have bounded treewidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Sintiari and Trotignon [20] answered this with a vehement no, providing a construction of (even-hole, K4)-free graphs with arbitrarily large treewidth, hence proving that there are t-clean (even-hole-free) graphs of arbitrarily large treewidth for every fixed t ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' In addition, graphs from this construction are rather sparse, in the sense that they exclude short holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='2 (Sintiari and Trotignon [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For all integers w, l ≥ 1, there exists an (even-hole, K4)-free graph Gw,l of treewidth more than w and with no hole of length at most l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Note that t-clean graphs for t ≤ 2 have empty vertex set or edge set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' But one might still hope for 3-clean graphs to have bounded treewidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This is in fact supported by a result from [7] asserting that 3-clean even-hole-free graphs have treewidth at most five.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' However, another construction by Sintiari and Trotignon [20] shows that being 3-clean fails to guarantee bounded treewidth in the more general class of theta-free graphs (see the next section for the definition of a theta;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' one may check that the every t-basic obstruction for t ≥ 3 contains either a theta or a triangle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Indeed, the treewidth of theta-free graphs remains unbounded even when forbidding short cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='3 (Sintiari and Trotignon [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For all integers w, g ≥ 1, there exists a theta-free graph Gw,g of treewidth more than w and girth more than g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' A natural question to ask then is what further conditions must be imposed to force bounded treewidth in even-hole-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For instance, graphs from both Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='3 have INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' 3 vertices of arbitrarily large degree, and so it was conjectured in [1] that (theta, triangle)- free graphs of bounded maximum degree have bounded treewidth and even-hole-free graphs of bounded maximum degree have bounded treewidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' These were proved in [3] and [4], respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' In the same paper [1], a stronger conjecture was made, asserting that basic obstructions are in fact the only obstructions to bounded treewidth in graphs of bounded maximum degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This was later proved in [16], which closed the line of inquiry into graph classes of bounded maximum degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='4 (Korhonen [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For all integers t, δ ≥ 1, there exists w = w(t, δ) such that every t-clean graph of maximum degree at most δ has treewidth at most w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Despite its generality, the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='4 is surprisingly short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' However, the case of proper hereditary classes containing graphs of unbounded maximum degree seems to be much harder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For graph classes G and H, let us say H modulates G if for every positive integer t, there exists a positive integer w(t) (depending on G and H) such that every t-clean H-free graph in G has treewidth at most w(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' An induced-subgraph analogue to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1 is therefore equivalent to a full characterization of graph classes H which modulate the class of all graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This remains out of reach, but the special case where |H| = 1 turns out to be more approachable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For a graph H and a graph class G, let us say H modulates G if {H} modulates G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Building on a method from [17], recently we characterized all graphs H which modulate the class of all graphs: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='5 (Abrishami, Alecu, Chudnovsky, Hajebi and Spirkl [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let H be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then H modulates the class of all graphs if and only if H is a subdivided star forest, that is, a forest in which every component has at most one vertex of degree more than two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' In general, for a hereditary class G containing t-clean graphs of arbitrarily large treewidth for small t, one may ask for a characterization of graphs H modulating G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Given Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='2, a natural class G to consider is the class of even-hole-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Note that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='2 shows that a graph H modulates even-hole-free graphs only if H is a chordal graph (that is, a graph with no hole) of clique number at most three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' As far as we know, the converse may also be true, that every chordal graph of clique number at most three modulates even-hole-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' In fact, in this paper we narrow the gap, showing that every chordal graph of clique number at most two, that is, every forest, modulates the class of even-hole-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For every forest H and every integer t ≥ 1, every even-hole-free graph of suffi- ciently large treewidth contains either H or a clique of cardinality t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This aligns with the observation [21] that every forest is contained in some graph Gw,l from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' As mentioned above, one way to improve on Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='6 is to push H towards being an arbitrary chordal graph of clique number three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Another way to strengthen Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='6 is to find a superclass G of even-hole-free graphs for which forests are the only graphs modulating G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' While the former remains open, we provide an appealing answer to the latter: our main result shows that forests are exactly the graphs which modulate the class of (theta, prism)-free graphs (see the next section for the definition of a prism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' again one may check that in (theta, prism)-free graphs, being t-clean is equivalent to being Kt-free for every positive integer t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let H be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then H modulates (theta, prism)-free graphs if and only if H is a forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' In other words, given a graph H, for every integer t ≥ 1, every (theta, prism)-free graph of sufficiently large treewidth contains either H or a clique of cardinality t, if and only if H is a forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let C be the class of all (theta, prism)-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' It is easily seen that C contains all even- hole-free graphs, and so Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='7 implies Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Note that the “only if” direction of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='7 follows immediately from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='3 as prisms contain triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Since every forest is an induced subgraph of a tree, in order to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='7, it suffices to prove 4 INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='8 below, which we do in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For a positive integer t and a tree F, we denote by Ct the class of all graphs in C with no clique of cardinality t (that is, t-clean graph in C), and by Ct(F) the class of all F-free graphs in Ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For every tree F and every integer t ≥ 1, there exists an integer τ(F, t) ≥ 1 such that every graph in Ct(F) has treewidth at most τ(F, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We conclude this introduction by sketching our proofs (the terms we use here are defined in later sections).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='8 begins with a two-step preparation which culminates in the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='2, a result we will also use in subsequent papers in this series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' As the first step, inspired by a result from [9], we show that for every graph G ∈ C which contains a pyramid with certain conditions on the apex and its neighbors, G admits a construction which we call a “(T, a)-strip-structure,” where a is the apex of the pyramid and T is an optimally chosen tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Roughly speaking, we show that G\\{a} can be partitioned into two induced subgraphs H and J where H is more or less similar to the line graph of the tree T and every vertex in J with a neighbor in H attaches at a pyramid lurking in H in a restricted way;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' we call the latter vertices “jewels”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' The proof of this theorem occupies Sections 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' The second step is to employ the previous result to show that if G ∈ Ct admits a (C, a)-strip-structure where C is a caterpillar, then every vertex in G \\ NG[a] can be separated from a by removing a few vertices (our proof works more generally when C is any tree of bounded maximum degree, but the caterpillar case suffices for our application).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We prove this in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' The central difficulty in the proof is to deal with the jewels separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This is surmounted in Section 5 where we prove several results concerning the properties of jewels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Most notably, we show that jewels only attach at “local areas of the line-graph-like part” of G, and that only a few jewels attach at each local area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This concludes the preparation for proving Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Next, we embark on the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We assume that G ∈ Ct has large treewidth, which together with results from Section 2 implies that G contains two vertices x, y joined by many pairwise internally disjoint induced paths P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' , Pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Now we analyze the structure of the graph G[P1 ∪ · · · ∪ Pm].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' It turns out that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' if m is large enough,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' then either there are many paths among Pi’s whose union H admits a (C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' x)-strip-structure for some caterpillar C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' or for some large value of d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' G[P1 ∪ · · · ∪ Pm] contains a tree S isomorphic to the complete bipartite graph K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' such that x is the vertex of degree d in S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' and for every leaf l of S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' there are many pairwise internally disjoint induced paths between l and y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' such that in addition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' paths corresponding to distinct leaves of S are also pairwise internally disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' The former case implies that y can be separated from x by removing few vertices, which using a result from Section 6, yields a contradiction with Menger’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' The latter case is the first step towards building the large tree in G as a subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We now iterate the argument we just described, applying it to each leaf l of S and y, obtaining larger and larger trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' The process is stopped once we reach a sufficiently large tree as a subgraph of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This, combined with the fact that G ∈ Ct and a result of Kierstead and Penrice [15], yields the desired tree F as an induced subgraph of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Section 2 covers preliminary definitions as well as some results from the literature used in our proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Section 3 investigates the behavior of pyramids in graphs from C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Section 4 is devoted to defining strip-structures and jewels, and showing how they arise from pyramids in graphs in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Section 5 takes a closer look at jewels for the strip-structures obtained in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' In Section 6 we show that admitting certain strip-structures weakens the connectivity of most vertices to the apex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Finally, in Section 7, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Preliminaries and results from the literature Let G = (V (G), E(G)) be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For a set X ⊆ V (G) we denote by G[X] the subgraph of G induced by X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For X ⊆ V (G)∪E(G), G\\X denotes the subgraph of G obtained by removing INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' 5 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Note that if X ⊆ V (G), then G \\ X denotes the subgraph of G induced by V (G) \\ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' In this paper, we use induced subgraphs and their vertex sets interchangeably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let x ∈ G and d be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We denote by N d G(x) the set of all vertices in G at distance d from some x, and by N d G[x] the set of all vertices in G at distance at most d from x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We write NG(x) for N 1 G(x) and NG[x] for N 1 G[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For an induced subgraph H of G, we define NH(x) = NG(x) ∩ H, NH[x] = NG[x] ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Also, for X ⊆ G, we denote by NG(X) the set of all vertices in G \\ X with at least one neighbor in X, and define NG[X] = NG(X) ∪ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let X, Y ⊆ G be disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We say X is complete to Y if all edges with an end in X and an end in Y are present in G, and X is anticomplete to Y if there are no edges between X and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' A path in G is an induced subgraph of G that is a path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' If P is a path in G, we write P = p1- · · · -pk to mean that V (P) = {p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' , pk} and pi is adjacent to pj if and only if |i−j| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We call the vertices p1 and pk the ends of P, and say that P is from p1 to pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' The interior of P, denoted by P ∗, is the set P \\ {p1, pk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' The length of a path is its number of edges (so a path of length at most one has empty interior).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Similarly, if C is a cycle, we write C = c1- · · · -ck-c1 to mean that V (C) = {c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' , ck} and ci is adjacent to cj if and only if |i − j| ∈ {1, k − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' The length of a cycle is its number edges (or equivalently, vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=') A theta is a graph Θ consisting of two non-adjacent vertices a, b, called the ends of Θ, and three pairwise internally disjoint paths P1, P2, P3 from a to b of length at least two, called the paths of Θ, such that P ∗ 1 , P ∗ 2 , P ∗ 3 are pairwise anticomplete to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For a graph G, by a theta in G we mean an induced subgraph of G which is a theta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' A prism is a graph Π consisting of two disjoint triangles {a1, a2, a3}, {b1, b2, b3} called the triangles of Π, and three pairwise disjoint paths P1, P2, P3 called the paths of Π, where Pi has ends ai, bi for each i ∈ {1, 2, 3}, and for distinct i, j ∈ {1, 2, 3}, aiaj and bibj are the only edges between Pi and Pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For a graph G, by a prism in G we mean an induced subgraph of G which is a prism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' A pyramid is a graph Σ consisting of a vertex a, a triangle {b1, b2, b3} and three paths P1, P2, P3 of length at least one with Pi from a to bi for each i ∈ {1, 2, 3} and otherwise pairwise disjoint, such that for distinct i, j ∈ {1, 2, 3}, bibj is the only edge between Pi \\ {a} and Pj \\ {a}, and at most one of P1, P2, P3 has length exactly one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We say that a is the apex of the pyramid and b1b2b3 is the base of the pyramid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' The pyramid Σ is said to be long if Pi has length more than one for every i ∈ {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For a graph G, by a pyramid in G we mean an induced subgraph of G which is a pyramid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Theta, pyramid and prism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' The dotted lines represent paths of length at least one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let us now mention a few results from the literature which we will use in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let G be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' By a separation in G we mean a triple (L, M, R) of pairwise disjoint subsets of vertices in G with L ∪ M ∪ R = G, such that neither L nor R is empty and L is anticomplete to R in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let x, y ∈ G be distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We say a set M ⊆ G \\ {x, y} separates x and y if there exists a separation (L, M, R) in G with x ∈ L and y ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Also, for disjoint sets X, Y ⊆ G, we say a set M ⊆ G \\ (X ∪ Y ) separates X and Y if there exists a separation (L, M, R) in G with X ⊆ L and Y ⊆ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' If X = {x}, we say that M separates x and Y to mean M separates X and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Recall the following well-known theorem of Menger [18]: 6 INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1 (Menger [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let k ≥ 1 be an integer, let G be a graph and let x, y ∈ G be distinct and non-adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then either there exists a set M ⊆ G \\ {x, y} with |M| < k such that M separates x and y, or there are k pairwise internally disjoint paths in G from x to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let k be a positive integer and let G be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' A strong k-block in G is a set B of at least k vertices in G such that for every 2-subset {x, y} of B, there exists a collection P{x,y} of at least k distinct and pairwise internally disjoint paths in G from x to y, where for every two distinct 2-subsets {x, y}, {x′, y′} ⊆ B of G, and every choice of paths P ∈ P{x,y} and P ′ ∈ P{x′,y′}, we have P ∩ P ′ = {x, y} ∩ {x′, y′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For a tree T and xy ∈ E(T), we denote by Tx,y the component of T − xy containing x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let G be a graph and (T, χ) be a tree decomposition for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For every S ⊆ T, let χ(S) = � x∈S χ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' By an adhesion of (T, χ) we mean the set χ(x) ∩ χ(y) = χ(Tx,y) ∩ χ(Ty,x) for some xy ∈ E(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For every x ∈ V (T), by the torso at x, denoted by ˆχ(x), we mean the graph obtained from the bag χ(x) by, for each y ∈ NT (x), adding an edge between every two non-adjacent vertices u, v ∈ χ(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' In [2], we used Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='4 and the following result from [13]: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='2 (Erde and Weißauer [13], see also [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let r be a positive integer, and let G be a graph containing no subdivision of Kr as a subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then G admits a tree decomposition (T, χ) for which the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Every adhesion of (T, χ) has cardinality less than r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For every x ∈ V (T), either ˆχ(x) has fewer than r2 vertices of degree at least 2r4, or ˆχ(x) has no minor isomorphic to K2r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' to prove the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='3 (Abrishami, Alecu, Chudnovsky, Hajebi and Spirkl [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let k, t ≥ 1 be integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then there exists an integer w = w(k, t) ≥ 1 such that every t-clean graph with no strong k-block has treewidth at most w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Note that for every t ≥ 3, every subdivision of Wt×t contains a theta and the line graph of every subdivision of Wt×t contains a prism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' It follows that for every t ≥ 1, every graph in Ct is t-clean, and so the following is immediate from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='3: Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For all integers k, t ≥ 1, there exists an integer β = β(k, t) such that every graph in Ct with no strong k-block has treewidth at most β(k, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' A vertex v in a graph G is said to be a branch vertex if v has degree more than two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' By a caterpillar we mean a tree C with maximum degree three such that there is a path P in C containing all branch vertices of C (our definition of a caterpillar is non-standard for two reasons: a caterpillar is often allowed to be of arbitrary maximum degree, and the path P from the definition often contains all vertices of degree more than one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' By a subdivided star we mean a graph isomorphic to a subdivision of the complete bipartite graph K1,δ for some δ ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' In other words, a subdivided star is a tree with exactly one branch vertex, which we call its root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For every graph H, a vertex v of H is said to be simplicial if NH(v) is a clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We denote by Z(H) the set of all simplicial vertices of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Note that for every tree T, Z(T) is the set of all leaves of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' An edge e of a tree T is said to be a leaf-edge of T if e is incident with a leaf of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' It follows that if H is the line graph of a tree T, then Z(H) is the set of all vertices in H corresponding to the leaf-edges of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' The following is proved in [2] based on (and refining) a result from [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='5 (Abrishami, Alecu, Chudnovsky, Hajebi and Spirkl [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For every integer h ≥ 1, there exists an integer µ = µ(h) ≥ 1 with the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let G be a connected graph with no clique of cardinality h and let S ⊆ G such that |S| ≥ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then either some path in G contains h vertices from S, or there is an induced subgraph H of G with |H ∩ S| = h for which one of the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' H is either a caterpillar or the line graph of a caterpillar with H ∩ S = Z(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' H is a subdivided star with root r such that Z(H) ⊆ H ∩ S ⊆ Z(H) ∪ {r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Jumps and jewels on pyramids with trapped apices For a graph G, an induced subgraph H of G and a vertex a ∈ H, we say a is trapped in H if we have N 2 G[a] ⊆ H, and;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' every vertex in NH(a) = NG(a) has degree two in H (and so in G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' The goal of this section is, for a graph G ∈ C, H ⊆ G and a pyramid Σ in H, to investigate the adjacency between Σ and a path in G \\ H, assuming that the apex of Σ is trapped in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This will be of essential use in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We begin with a few definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let G be a graph and let Σ be a pyramid in G with apex a, base b1b2b3 and paths P1, P2, P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' A set X ⊆ Σ is said to be local (in Σ) if either X ⊆ Pi for some i ∈ {1, 2, 3} or X ⊆ {b1, b2, b3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let P be a path in G \\ Σ with (not necessarily distinct) ends p1, p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For i ∈ {1, 2, 3}, we say P is a corner path for Σ at bi if p1 has at least one neighbor in Pi \\ {bi};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' p2 is complete to {b1, b2, b3} \\ {bi}, and;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' except for the edges between {p1, p2} and Σ described in the above two bullets, there is no edge with an end in P and an end in Σ \\ {bi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' By a corner path for Σ we mean a corner path for Σ at one of b1, b2 or b3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let p ∈ G \\ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then p is said to be narrow for Σ if NΣ(p) is local in Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Otherwise, we say p is wide for Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For i ∈ {1, 2, 3}, we say p is a jewel for Σ at bi if p is anticomplete to Pi (in particular, p is anticomplete to a), and for every j ∈ {1, 2, 3} \\ {i}, we have NPj(p) = NPj[bj].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' By a jewel for Σ we mean a jewel for Σ at one of b1, b2 or b3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Note that if p is either a corner path or a jewel for Σ, then p is wide for Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' The following lemma establishes a converse to this fact for graphs in C and pyramids with a trapped apex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let G ∈ C be graph, let H ⊆ G and let a ∈ H be trapped in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let Σ be a pyramid in H with apex a, base b1b2b3 and paths P1, P2, P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let p ∈ G \\ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then p is wide for Σ if and only if p is either a corner path for Σ or a jewel for Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We only need to prove the “only if” direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Assume that p ∈ G \\ H is wide for Σ and p is not a corner path for Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Since a is trapped in H and p ∈ G \\ H, it follows that Σ is long and p is anticomplete to NΣ[a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' First, we show that: (1) There exists i ∈ {1, 2, 3} for which p is anticomplete to Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Suppose for a contradiction that p has a neighbor in each of P1, P2, P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Since p is wide for Σ and p is not a corner path for Σ, we may assume without loss of generality that p has a neighbor in P ∗ 1 and a neighbor in P ∗ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For each i ∈ {1, 2, 3}, traversing Pi from a to bi, let xi be the first neighbor of p in Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Since a is trapped, it follows that x1 ∈ P ∗ 1 , x2 ∈ P ∗ 2 and x3 ∈ P3\\NΣ[a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' But then there is a theta in G with ends a, p and paths a-Pi-xi-p for i ∈ {1, 2, 3}, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This proves (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' By (1) and without loss of generality, we may assume that p is anticomplete to P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Note that since p is wide for Σ, it follows that for every j ∈ {1, 2}, p has a neighbor in Pj, and there exists j ∈ {1, 2} for which p has a neighbor in P ∗ j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For each j ∈ {1, 2}, traversing Pj from a to bj, let xj and yj be the first and the last neighbor of p in Pj, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then we have xj ∈ P ∗ j \\NPj(a) for some j ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' In fact, the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' (2) For every j ∈ {1, 2}, we have xj ∈ P ∗ j \\ NPj(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then since p is wide for Σ, we may assume without loss of generality that p has a neighbor in P ∗ 1 and we have x2 = y2 = b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' But now there is a theta in G with ends a, b2 and paths a-P1-x1-p-b2, a-P2-b2 and a-P3-b3-b2, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This proves (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' (3) For every j ∈ {1, 2}, NPj(p) is a clique of cardinal ity two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then we may assume without loss of generality that either x1 = y1 or x1 and y1 are distinct and non-adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' By (2), for every j ∈ {1, 2}, we have xj ∈ P ∗ j \\NPj(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Therefore, 8 INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' if x1 = y1, then there is a theta in G with ends a, x1 and paths a-P1-x1, a-P2-x2-p-x1 and a-P3-b3-b1-P1-x1, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Thus, x1 and y1 are distinct and non-adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' But now there is a theta in G with ends a, p and paths a-P1-x1-p, a-P2-x2-p and a-P3-b3-b1-P1-y1-p, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This proves (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' The proof is almost concluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' By (3), for every j ∈ {1, 2}, we have NPj(p) = {xj, yj} and xj is adjacent to yj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' If yj ∈ P ∗ j for some j ∈ {1, 2}, then there is a prism in G with triangles xjyjp and b1b2b3 and paths xj-Pj-a-P3-b3, yj-Pj-bj and p-y3−j-P3−j-b3−j, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Hence, we have yj = bj for every j ∈ {1, 2}, and so p is a jewel corner for Σ at bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This completes the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' ■ We can now prove the main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let G ∈ C be a graph, let H ⊆ G and let a ∈ H be trapped in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let Σ be a pyramid in H with apex a, base b1b2b3 and paths P1, P2, P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let P be a path in G \\ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then one of the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' NΣ(P) is local in Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' P contains a corner path for Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' P contains a jewel for Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Suppose for a contradiction that there exists a path P in G \\ H for which none of the outcomes of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='2 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We choose such a path P with |P| as small as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' It follows that NΣ(P) is not local in Σ, NΣ(X) is local in Σ for every connected set X ⊊ P, P contains no corner path for Σ and P contains no jewel for Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Therefore, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1, we have |P| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Since a is trapped in H and P ⊆ G\\H, it follows that Σ is long and P is anticomplete to NΣ[a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For every i ∈ {1, 2, 3}, let P ′ i = Pi \\ NPi[a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Since NΣ(P) is not local and P is minimal subject to this property, we may assume without loss of generality that NΣ(p1) ⊆ P ′ 1 and p1 has a neighbor in P ′ 1 \\ {b1}, and;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' p2 has a neighbor in P ′ 2, and either NΣ(p2) ⊆ P ′ 2, or NΣ(p2) ⊆ {b1, b2, b3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' It follows from the choice of P that P ∗ is anticomplete to Σ\\{b1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For each i ∈ {1, 2}, traversing Pi from a to bi, let xi and yi be the first and the last neighbor of pi in Pi, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' So we have x1 ∈ P ′ 1 \\ {b1}, y1 ∈ P ′ 1 and x2, y2 ∈ P ′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' In fact, the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' (4) We have x2 ∈ P ′ 2 \\ {b2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then we have x2 = y2 = b2, and so b2 ∈ NΣ(p2) ⊆ {b1, b2, b3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Note that if p2 is adjacent to b3, then P is a corner path for Σ at b1, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' So p2 is not adjacent to b3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' But now there is a theta in G with ends a, b2 and paths a-P1-x1-p1-P-p2-b2, a-P2-b2 and a-P3-b3-b2, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This proves (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' In view of (4) and the choice of P, we conclude that P ∗ is anticomplete to Σ, and for every i ∈ {1, 2}, we have NΣ(pi) = NP ′ i (pi), xi ∈ P ′ i \\ {bi} and yi ∈ P ′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' (5) For every i ∈ {1, 2}, xi and yi are distinct and adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then we may assume without loss of generality that either x1 = y1 or x1 and y1 are distinct and non-adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' In the former case, there is a theta in G with ends a, x1 and paths a-P1-x1, a-P2-x2-p2-P-p1-x1 and a-P3-b3-b1-P1-x1, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' It follows that x1 and y1 are distinct and non-adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' But then there is a theta in G with ends a, p1 and paths a-P1-x1-p1, a-P2-x2-p2-P-p1 and a-P3-b3-b1-P1-y1-p1, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This proves (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' By (5), for every i ∈ {1, 2}, we have NPi(p) = {xi, yi} and xi is adjacent to yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' But now there is a prism in G with triangles p1x1y1 and p2x2y2 and paths P, x1-P1-a-P2-x2 and y1-P1-b1-b2-P2-y2, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This completes the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' ■ INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Strip structures with an ornament of jewels The main result of this section, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='2, provides a description of the structure of graphs in C which have an induced subgraph containing a pyramid with a trapped apex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We first set up a framework that allows us to think of a pyramid with apex a as a special case of a construction similar to the line graph of a tree T, which we call a “(T, a)-strip-structure.” We start with an induced subgraph W of G that admits an “optimal” (T, a)-strip-structure in G in a certain sense, and show that the rest of the graph fits into the same construction, except for vertices which are jewels for certain canonically positioned pyramids in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' First, we need to properly define a strip-structure (this is similar to [8], [9] and [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' A tree T is said to be smooth if T has at least three vertices and every vertex of T is either a branch vertex or a leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let G be a graph, let a ∈ G, let T be a smooth tree, and let η : V (T) ∪ E(T) ∪ (E(T) × V (T)) → 2G\\{a} be a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For every S ⊆ V (T), we define η(S) = � v∈S,e∈E(T[S])(η(v) ∪ η(e)) and η+(S) = η(S) ∪ {a}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For every vertex v ∈ V (T), we define Bη(v) to be the union of all sets η(e, v) taken over all edges e ∈ E(T) incident with v (we often omit the subscript η unless there is ambiguity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' The function η is said to be a (T, a)-strip-structure in G if the following conditions are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' (S1) For all distinct o, o′ ∈ V (T) ∪ E(T), we have η(o) ∩ η(o′) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' (S2) If l ∈ V (T) is a leaf of T, then η(l) is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' (S3) For all e ∈ E(T) and v ∈ V (T), we have η(e, v) ⊆ η(e) and η(e, v) ̸= ∅ if and only if e is incident with v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' (S4) For all distinct edges e, f ∈ E(T) and every vertex v ∈ V (T), η(e, v) is complete to η(f, v), and there are no other edges between η(e) and η(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' In particular, if e and f share no end, the η(e) is anticomplete to η(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' (S5) For every e ∈ E(T) with ends u, v, define η◦(e) = η(e) \\ (η(e, u) ∪ η(e, v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then for every vertex x ∈ η(e), there is a path in η(e) from x to a vertex in η(e, u) with interior contained in η◦(e), and there is a path in η(e) from x to a vertex in η(e, v) with interior contained in η◦(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' (S6) For all v ∈ V (T) and e ∈ E(T), η(v) is anticomplete to η(e) \\ η(e, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' In other words, we have Nη(T)(η(v)) ⊆ Bη(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' (S7) For every v ∈ V (T) and every connected component D of η(v), we have NBη(v)(D) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' (S8) For every leaf l ∈ V (T) of T, assuming e ∈ E(T) to be the leaf-edge of T incident with l, a is complete to η(e, l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Also, a has no other neighbors in η(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let S ⊆ η(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We say that S is local in η if S ⊆ η(e) for some e ∈ E(T) or S ⊆ Bη(v) ∪ η(v) for some v ∈ V (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' The following lemma shows that every non-local subset contains a 2-subset (that is, a subset of cardinality two) which is non-local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let G be a graph and a ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let T be a smooth tree and η be a (T, a)-strip- structure in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Assume also that C ⊆ η(T) is not local in η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then there is a 2-subset of C which is not local in η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' First, suppose there exists a vertex x ∈ C ∩ η◦(e) for some e ∈ E(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Since C is not local, there exists y ∈ C \\ η(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Now {x, y} is a 2-subset of C which is not local in η, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Therefore, we may assume that C ⊆ � v∈V (T)(B(v) ∪ η(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Since the empty set is local in η, we have C ̸= ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' thus, we may pick x ∈ C, v ∈ V (T) and e ∈ E(T) such that x ∈ η(e, v) ∪ η(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' If there exists a vertex y ∈ C \\ (η(e) ∪ B(v) ∪ η(v)), then {x, y} is a 2-subset of C which is not local in η, and so we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Consequently, we may assume that C ⊆ η(e) ∪ B(v) ∪ η(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Since C is not local, there exist x′ ∈ η(e) \\ (B(v) ∪ η(v))) and y′ ∈ (B(v) ∪ η(v)) \\ η(e) such that {x′, y′} ⊆ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Now {x′, y′} is a 2-subset of C which is not local in η, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This completes the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' ■ In order to state and prove the main result of this section, we need to define several notions related to strip-structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' From here until the statement of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='2, let us fix a graph G, a vertex a ∈ G, a smooth tree T and a (T, a)-strip-structure η in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' 10 INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For every edge e ∈ E(T) with ends u, v, by an η(e)-rung, we mean a path P in η(e) ⊆ η(T) for which either |P| = 1 and P ⊆ η(e, u) ∩ η(e, v), or P has an end in η(e, u) \\ η(e, v) and an end in η(e, v) \\ η(e, u) and we have P ∗ ⊆ η◦(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Equivalently, a path P in η(e) is an η(e)-rung if P has an end in η(e, u) and an end in η(e, v) and we have |P ∩ η(e, u)| = |P ∩ η(e, v)| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' It follows from (S5) that every vertex in η(e) \\ η◦(e) is contained in an η(e)-rung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' In particular, if either η(e, u) ⊆ η(e, v) or η(e, v) ⊆ η(e, u), then η(e, u) = η(e, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' An η(e)-rung is said to be long if it is of non-zero length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For every edge e ∈ E(T), let ˜η(e) be the set of vertices in η(e) that are not in any η(e)-rung (so ˜η(e) ⊆ η◦(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=') We say that η is tame if η(v) = ∅ for every v ∈ V (T), and;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' ˜η(e) = ∅ for every e ∈ E(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' In other words, η is tame if and only if every vertex in η(T) is in an η(e)-rung for some e ∈ E(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For a (T, a)-strip-structure η′ in G, we write η ≤ η′ to mean that for every o ∈ V (T)∪E(T)∪ (E(T)×V (T)), we have η(o) ⊆ η′(o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We say a (T, a)-strip-structure η is substantial if for every e ∈ E(T), there exists a long η(e)-rung in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Equivalently, η is substantial if for every edge e ∈ E(T) with ends u, v, we have η(e, u) ̸= η(e, v), and so η(e, u) \\ η(e, v), η(e, v) \\ η(e, u) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' One may observe that since T has at least three vertices, if η is substantial and η ≤ η′, then η′ is substantial too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We say η is rich if a is trapped in η+(T), and;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' for every leaf l ∈ V (T) of T, assuming e ∈ E(T) to be the leaf-edge of T incident with l, we have |η(e, l)| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' It follows that if there exists a rich (T, a)-strip-structure η in G, then T has exactly |NG(a)| leaves, and for every leaf l ∈ V (T) of T, assuming e ∈ E(T) to be the leaf-edge of T incident with l and v ∈ V (T) to be the unique neighbor of l in T, we have η(e, v) ∩ η(e, l) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' By a seagull in T we mean a triple (v, e1, e2) where v ∈ V (T) and e1, e2 are two distinct edges of T incident with v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' By a claw in T we mean a 4-tuple (v, e1, e2, e3) where v ∈ V (T) and e1, e2, e3 are three distinct edges of T incident with v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let (v, e1, e2, e3) be a claw in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' By an η-pyramid at (v, e1, e2, e3), we mean a pyramid Σ with apex a, base b1b2b3 and paths P1, P2, P3, satisfying the following for each i ∈ {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' bi ∈ η(ei, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' There exists a leaf li of T with the following properties: (1) li belongs to the component of T \\ {ei} not containing v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' (2) Let Λi be the unique path in T from v to li (so ei ∈ E(Λi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then Pi = Γi ∪ {a}, where Γi is a path in � e∈E(Λi) η(e) such that Ri = Γi ∩ η(ei) is a long η(ei)-rung and Γi ∩ η(e) is a η(e)-rung for each e ∈ E(Λi) \\ {ei}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' In particular, assuming ui to be the ends of ei distinct from v and ci to be the unique vertex in NRi(bi) = NPi(bi) for each i ∈ {1, 2, 3}, we have bi ∈ η(ei, v) \\ η(ei, ui) and ci ∈ η(ei) \\ η(ei, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For a branch vertex v ∈ V (T), by an η-pyramid at v we mean an η-pyramid at (v, e1, e2, e3) for some claw (v, e1, e2, e3) in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Also, by an η-pyramid we mean an η-pyramid at v for some branch vertex v ∈ V (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' It follows that every η-pyramid is a long pyramid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Also, if η is substantial, then for every claw (v, e1, e2, e3) in T there is a η-pyramid at (v, e1, e2, e3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let (v, e1, e2) be a seagull in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' A vertex p ∈ G\\η+(T) is said to be a jewel for η at (v, e1, e2) if for some edge e3 ∈ E(T)\\{e1, e2} incident with v, there exists an η-pyramid Σ at (v, e1, e2, e3) with base b1b2b3 where bi ∈ η(ei, v) for each i ∈ {1, 2, 3}, such that p is a jewel for Σ at b3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' In particular, for each i ∈ {1, 2}, p is adjacent to bi and the unique vertex ci in NPi(bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Therefore, since Σ is an η-pyramid at (v, e1, e2, e3), assuming ui to be the end of ei distinct from v, it follows that p has a neighbor bi ∈ η(ei, v) \\ η(ei, ui) and a neighbor ci ∈ η(ei) \\ η(ei, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For a vertex v ∈ V (T), by a jewel for η at v we mean a jewel for η at (v, e1, e2) for some seagull (v, e1, e2) in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Also, by a jewel for η we mean a jewel for η at v for some branch vertex v ∈ V (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We denote by Jη the set of all jewels for η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' It follows that Jη ⊆ G \\ η+(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' 11 We are now in a position to prove the main result of this section: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let G ∈ C, let a ∈ G and let T be a smooth tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Suppose that there exists a tame, substantial and rich (T, a)-strip-structure in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then there is a substantial and rich (T, a)-strip-structure ζ in G for which G \\ (ζ+(T) ∪ Jζ) is anticomplete to ζ+(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let η be a tame, substantial and rich (T, a)-strip-structure in G such that η(T) is maximal with respect to inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let M = G \\ (η+(T) ∪ Jη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' (6) Let P be a path in M with ends p1 and p2 such that there exists x1 ∈ Nη(T)(p1) and x2 ∈ Nη(T)(p2) for which {x1, x2} is not local in η, and such that |P| ≥ 1 is as small as possible subject to this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then there exists {j1, j2} = {1, 2} and f = v1v2 ∈ E(T) such that xj1 ∈ B(vj1) \\ η(f) and xj2 ∈ (B(vj2) ∪ η(f)) \\ B(vj1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For each i ∈ {1, 2}, let ei ∈ E(T) such that xi ∈ η(ei) (hence e1 ̸= e2) and si be an end of ei such that there exists a path Λ0 (possibly of length zero) from s1 to s2 in T \\ {e1, e2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We claim that there is a vertex v ∈ Λ0 such that B(v) ∩ {x1, x2} = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Suppose first that s1 ̸= s2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' let v1 be unique neighbor of s1 in Λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then we have x1 /∈ B(v1) and x2 /∈ B(s1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Also, since f = s1v1 does not satisfy (6), we have either x1 /∈ B(s1) or x2 /∈ B(v1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' But then either v = s1 or v = v1 satisfies the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Thus, we may assume that v = s1 = s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Note that since neither e1 nor e2 satisfies (6), we have x1 /∈ B(s1) and x2 /∈ B(s2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' In other words, we have B(v) ∩ {x1, x2} = ∅, and the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Henceforth, let v be as promised by the above claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For each i ∈ {1, 2}, let ui be the end of ei distinct from si (hence u1 ̸= u2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let Λ = u1-s1-Λ0-s2-u2 and let u′ 1, u′ 2 be the neighbors of v in Λ such that Λ traverses u1, u′ 1, v, u′ 2, u2 in this order (so either of u1 = u′ 1 and u2 = u′ 2 is possible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let e′ i = u′ iv for each i ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Since T is smooth, there exists a vertex u′ 3 ∈ NT (v) \\ Λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' let e′ 3 = u′ 3v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For each i ∈ {1, 2, 3}, let Ti be the component of T \\(NT (v)\\{u′ i}) containing v (so e′ i ∈ E(Ti)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then since B(v)∩{x1, x2} = ∅ and since η is tame and substantial, there exists an η-pyramid Σ at (v, e′ 1, e′ 2, e′ 3) with apex a, base b1b2b3 and paths P1, P2, P3 such that we have bi ∈ η(e′ i, v) and Pi \\ {a, bi} ⊆ η(Ti) \\ B(v) for each i ∈ {1, 2, 3}, and;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' xi ∈ P ∗ i for each i ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' In particular, the second bullet above implies that NΣ(P) is not local in Σ and P is not a corner path for Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Since P ⊆ M, we have P ∩ Jη = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Thus, Σ being an η-pyramid, it follows that P contains no jewel for Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Also, since η is rich, a is trapped in η+(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Therefore, applying Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='2 to G, H = η+(T), a, Σ and P, we deduce that P contains a corner path for Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' On the other hand, note that by the second bullet above, for every vertex x ∈ Σ \\ {a}, either {x, x1} or {x, x2} is not local in η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' From this, the minimality of |P| and the fact that η is rich, it follows that P ∗ is anticomplete to Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' But then P is a corner path for Σ, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This proves (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' (7) Let P be a path in M with ends p1 and p2 such that there exists x1 ∈ Nη(T)(p1) and x2 ∈ Nη(T)(p2) for which {x1, x2} is not local in η, and such that |P| ≥ 1 is as small as possible subject to this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let f = v1v2 ∈ E(T) and {j1, j2} = {1, 2} be as guaranteed by (6) applied to P, x1 and x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then we have Nη(T)(P ∗) ⊆ η(f, vj1) and Nη(T)({p1, p2}) ⊆ η(f)∪B(v1)∪B(v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Without loss of generality, we may assume that j1 = 1 and j2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Note that by the minimality of |P|, we have Nη(T)(P ∗) ⊆ η(f, v1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Therefore, one of p1 and p2 has a neighbor in η(T) \\ (η(f) ∪ B(v1) ∪ B(v2));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' say p1 is adjacent to x′ 1 ∈ η(T) \\ (η(f) ∪ B(v1) ∪ B(v2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For each i ∈ {1, 2}, let Ti be the component of T \\ {f} containing vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' It follows that there exists j ∈ {1, 2} such that x′ 1 ∈ η(Tj) \\ B(vj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Assume that |P| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' By the minimality of |P|, we have j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' But then P, x′ 1 and x2 violate (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We deduce that |P| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' But now P, x′ 1 and x3−j violate (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This proves (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' 12 INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' (8) Let P be a path in M with ends p1 and p2 such that there exists x1 ∈ Nη(T)(p1) and x2 ∈ Nη(T)(p2) for which {x1, x2} is not local in η, and such that |P| ≥ 1 is as small as possible subject to this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Suppose that there exist {k1, k2} = {1, 2}, f = v1v2 ∈ E(T) and e1 ∈ E(T) \\ {f} incident with vk1 such that pk1 has a neighbor in η(e1, vk1) and pk2 has a neighbor in (B(vk2) ∪ η(f)) \\ B(vk1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then pk1 is complete to B(vk1) \\ (η(e1, vk1) ∪ η(f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Due to symmetry, we may assume that k1 = 1 and k2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let e3 ∈ E(T)\\{e1, f} be incident with v1 and let b3 ∈ η(e3, v1) be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We need to show that p1 is adjacent to b3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Suppose for a contradiction that p1 and b3 are non-adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let b1 ∈ η(e1, v1) be adjacent to p1 and let x ∈ (B(v2) ∪ η(f)) \\ B(v1) be adjacent to p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let T2 be the component of T \\ (NT (v1) \\ {v2}) containing v1 (so f ∈ E(T2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Also, for each i ∈ {1, 3}, let ui be the end of ei distinct from v1 and let Ti be the component of T \\ (NT (v1) \\ {ui}) containing v1 (so ei ∈ E(Ti)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' By (6) and (7), there exists an edge f ′ = v′ 1v′ 2 ∈ E(T) such that Nη(T)({p1, p2}) ⊆ η(f ′) ∪ B(v′ 1) ∪ B(v′ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This, along with the minimality of |P|, implies that p1 is anticomplete to (η(T1)∪η(T3))\\B(v1), P \\ {p1} is anticomplete to η(T1) ∪ η(T3) and P \\ {p2} is anticomplete to η(T2) \\ B(v1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Since p2 has a neighbor x ∈ (B(v2) ∪ η(f)) \\ B(v1) and since η is tame, there exists a path P2 in G from a to p2 with P ∗ 2 ⊆ η(T2) \\ B(v1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Also, for each i ∈ {1, 3}, there exists a path Pi in G from a to bi with P ∗ i ⊆ η(Ti) \\ B(v1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Note that since η is rich, P anticomplete to NG[a];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' in particular, P1 has length at least two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' But now there is a theta in G with ends a and b1 and paths P1, a-P2-p2-P-p1-b1 and b1-b3-P3-a, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This proves (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' The following is immediate from (8) and the fact that T is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' (9) Let P be a path in M with ends p1 and p2 such that there exists x1 ∈ Nη(T)(p1) and x2 ∈ Nη(T)(p2) for which {x1, x2} is not local in η, and such that |P| ≥ 1 is as small as possible subject to this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Suppose that there exist {k1, k2} = {1, 2} and f = v1v2 ∈ E(T) such that xk1 ∈ B(vk1) \\ (η(f)) and xk2 ∈ (B(vk2) ∪ η(f)) \\ B(vk1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then pk1 is complete to B(vk1) \\ η(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We now deduce: (10) Let D be a component of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then Nη(T)(D) is local in η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1, there exist x1, x2 ∈ Nη(T)(D) such that {x1, x2} is not local in η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For each i ∈ {1, 2}, let pi be a neighbor of xi in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Since D is connected, there exists a path P in D ⊆ M from p1 to p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' In other words, there exists a path P in M with ends p1, p2 along with x1 ∈ Nη(T)(p1) and x2 ∈ Nη(T)(p2) such that {x1, x2} is not local in η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Now, let P be a path in M with ends p1 and p2 such that there exists x1 ∈ Nη(T)(p1) and x2 ∈ Nη(T)(p2) for which {x1, x2} is not local in η, and such that |P| ≥ 1 is as small as possible subject to this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' So we can apply (6) to P, x1 and x2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' let {j1, j2} = {1, 2} and f = v1v2 ∈ E(T) be as in (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We may assume without loss of generality that j1 = 1 and j2 = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' thus, v1 is a branch vertex of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' It follows from (7) that Nη(T)(P ∗) ⊆ η(f, v1) and Nη(T)({p1, p2}) ⊆ η(f) ∪ B(v1) ∪ B(v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' By (9) applied to k1 = 1 and k2 = 2, p1 is complete to B(v1) \\ η(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Also, from (9) applied to k1 = 2 and k2 = 1, it follows that either p2 is complete to B(v2) \\ η(f) and B(v2) \\ η(f) ̸= ∅, or p2 is anticomplete to B(v2) \\ η(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Note that if |P| > 1, then by the minimality of |P|, we have Nη(T)(p1) ⊆ B(v1) and Nη(T)(p2) ⊆ (B(v2)∪η(f))\\B(v1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let us define η′ : V (T)∪E(T)∪(E(T)×V (T)) ⊆ 2G\\{a} as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let η′(f) = η(f) ∪ P and let η′(f, v1) = η(f, v1) ∪ {p1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let η′(f, v2) = η(f, v2) ∪ {p2} if p2 is complete to B(v2) \\ η(f) and B(v2) \\ η(f) ̸= ∅, and;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' η′(f, v2) = η(f, v2) if p2 is anticomplete to B(v2) \\ η(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let η′ = η elsewhere on V (T) ∪ E(T) ∪ (E(T) × V (T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then since η is tame, substantial and rich, and p2 is adjacent to x2 ∈ B(v2) ∪ η(f)) \\ B(v1), it is straightforward to check that η′ is also a tame, substantial and rich (T, a)-strip-structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' But we have η′(T) = η(T) ∪ P, a contradiction with the maximality of η(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This proves (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' The proof is almost concluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let X be the union of all the components D of M such that D is anticomplete to η+(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Since η is rich, it follows that for every component D of M \\ X, a INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' 13 is anticomplete to X and Nη+(T)(D) = Nη(T)(D) is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' By (10), for every component D of M \\ X, Nη(T)(D) is local in η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let D be the set of all components D of M \\ X for which we have Nη+(T)(D) ⊆ Bη(v) for some v ∈ V (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Breaking the ties arbitrarily and by the definition of X, we may write D = � v∈V (T) Dv, where for all distinct u, v ∈ V (T), we have Du ∩ Dv = ∅, and;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' for all v ∈ V (T) and every D ∈ Dv, we have Nη+(T)(D) ⊆ Bη(v) and Nη+(T)(D) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Also, for every e = uv ∈ E(T), let De be the set of all components D of M \\ X for which we have Nη+(T)(D) ⊆ η(e) and either Nη(T)(D) ∩ η◦(e) ̸= ∅, or;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Nη(T)(D) ∩ (η(e, u) \\ η(e, v)) ̸= ∅ and Nη(T)(D) ∩ (η(e, v) \\ η(e, v)) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' From the definition of X, it follows that every component of M \\ X belongs to exactly one of the sets {Dv, De : v ∈ V (T), e ∈ E(T)} (note that since η is rich, a is anticomplete to each such component).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let ζ : V (T) ∪ E(T) ∪ (E(T) × V (T)) ⊆ 2G\\{a} be defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For all v ∈ V (T) and e ∈ E(T), let ζ(v) = � D∈Dv D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' ζ(e) = η(e) ∪ (� D∈De D), and;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' ζ(e, v) = η(e, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' It is easily seen that ζ satisfies the conditions (S1-S8) from the definition of a (T, a)-strip- structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' In particular, since η is rich, ζ satisfies (S2), and from the definitions of X, Dv’s and De’s, it follows that ζ satisfies (S5) and (S7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Also, we have η ≤ ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Now, since η is substantial and rich, since η ≤ ζ and from the definitions of X and ζ, it follows that ζ is a substantial and rich (T, a)-strip-structure with Jζ = Jη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Moreover, note that we have ζ+(T) = η(T)+ ∪ (M \\ X), and so G \\ (ζ+(T) ∪ Jζ) = G \\ (ζ+(T) ∪ Jη) = X is anticomplete to ζ+(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This completes the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' ■ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Jewels under the loupe Here we revisit jewels for strip-structures, establishing several results about their proper- ties in various settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This will help attune Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='2 for its application in the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' First we need to introduce some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let G be a graph and let a ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let T be a smooth tree and let ζ be a (T, a)-strip-structure in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let v ∈ V (T) and let e ∈ E(T) be incident with v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We denote by ζe(v) the set of all components D of ζ(v) for which we have NB(v)(D) ⊆ η(e, v), or equivalently, Nζ(T)\\ζ(e,v)(D) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let (v, e1, e2) be a seagull in T and let ui be the end of ei distinct from v for each i ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We define ζ(v, e1, e2) = ζ(e1) ∪ ζ(e2) ∪ ζe1(u1) ∪ ζe2(u2) ∪ ζ(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We denote by Jζ,(v,e1,e2) the set of all jewels for ζ at (v, e1, e2), and for every vertex v ∈ V (T), Jζ,v stands for the set of all jewels for η at v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' It follows that Jζ,v = ∅ if v is a leaf of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' The first result in this section describes, for a (T, a)-strip-structure in a theta-free graph, the attachments of jewels at a vertex of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let G be a theta-free graph and let a ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let T be a smooth tree and let ζ be a (T, a)-strip-structure in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let (v, e1, e2) be a seagull in T and let x ∈ Jζ,(v,e1,e2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We have Nζ+(T)(x) ⊆ ζ(v, e1, e2), and so Nζ+(T)(Jζ,(v,e1,e2)) ⊆ ζ(v, e1, e2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Consequently, for every vertex v ∈ V (T), we have Nζ+(T)(Jζ,v) ⊆ ζ(NT [v]), and for every two distinct vertices v, v′ ∈ V (T), we have Jζ,v ∩ Jζ,v′ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' 14 INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Assume that ζ is rich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let i ∈ {1, 2} and let R be a long ζ(ei)-rung, let r be the end of R in ζ(ei, v) and let r′ be the unique neighbor of r in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then either x is anticomplete to R or NR(x) = {r, r′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Note that v is a branch vertex of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For each i ∈ {1, 2}, let ui be the end of ei distinct from v and let Ti be the component of T \\(NT (v)\\{ui}) containing v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let T ′ be the component of T \\{u1, u2} containing v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let x ∈ Jζ,(v,e1,e2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Since x ∈ Jζ,(v,e1,e2) is a jewel for ζ, there exists an edge e3 ∈ E(T)\\{e1, e2} incident with v and a ζ-pyramid Σ at (v, e1, e2, e3) with apex a, base b1b2b3 and paths P1, P2, P3 such that x is a jewel for Σ at b3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' In particular, for each j ∈ {1, 2, 3}, Pj ∩ ζ(ej) is a long ζ(ej)-rung Rj with bj as its end in ζ(ej, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Also, x is anticomplete to P3 (and so x is not adjacent to a), and for each j ∈ {1, 2}, assuming cj to be the unique vertex in NRj(bj) = NPj(bj), x is adjacent to bj ∈ ζ(ej, v) \\ ζ(ej, uj) and cj ∈ ζ(ej) \\ ζ(ej, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Therefore, there exist paths Qi, Si of length more than one in G from a to x for which we have bi ∈ Q∗ i ⊆ (ζ(T ′) \\ ζ(v)) ∪ (ζ(ei, v) \\ ζ(ei, ui)) and ci ∈ S∗ i ⊆ ζ(Ti) \\ (B(v) ∪ ζ(ui) ∪ ζ(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' To prove the first assertion of the Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1, assume for a contradiction that x has a neighbor y ∈ ζ+(T) \\ ζ(v, e1, e2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Since x is not adjacent to a, we have y ∈ ζ(T) \\ ζ(v, e1, e2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' First, assume that y ∈ ζ(T ′)\\ζ(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then by (S5) and (S7) from the definition of a strip-structure, there exists a path Q′ of length more than one in G from a to x with Q′∗ ⊆ ζ(T ′) \\ ζ(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' But now there is a theta in G with ends a, x and paths a-S1-x, a-S2-x and a-Q′-x, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' It follows that y ∈ ζ(T1 ∪ T2) \\ ζ(v, e1, e2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' In other words, for some i ∈ {1, 2}, we have y ∈ ζ(Ti) \\ (ζ(ei) ∪ ζei(ui) ∪ ζ(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' As a result, by (S5) and (S7) from the definition a strip- structure, and by the definition of ζei(ui), there exists a path S′ i of length more than one in G from a to x with S′∗ i ⊆ ζ(Ti)\\(ζ(ei)∪ζei(ui)∪ζ(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' But now assuming i′ ∈ {1, 2} to be distinct from i, there is a theta in G with ends a, x and paths a-Qi-x, a-S′ i-x and a-Si′-x, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This proves the the first assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Next we prove the second assertion of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' By symmetry, we may assume that i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Assume that x has a neighbor y ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let P ′ 1 = (P1 \\ R1) ∪ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let Σ′ be the pyramid with apex a, base rb2b3 and paths P ′ 1, P2 and P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Recall that since ζ is rich, a is trapped in ζ+(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Also, Σ′ is a pyramid in ζ+(T), x is adjacent to y ∈ P ′ 1, x is adjacent to b2, c2 ∈ P2 and x is anticomplete to P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' It follows that x is a wide vertex for Σ′ which is not a corner path for Σ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Now applying Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1 to G, a, H = ζ+(T), Σ′ and p = x, we deduce that x is a jewel for Σ′ at b3, and so NR(x) = NP ′ 1(x) = {r, r′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This completes the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' ■ Our next goal is to show that for every rich (T, a)-strip-structure in a graph G ∈ Ct, there are only a few jewels at each vertex of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let us begin with a lemma, asserting that for a rich (T, a)-strip-structure ζ in a theta-free graph, each set Bζ(v) is almost a clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let G be a theta-free graph and a ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let T be a smooth tree and ζ be a rich (T, a)-strip-structure in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then for every v ∈ V (T), there exists at most one edge f ∈ E(T) such that η(f, v) is not a clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Suppose for a contradiction that there are two distinct edges f1, f2 ∈ E(T) incident with v, and for each i ∈ {1, 2}, there exist xi, yi ∈ ζ(fi, v) such that xi is not adjacent to yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then v is not a leaf of T and H = x1-x2-y1-y2-x1 is a hole of length four in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Since ζ is rich, a is anticomplete to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let f1 = u1v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let l1 be a leaf of T which belongs to the component of T \\ {v} containing u1, and let Λ1 be the unique path in T from v to l1 (so f1 ∈ E(Λ1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let Rx1 be an ζ(f)-rung containing x1 and let Ry1 be an ζ(f)-rung containing y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Since ζ is rich, H1 = Rx1 ∪ Rx2 ∪ B(u1) is a connected induced subgraph of G, and so there is a path Q in H1 from x1 to y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' It follows that Q has length more than one and Q∗ ⊆ (B(u1) ∪ ζ(f1))\\B(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' But now there is a theta in G with ends x1, y1 and paths Q, x1-x2-y1 and x1-y2-y1, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This completes the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' ■ Recall the following classical result of Ramsey (see, for instance, [5] for an explicit bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=') INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' 15 Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='3 (See [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For all integers a, b ≥ 1, there exists an integer R = R(a, b) ≥ 1 such that every graph G on at least R(a, b) vertices contains either a clique of cardinality a or a stable set of cardinality b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We can now prove the second main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For all positive integers t, δ, there exists a positive integer j = j(t, δ) with the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let G ∈ Ct be a graph and let a ∈ G and let T be a smooth tree of maximum degree δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let ζ be a rich (T, a)-strip-structure in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then for every vertex v ∈ V (T), we have |Jζ,v| < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let j = j(t, δ) = �δ 2 �R(t, 3) with R(·, ·) as in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then in order to prove |Jζ,v| < j, it is enough to show that |Jζ,(v,e1,e2)| < R(t, 3) for every seagull (v, e1, e2) in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Suppose for a contradiction that |Jζ,(v,e1,e2)| ≥ R(t, 3) for some seagull (v, e1, e2) in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then v is a branch vertex of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For each i ∈ {1, 2}, let ui be the end of ei different from v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Since G ∈ Ct, it follows from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='3 that Jζ,(v,e1,e2) contains a stable set X of cardinality three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For every x ∈ X, since x is a jewel for ζ at (v, e1, e2), it follows that for every i ∈ {1, 2}, there exists a long ζ(ei)-rung Rx i such that Qx i = Rx i \\ ζ(ei, v) is a path in ζ(ei) \\ ζ(ei, v) from a neighbor of x to a vertex in ζ(ei, ui) \\ ζ(ei, v);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' in particular, Rx i contains a neighbor of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Therefore, for each i ∈ {1, 2}, we may pick a non-empty set Ri of long ζ(ei)-rungs such that every vertex in X has a neighbor in at least one rung in Ri, and with Ri minimal with respect to inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We deduce: (11) There exists i ∈ {1, 2} with |Ri| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then for every i ∈ {1, 2}, there exists a long ζ(ei)-rung Si such that every vertex in X has a neighbor in Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let si be the end of Si in ζ(ei, v) and s′ i be unique neighbor of si in Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' By the second assertion of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1, X is complete to {s′ 1, s′ 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' But now X ∪ {s′ 1, s′ 2} is a theta in G with ends s′ 1, s′ 2, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This proves (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' By (11) and due to symmetry, we may assume that |R1| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This, together with the minimality of R1, implies that there exist distinct vertices x, y ∈ X as well as distinct long ζ(e1)-rungs Rx, Ry ∈ R1 such that x has a neighbor in Rx, y has a neighbor in Ry, x is anticomplete to Ry, and y anticomplete to Rx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let rx and ry be the ends of Rx and Ry in ζ(e1, v), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let r′ x be the unique neighbor of rx in Rx and r′ y be the unique neighbor of ry in Ry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' so we have r′ x, r′ y ∈ ζ(e1) \\ ζ(e1, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' By the second assertion of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1, we have NRx∪Ry(x) = {rx, r′ x} and NRx∪Ry(y) = {ry, r′ y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' It follows that rx, r′ x ∈ Rx \\ Ry and ry, r′ y ∈ Ry \\ Rx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Also, rx is anticomplete to Ry \\ {ry}, as otherwise (Ry \\ {ry}) ∪ {rx} contains a long ζ(e1)-rung R with NR(x) = {rx}, which violates the second assertion of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Similarly, ry is anticomplete to Rx \\ {rx}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Now, let G1 = G[(B(u1)\\ζ(e1, u1))∪((Rx∪Ry)\\{rx, ry})] and let G2 = G[(B(u2)\\ζ(e2, u2))∪ Qx 2 ∪ Qy 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Since ζ is rich, the second bullet in the definition of a rich strip-structure implies that G1 and G2 are connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Consequently, there exists a path Q1 in G1 from r′ x to r′ y, and there exists a path Q2 from x to y with Q∗ 2 ⊆ G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Also, since v is a branch vertex of T, we may choose an edge e3 ∈ E(T) \\ {e1, e2} incident with v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' By the first assertion of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1, {x, y} is anticomplete to ζ(e3, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let Q3 be a path from rx to ry with Q∗ 3 ⊆ ζ(e3, v) (thus |Q3| ∈ {2, 3}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' But now there is a prism with triangles xrxr′ x and yryr′ y and paths Q1, Q2, Q3, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This completes the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' ■ Our last theorem in this section examines the connectivity within G \\ ζ+(T) for a (T, a)- strip-structure ζ arising from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We need the following lemma, the proof of which is similar to that of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let G be a theta-free graph and let a ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let T be a smooth tree and let ζ be a (T, a)-strip-structure in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let v, v′ ∈ V (T) be distinct and let P be a path in G \\ ζ+(T) with 16 INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' ends x, x′ such that x ∈ Jζ,v, x′ ∈ Jζ,v′ and P ∗ is anticomplete to ζ+(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then v and v′ are adjacent in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Note that by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1, x and x′ are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let Λ be the path in T from v to v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then Λ has length more than one, and so there are two distinct edges f, f ′ ∈ E(Λ) such that f is incident with v and f ′ is incident with v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let u be the end of f distinct from v and u′ be the end of f ′ distinct from v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let (v, e1, e2) and (v′, e′ 1, e′ 2) be two seagulls in G such that x ∈ Jζ,(v,e1,e2) and x′ ∈ Jζ,(v′,e′ 1,e′ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For each i ∈ {1, 2}, let ui be the end of ei distinct from v and let u′ i be the end of e′ i distinct from v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Without loss of generality, we may assume that u2, u′ 2 /∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let T2 be the component of T \\ (NT (v) \\ {u2}) containing v and let T ′ 2 be the component of T \\(NT (v′)\\{u′ 2}) containing v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let T ′ be the component of T \\{u′, u′ 2} containing v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Since x is a jewel for ζ at (v, e1, e2), it follows that x is not adjacent to a, and x has a neighbor c ∈ ζ(e2) \\ ζ(e2, v) ⊆ ζ(T2) \\ (B(v) ∪ ζ(u2) ∪ ζ(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Therefore, there exists a path Q of length more than one in G from a to x for which we have c ∈ Q∗ ⊆ ζ(T2) \\ (B(v) ∪ ζ(u2) ∪ ζ(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Also, since x′ is a jewel for ζ at (v′, e′ 1, e′ 2), it follows that x′ is not adjacent to a, and x′ has a neighbor b′ ∈ B(v′)\\(ζ(f ′, u′)∪ζ(e′ 2, v′)) and a neighbor c′ ∈ ζ(e′ 2)\\ζ(e′ 2, v′) ⊆ ζ(T ′ 2)\\(B(v′)∪ζ(u′ 2)∪ζ(v′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Therefore, there exist paths P ′, Q′ of length more than one in G from a to x′ for which we have b′ ∈ P ′∗ ⊆ (ζ(T ′) \\ ζ(v′)) ∪ (ζ(f ′, v′) \\ ζ(f ′, u′)) and c′ ∈ Q′∗ ⊆ ζ(T ′ 2) \\ (B(v′) ∪ ζ(u2) ∪ ζ(v′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' But now there is a theta in G with ends a, x′ and paths a-P ′-x′, a-Q′-x′ and a-Q-x-P-x′, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This proves Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' ■ Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let t, δ ≥ 1 be integers and let j(t, δ) be as in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let G ∈ Ct be a graph and let a ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let T be a smooth tree of maximum degree δ and let v ∈ V (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let ζ be a rich (T, a)-strip-structure in G such that G \\ (ζ+(T) ∪ Jζ) is anticomplete to ζ+(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let x ∈ G \\ (ζ+(T) ∪ Jζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then there exists Sx ⊆ G \\ (ζ+(T) ∪ {x}) such that |Sx| < 2j(t, δ) and Sx separates x and Jζ \\ ({x} ∪ Sx) in G \\ ζ+(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Consequently, Sx separates x and ζ+(T) in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1, {Jζ,v : v ∈ V (T)} is a partition of Jζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let G′ be the graph obtained from G\\ζ+(T) by contracting the set Jζ,v into a vertex zv for each v ∈ V (T) with Jζ,v ̸= ∅, and then adding a new vertex z such that NG′(z) = {zv : v ∈ V (T), Jζ,v ̸= ∅}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We claim that there is a set Y ⊆ G′ \\ {x, z} of cardinality at most two which separates x and z in G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1, there are three pairwise internally disjoint paths in G′ from x to z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Thus, there exist S ⊆ T with |S| = 3 as well as three paths {Pv : v ∈ S} in G \\ ζ+(T) all having x as an end and otherwise disjoint, such that for each v ∈ S, Pv has an end yv ∈ Jζ,v distinct from x, and we have P ∗ v ⊆ G\\(ζ+(T) ∪ Jζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' As a result, for all distinct v, v′ ∈ S, Pv,v′ = yv-Pv-x-Pv′-yv′ is a path in G\\ζ+(T) from yv ∈ Jζ,v to yv′ ∈ Jζ,v′ such that P ∗ v,v′ ⊆ G\\(ζ+(T)∪Jζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' In particular, P ∗ v,v′ is anticomplete to ζ+(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' But then by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='5, S is a clique in T, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' The claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let Y be as in the above claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For each y ∈ Y , if y = zv for some v ∈ V (T), then let Ay = Jζ,v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Otherwise, let Ay = {y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let Sx = � y∈Y Ay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then Sx ⊆ G\\(ζ+(T)∪{x}) separates x and Jζ \\({x}∪Sx) in G\\ζ+(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Also, by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='4, we have |Sx| < 2j(t, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This completes the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' ■ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Strip structures and connectivity In this section, we investigate the connectivity implications of the presence of certain (T, a)- strip-structures in graphs from Ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' The main result is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For all integers t, δ ≥ 1, there exists an integer σ = σ(t, δ) ≥ 1 with the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let G ∈ Ct be a graph and let a ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let T be a smooth tree of maximum degree δ and let ζ be a rich (T, a)-strip-structure in G such that G \\ (ζ+(T) ∪ Jζ) is anticomplete to ζ+(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then for every vertex x ∈ G \\ NG[a], there exists a set Sx ⊆ G \\ {a, x} with |Sx| < σ such that S separates a and x in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' 17 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let j(t, δ) be as in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We claim that σ = σ(t, δ) = 2δ(j(t, δ) + t) satisfies Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For every vertex v ∈ V (T), we define Cv = B(v) if v is a leaf of T and Cv = ∅ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Also, for every vertex v ∈ V (T), let Kv be a maximal clique of G contained in B(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Thus, we have |Kv| < t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Moreover, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='2 along with the assumption that ζ is rich implies if v is a leaf of T, then we have Kv = B(v) = Cv (and so |Kv| = 1), and if v is a branch vertex of T, then Kv contains all but possibly one of the sets η(f, v) for f ∈ E(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For every S ⊆ T, we define MS = � w∈NT (S) Jη,w, NS = � w∈NT (S) Kw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Also, we write Mv for M{v} and Nv for N{v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For every v ∈ V (T), let Ov = Mv ∪ Nv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' The following is immediate from Theorems 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='4 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' (12) For every v ∈ V (T), we have Ov ⊆ G \\ (Jζ,v ∪ {a});' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' |Ov| < δ(j(t, δ) + t) ≤ σ, and;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Ov separates a and Jζ,v in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Now, for every x ∈ G \\ NG[a], we define Sx as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' First, assume that x ∈ ζ(T) \\ NG[a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then either x ∈ ζ(e) for some edge e = uv ∈ E(T), or x ∈ ζ(v) for some branch vertex v ∈ V (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' In the former case, let Ex = Mu ∪ Mv, Ix = N{u,v} ∪ Cu ∪ Cv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' In the latter case, let Ex = Mv ∪ Jζ,v Ix = Nv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let Sx = Ex ∪ Ix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Observe that since x ∈ G \\ NG[a], we have Sx ⊆ G \\ {a, x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Also, by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='4, we have |Ex| ≤ 2δj(t, δ) and so |Sx| < 2δ(j(t, δ) + t) = σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Moreover, from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1 and the fact that ζ is rich, it is easy to check that for every path P in G from a to x, if P ⊆ ζ+(T), then P contains a vertex from Ix, and otherwise P contains a vertex from either Ix or Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Therefore, Sx separates a and x in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Next, assume that x ∈ Jζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1, there exists a unique vertex v ∈ V (T) such that x ∈ Jζ,v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let Sx = Ov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then by (12), we have Sx ⊆ G \\ {a, x}, |Sx| < σ and Sx separates a and x in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Finally, assume that x ∈ G\\(ζ+(T)∪Jζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then letting Sx to be as in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='6, it follows from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='6 that Sx ⊆ G \\ {a, x}, |X| < 2j(t, δ) ≤ σ and Sx separates a and x in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This completes the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' ■ Our application of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1 though is confined to the case where T is a caterpillar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' More precisely, for a graph G and a vertex a ∈ G, an induced subgraph H ⊆ G \\ {a} is said to be an a-seed in G if the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' There exists a caterpillar C such that H is the line graph of a 1-subdivision of C and NG(a) = Z(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' The vertex a is trapped in H ∪ {a}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' It follows that Z(H) is the set of all degree-one vertices of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We now combine Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='2 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1 to deduce the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' 18 INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For every integer t ≥ 1, there exists an integer s = s(t) ≥ 1 with the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let G ∈ Ct be a graph and a ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Assume that there is an a-seed in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then for every vertex x ∈ G \\ NG[a], there exists Sx ⊆ G \\ {a, x} with |Sx| < s such that Sx separates a and x in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let σ(·, ·) be as in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We show that s = s(t) = σ(t, 3) satisfies Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Pick an a-seed H in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let T be the unique smooth caterpillar with |NG(a)| leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then T has maximum degree three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Also, one may immediately observe that there is a tame, substantial and rich (T, a)-strip-structure η in G with η(T) = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Now we can apply Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='2 to G, a and T, deducing that there exists a substantial and rich (T, a)-strip-structure ζ in G such that G \\ (ζ+(T) ∪ Jζ) is anticomplete to ζ+(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Hence, by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1 applied to G, a, T and ζ, for every vertex x ∈ G \\ NG[a], there exists Sx ⊆ G \\ {a, x} with |Sx| < s such that Sx separates a and x in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This completes the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' ■ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' From blocks to trees In this section, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We begin with a result which captures the use of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='2 in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For a positive integer n, we write [n] = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For all integers ν, t ≥ 1, there exists an integer ψ = ψ(t, ν) ≥ 1 with the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let G ∈ Ct, let a, b ∈ G be distinct and non-adjacent and let {Pi : i ∈ [ψ]} be a collection of ψ pairwise internally disjoint paths in G from a to b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For each i ∈ [ψ], let ai be the neighbor of a in Pi (so ai ̸= b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then there exists I ⊆ [ψ] with |I| = ν for which the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' {ai : i ∈ I} ∪ {b} is a stable set in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For all i, j ∈ I with i < j, ai has a neighbor in P ∗ j \\ {aj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let s = s(t) be as in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='2 and let µ = µ(max{2s + 1, t}), where µ(·) is as in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let R(·, ·) be as in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For every integer p ≥ 1, let Rtourn(p) be the smallest positive integer n such that every tournament on at least n vertices contains a transitive tournament on p vertices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' the existence of Rtourn(p) follows easily from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='3 (in fact, one may observe that Rtourn(p) ≤ R(p, p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let γ = R(Rtourn(ν + 1), µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We prove that ψ = ψ(t, ν) = R(γ, t) satisfies Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' , Pψ be ψ pairwise internally disjoint paths in G from a to b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Since G is Kt-free, it follows from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='3 and the definition of ψ that there exists a stable set N ⊆ {ai : i ∈ [ψ]} in G with |N| = γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' we may assume without loss of generality that N = {ai : i ∈ [γ]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let D be a directed graph with V (D) = N such that for distinct i, j ∈ [γ], there is an arc from ai to aj in D if and only if xi has a neighbor in P ∗ j \\ {aj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Note that D may contain both arcs (ai, aj) and (aj, ai), and so the undirected underlying graph of D might not be simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let D− be the simple graph obtained from the undirected underlying graph of D by removing one of every two parallel edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' (13) D− contains no stable set of cardinality µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Suppose for a contradiction that D− contains a stable set S of cardinality µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We may assume without loss of generality that S = {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' , aµ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let G1 = G[(�µ j=1 Pj) \\ {a}].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Note that by the definition of D, for every i ∈ [µ], we have NG1(ai) = NPi(ai) \\ {a}, and in particular |NG1(ai)| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Since G1 is connected and Kt-free, and since and |S| = µ = µ(max{2s + 1, t}), we can apply Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='5 to G1 and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Note that every vertex in S has a unique neighbor in G1, and so no path in G1 contains max{2s + 1, t} ≥ 3 vertices from S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Consequently, there is an induced subgraph H1 of G1 with |H1 ∩ S| = 2s + 1 for which one of the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' H1 is either a caterpillar or the line graph of a caterpillar with H1 ∩ S = Z(H1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' H1 is a subdivided star with root r1 such that Z(H1) ⊆ H1 ∩ S ⊆ Z(H1) ∪ {r1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' 19 If H1 is a caterpillar, then G[H1∪{a}] contains a theta with ends a and a′ for every vertex a′ ∈ H1 of degree more than two, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Also, if the second bullet above holds, then since every vertex in S is of degree one in G1, we have H1 ∩ S = Z(H1), and so r1 is not adjacent to a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' But then G[H1 ∪ {a}] contains a theta with ends x and r1, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' It follows that H1 is the line graph of a caterpillar with |H1 ∩ S| = 2s + 1 and H1 ∩ S = Z(H1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This, together with the fact that every vertex in H1 ∩ S ⊆ S has a unique neighbor in H1 ⊆ G, implies that H1 contains the line graph H2 of a 1-subdivision of a caterpillar with |H2 ∩ S| = s and H2 ∩ S = Z(H2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let S2 = H2 ∩ S = Z(H2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' then S2 is the set of all vertices of degree one in H2, and we may assume without loss of generality that S2 = {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' , as}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let G2 = G[H2 ∪(�s j=1 Pj)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' It follows that G2 ∈ Ct, NG2(a) = S2 = Z(H2) and a is trapped in H2 ∪ {a}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Therefore, H2 is an a-seed in G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Since b ∈ G2 \\ NG2[a], applying Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='2 to G2 and a, we deduce that there exists Sb ⊆ G2 \\{a, b} such that |Sb| < s and Sb separates a and b in G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' But P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' , Ps are s pairwise internally disjoint paths in G2 from a to b, a contradiction with Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This proves (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' By (13), Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='3 and the definition γ, D− contains a clique of cardinality Rtourn(ν + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This, along with the definition of Rtourn(·), implies that D contains (as a subdigraph) a transitive tournament K on ν + 1 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We may assume without loss of generality that V (K) = {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' , aν+1} such that for distinct i, j ∈ [ν + 1], (ai, aj) is an arc in K if i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' From the definition of D, it follows that {a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' , aν+1, b} is a stable set in G, and for all i, j ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' , ν+1} with i < j, ai has a neighbor in P ∗ j \\{aj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Hence, I = {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' , ν + 1} satisfies Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' ■ For positive integers d and r, let T r d denote the rooted tree in which every leaf is at distance r from the root, the root has degree d, and every vertex that is neither a leaf nor the root has degree d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We need a result from [15]: Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='2 (Kierstead and Penrice [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For all integers d, r, s, t ≥ 1, there exists an integer f = f(d, r, s, t) ≥ 1 such that if G contains T f f as a subgraph, then G contains one of Ks,s, Kt and T r d as an induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' The following lemma is the penultimate step in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For all integers d, r, t ≥ 1, there exists an integer m = m(d, r, t) with the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let G ∈ Ct be a graph, let a, b ∈ G be non-adjacent and let {Pi : i ∈ [m]} be a collection of m pairwise internally disjoint paths in G from a to b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then G[�m j=1 Pj] contains a subgraph J isomorphic to T r d such that a ∈ J and a has degree d in J (that is, a is the root of J), and we have b /∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let d, t ≥ 1 be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let m1 = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For every integer r > 1, let mr = ψ(t, (mr−1 + 1)d) where ψ(·, ·) is as in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We prove by induction on r ≥ 1 that m(d, r, t) = mr satisfies Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' , Pmr be mr pairwise internally disjoint paths in G from a to b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Since a and b are not adjacent, it follows that for each i ∈ [mr], we have P ∗ i ̸= ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' let ai be the neighbor of a in Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' In particular, we have b /∈ {ai : i ∈ [mr]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Suppose first that r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then we have |{ai : i ∈ [m1]}| = m1 = d, and so G[{ai : i ∈ [mr]} ∪ {a}] contains a (spanning) subgraph J isomorphic to T 1 d such that a ∈ J and a has degree d in J, and we have b /∈ J, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Therefore, we may assume that r ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Since mr = ψ(t, (mr−1 + 1)d), we can apply Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1 to a, b and {Pi : i ∈ [mr]}, obtaining I ⊆ [mr] with |I| = (mr−1 + 1)d which satisfies the two outcomes of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Without loss of generality, we may assume that I = [(mr−1 + 1)d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' It follows that {a1, · · · , a(mr−1+1)d, b} is a stable set in G, and for all i, j ∈ [(mr−1 + 1)d] with i < j, ai has a neighbor in P ∗ j \\ {aj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For every i ∈ [d], let a′ i = a(i−1)mr−1+i and let Ai = {(i − 1)mr−1 + i + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' , (i − 1)mr−1 + i + mr−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' In particular, we have |Ai| = mr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Then for each i ∈ [d] and each j ∈ Ai, a′ i has a neighbor in P ∗ j \\ {aj}, and so there exists a path Qj in G from a′ i to b with Q∗ j ⊆ P ∗ j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Now, for every i ∈ [d], a′ i and b are non-adjacent, and {Qj : j ∈ Ai} is a collection of mr−1 pairwise internally disjoint 20 INDUCED SUBGRAPHS AND TREE DECOMPOSITIONS VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' paths in G from a′ i to b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' It follows from the induction hypothesis that G[� j∈Ai Qj] contains a subgraph Ji isomorphic to T r−1 d such that a′ i ∈ Ji and a′ i has degree d in Ji, and we have b /∈ Ji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' But now G[(�d i=1 V (Ji)) ∪ {a}] ⊆ G[�mr j=1 Pj] contains a (spanning) subgraph J isomorphic to T r d such that a ∈ J and a has degree d in J, and we have b /∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This completes the proof of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' ■ Finally, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='8, which we restate: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' For every tree F and every integer t ≥ 1, there exists an integer τ(F, t) ≥ 1 such that every graph in Ct(F) has treewidth at most τ(F, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let d and r be the maximum degree and the radius of F, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' It follows that T r d contains F as an induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let f = f(d, r, 3, t) be as in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='2 and let m = m(f, f, t) be as in Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Let β(·, ·) be as in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' We claim that τ(F, t) = β(max{m, t + 1}, t) satisfies Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Suppose for a contradiction that tw(G) > τ for some G ∈ Ct(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' By Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='4, G contains a max{m, t + 1}-block B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Consequently, since G is Kt-free, there are two distinct and non-adjacent vertices a, b ∈ B, and m pairwise internally disjoint paths P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' , Pm in G from a to b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' It follows from Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='3 that G contains T f f as a subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Also, since G ∈ Ct(F) ⊆ Ct, G is (K3,3, Kt)-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' But now by Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='2, G contains T r d , and so F, as an induced subgraph, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' ■ References [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Aboulker, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Adler, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' J.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Basic obstructions in H-free graphs.” arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='02737, (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' [3] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Abrishami, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Chudnovsky, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Dibek, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Hajebi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Rzążewski, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Spirkl, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Vušković, “Induced subgraphs and tree decompositions II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content=' Toward walls and their line graphs in graphs of bounded degree.” arXiv:2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/19A0T4oBgHgl3EQfMv_l/content/2301.02138v1.pdf'} +page_content='01162, (2021).' metadata={'source': 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/dev/null +++ b/1tA0T4oBgHgl3EQfMv-f/content/tmp_files/2301.02137v1.pdf.txt @@ -0,0 +1,2100 @@ +arXiv:2301.02137v1 [nucl-th] 5 Jan 2023 +Reduced nuclear helicity amplitudes for deuteron +electrodisintegration and other processes +J. Flores and S. S. Chabysheva +Department of Physics, University of Idaho, Moscow ID 83844 USA +J. R. Hiller +Department of Physics, University of Idaho, Moscow ID 83844 USA and +Department of Physics and Astronomy, +University of Minnesota-Duluth, Duluth, Minnesota 55812 USA +(Dated: January 6, 2023) +1 + +Abstract +We extend the original idea of reduced nuclear amplitudes to capture individual helicity ampli- +tudes and discuss various applications to exclusive processes involving the deuteron. Specifically, +we consider deuteron form factors, structure functions, tensor polarization observables, photodisin- +tegration, and electrodisintegration. The basic premise is that nuclear processes at high momentum +transfer can be approximated by tree graphs for point-like nucleons supplemented by empirical form +factors for each nucleon. The latter represent the internal structure of the nucleon, and incorporate +nonperturbative physics, which can allow for early onset of scaling behavior. The nucleon form +factors are evaluated at the net momentum transfer experienced by the given nucleon, with use of +GE for a no-flip contribution and GM for a helicity-flip contribution. Results are compared with +data where available. The deuteron photodisintegration asymmetry Σ is obtained with a value of +Σ(90◦) ≃ −0.06, which is much closer to experiment than the value of -1 originally expected. The +method also provides an estimate of the momentum transfer values required for scaling onset. We +find that the deuteron structure function B is a good place to look, above momentum transfers of +10 GeV2. +I. +INTRODUCTION +With the advent of the upgraded electron accelerator at the Thomas Jefferson National +Accelerator Facility, scattering experiments with polarized beams and targets at high energy +and high momentum transfer become possible. In the regime of high momentum transfer to +all relevant nucleons, quantum chromodynamics (QCD) implies that the internal structure of +every nucleon is important. Until ab initio QCD (lattice) calculations for nuclear scattering +processes are available for more than very simple processes, one is led to consider models +that can represent the basic physics. +One such approach is the reduced nuclear amplitude (RNA) analysis pioneered by Brod- +sky and Chertok [1]. In addition to their application to a generic deuteron form factor, +the approach has been applied to deuteron disintegration [2], pion photoproduction [3], and +photodisintegration of 3He [4]. As originally developed, a nuclear process was modeled as +a tree-level amplitude multiplied by a generic form factor for each nucleon, with each form +factor evaluated at the net momentum transferred to that nucleon. In order to model the +behavior of polarization observables [5–15], we extend this approach to a reduced nuclear +helicity amplitude (RNHA) method to combine a tree-level helicity amplitude for point-like +nucleons with the appropriate form factor for each nucleon. When the nucleon does (not) +flip its helicity, we use the electric (magnetic) form factor GEN (GMN). As a check on the +procedure, virtual photon absorption by a single nucleon in the RNHA approach is consistent +with the definitions of GEN and GMN. +A caveat in applications of the RNA approach is that the normalization is not determined +by the model and is fixed to data at infinite momentum transfer by the coefficient of the +leading power-law behavior. This means that the normalization cannot be determined in +practice; fitting to a data point at some intermediate kinematics will give the wrong nor- +malization and the wrong magnitude at higher momentum transfer. Instead, ratios need to +be considered, so that the normalization becomes irrelevant. +The primary criterion for the asymptotic region is in the momentum transfer to each +nucleon. For every nucleon in the process, the momentum transfer must be above some +common threshold, which is at least 1 GeV2. For example, for deuteron photodisintegration, +2 + +the momentum transfer to a nucleon is −tN = −(pN − p/2)2, where pN is the final four- +momentum of the nucleon and p is the initial deuteron momentum. When expressed in +terms of the photon energy Eγ and the final nucleon angle θ, the constraint to be above 1 +GeV2 becomes [16] +mNEγ +� +1 − +� +Eγ +mN + Eγ +| cos θ| +� +≥ 1 GeV2. +(1.1) +This relationship is illustrated in Fig. 1. +Notice that away from 90◦, the lower limit is +quite high. For electrodisintegration, only the most recent data [17, 18] begins to reach this +θ (deg) +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +Eγ (GeV) +0 +2 +4 +6 +8 +10 +FIG. 1. Angular dependence of the scale for large momentum transfer in deuteron photodisinte- +gration. +threshold. +Here we will focus on deuteron processes, including photodisintegration and electrodisin- +tegration. For recent reviews of deuteron studies at high momentum transfer, see [16, 19, 20]. +Elastic electron scattering data at high momentum transfer is presented in [21–25]. Recent +photodisintegration data can be found in [26–29], and for electrodisintegration data, in +[17, 18, 30–33]. Other analyses of deuteron processes include hidden-color contributions to +deuteron form factors [34], the hard rescattering mechanism [35], quark-gluon strings [36], +the Moscow NN potential [37], and AdS/QCD models [38, 39]. +One recent experiment [17] used the 10.6 GeV electron beam at JLab and the Hall +C spectrometers to measure electron scattering from a liquid deuterium target. The final +electron and the proton were detected, with the kinematics restricted to the exclusive process +ed → e′pn. One spectrometer measured the final electron at a nominal 12.2◦ degrees from +the beam direction, with a momentum of 8.5-9.1 GeV such that the recorded events had a +distribution of momentum transfer squared reaching 5 GeV2. The events studied were taken +from a bin of 4.5±0.5 GeV2 in the tail of the distribution; however, the nominal transfer +was 4.2 GeV2, because the majority of the events were in the lower half of the bin. +A second spectrometer measured the proton momentum at a range of angles to the +beam direction, tuned to select events where the (missing) neutron had an angle relative to +the direction of the momentum transfer that fell within a chosen bin. In the one-photon +exchange approximation, which we assume, the momentum transferred is, of course, the +photon momentum. The published neutron angles are binned at 35◦, 45◦ and 75◦, with the +first two selected to minimize final-state interactions. For our purposes, the importance of +3 + +these two angles is that the momentum transferred to the neutron reaches 1 GeV in a zero- +binding approximation, so that, rather than focus on the internal structure of the deuteron, +we can consider the response to a large momentum transfer to all the nucleons involved and +we can see that experiments may be approaching the threshold where our model can be +applied. +The RNHA model is constructed in detail in Sec. II for two-nucleon processes. In the +remainder of the paper, we consider various processes for the deuteron. In Sec. III, the +form factors,1 structure functions, and tensor polarization observables of elastic electron +scattering from the deuteron are obtained. Photodisintegration and electrodisintegration +are analyzed in Secs. IV and V. Within the zero-binding approximation, elastic scattering +and photodisintegration live at edges of the kinematic range of electrodisintegration and +are essentially special cases that provide introductory examples. +Section VI contains a +summary of the results and suggestions for additional applications. Many details of the +electrodisintegration helicity amplitudes are left to an appendix. +II. +CONSTRUCTION OF THE MODEL +The basic process for a two-nucleon system to absorb a photon and exchange momentum +between the nucleons is illustrated in Fig. 2. These diagrams are modeled on the primitive +process of γ∗ff → ff, with f representing a point-like nucleon.2 The structure of each +nucleon is then introduced by combining the Feynman amplitude for each diagram with the +appropriate form factor for each nucleon, evaluated at the net momentum transfer for that +nucleon. For a deuteron process in the zero-binding limit, the initial nucleons share the +initial deuteron momentum p equally, so that pp = pn = p/2. We also neglect the nucleon +mass difference, setting mp = mn ≡ m. The distinction between different photon-absorption +processes is then in the nature of the photon, being either real or virtual, and in the outcome +for the final nucleons, bound as a deuteron or not. +The tree-level amplitudes for the four diagrams in Fig. 2 are +Mν +a (λ′ +p, λ′ +n, λp, λn) = Aµν +p (p/2 + q; λ′ +p, λp) +1 +(p′ +n − p/2)2Bnµ(λ′ +n, λn), +(2.1) +Mν +b (λ′ +p, λ′ +n, λp, λn) = Aνµ +p (p′ +p − q; λ′ +p, λp) +1 +(p′n − p/2)2Bnµ(λ′ +n, λn), +(2.2) +Mν +c (λ′ +p, λ′ +n, λp, λn) = Aµν +n (p/2 + q; λ′ +n, λn) +1 +(p′p − p/2)2Bpµ(λ′ +p, λp), +(2.3) +Mν +d (λ′ +p, λ′ +n, λp, λn) = Aνµ +n (p′ +n − q; λ′ +n, λn) +1 +(p′ +p − p/2)2Bpµ(λ′ +p, λp), +(2.4) +where +Aµν +N (p; λ′ +N, λN) = ¯u′ +Nγµ ̸p + m +p2 − m2γνuN, Bµ +N(λ′ +N, λN) = ¯u′ +NγµuN, +(2.5) +with uN (¯u′ +N) the initial (final) spinor for the nucleon N with helicity λN (λ′ +N). The sub- +amplitude AN represents the fermion line that absorbs the photon, and BN represents the +1 For discussion specifically in terms of perturbative QCD, see [34]. +2 In [2], the primitive process was γ∗q¯q → q¯q, with q corresponding to a point-like proton and ¯q to a point- +like neutron, and direct interaction of the photon with the neutron was neglected. Here we amend and +extend this, to retain information about helicity states of the fermions and include photon absorption by +the neutron. +4 + +pp, λp +pn, λn +p′ +p, λ′ +p +p′ +n, λ′ +n +q, λγ +pp, λp +pn, λn +p′ +p, λ′ +p +p′ +n, λ′ +n +q, λγ +(a) +(b) +pp, λp +pn, λn +p′ +p, λ′ +p +p′ +n, λ′ +n +q, λγ +pp, λp +pn, λn +p′ +p, λ′ +p +p′ +n, λ′ +n +q, λγ +(c) +(d) +FIG. 2. Tree graphs for deuteron processes that absorb a photon of momentum q and helicity +λγ. The initial (final) nucleon momentum and helicity are pN (p′ +N) and λN (λ′ +N), with N = p or +n. The two nucleons exchange momentum via a vector particle. The four diagrams differ in the +nature of the photon-absorbing nucleon and the order of this absorption and momentum transfer +between nucleons. +other fermion line. Calculation of these sub-amplitudes can be checked against the trace +theorem for sums over helicities: +� +λN,λ′ +N +Aν′µ′∗ +N +(p; λ′ +N, λN)Aµν +N (p; λ′ +N, λN) = Tr +� +γν′ ̸p + m +p2 − m2γµ′(̸p′ +N + m)γµ ̸p + m +p2 − m2γν(̸pN + m) +� +, +(2.6) +� +λN,λ′ +N +Bµ′∗ +N (λ′ +N, λN)Bµ +N(λ′ +N, λN) = Tr +� +γµ′(̸p′ +N + m)γµ(̸pN + m) +� +. +(2.7) +The full amplitude is constructed from the MX by combining them with form factors for +each nucleon. For a deuteron with initial helicity λd, we have +Mν(λ′ +p, λ′ +n, λd) = +� +λp,λn +Cλd +λpλn + + +� +X=a,b,c,d +Mν +X(λ′ +p, λ′ +n, λp, λn) + + Gpλ′pλp(Q2 +p)Gnλ′nλn(Q2 +n), +(2.8) +5 + +where Q2 +N = −(p′ +N − pN)2, +Cλd +λpλn = + + + +δλp± 1 +2δλn± 1 +2, +λd = ±1 +1 +√ +2 +� +δλp 1 +2δλn− 1 +2 + δλp− 1 +2δλn+ 1 +2 +� +, λd = 0, +(2.9) +and +GNλ′λ = +� +GEN, λ′ = λ +GMN, λ′ = −λ. +(2.10) +The form factors GEN and GMN represent the internal structure of the nucleons. They can +be represented by data or empirical fits. For simplicity, we use the fits [40] +GEp ≃ +� +1 + Q2 +N +m2 +0 +�−2 +, GMp ≃ µpGEp, GMn ≃ µnGEp, GEn ≃ − +µnτ +1 + 5.6τ GEp, +(2.11) +where m2 +0 = 0.71 GeV2, τ = +Q2 +N +4m2 , µp = 2.79, and µn = −1.91. To limit the analysis to a +single mass scale, we take the parameter m0 to be proportional to the nuclear mass, with +m2 +0 = 0.80 m2. We do not attempt to compute or assign an overall normalization to Mν, +and the running of the strong coupling constant is not included. +The initial nucleon spinor, for a deuteron traveling along the negative z direction, is [41] +uN = +̸p/2 + m +� +Ed/2 + m +� +φ(λN)(−ˆz) +0 +� +, +(2.12) +with +φ(1/2)(−ˆz) = +� +0 +1 +� +, φ(−1/2)(−ˆz) = +� +1 +0 +� +. +(2.13) +The final nucleon spinor is +u′ +N = ̸p′ +N + m +� +E′ +N + m +� +φ(λ′ +N)(ˆp′ +N) +0 +� +, +(2.14) +with +φ(1/2)(ˆp′ +N) = +� +cos(θN/2) +eiφN sin(θN/2) +� +, φ(−1/2)(ˆp′ +N) = +� +−e−iφN sin(θN/2) +cos(θN/2) +� +, +(2.15) +where θN and φN are the polar and azimuthal angles of the outgoing momentum of the +particular nucleon. +As discussed in the Introduction, the overall normalization of the RNHA amplitude is +unknown. For comparison with data, we consider quantities which are themselves ratios or +a ratio of the model to data. +III. +ELASTIC ELECTRON SCATTERING +A. +Form factors +The three deuteron form factors, GC, GM, and GQ, are readily obtained from the hadronic +helicity amplitudes of elastic electron-deuteron scattering in the Breit frame [42]. The kine- +matics are shown in Fig. 3. The photon four-momentum is q = (0, 0, 0, qz) and the initial +6 + +(final) deuteron four-momentum is p = (Ed, 0, 0, −qz/2) (p′ = (Ed, 0, 0, qz/2)), with q2 +z = Q2 +and Ed = +� +Q2/4 + m2 +d. In the zero-binding limit,3 md = 2m and the individual nucleon +four-momenta are pp = pn = p/2 and p′ +p = p′ +n = p′/2. The hadronic matrix elements are +given by +Gµ +λ′ +d,λd = +� +λ′p,λ′n +C +λ′ +d +λ′pλ′nMµ(λ′ +p, λ′ +n, λd). +(3.1) +The initial spinors are as in (2.12); the final spinors are specified by +u′ +N = +̸p′/2 + m +� +Ed/2 + m +� +φ(λ′ +N)(ˆz) +0 +� +. +(3.2) +⃗q, λγ +−⃗q/2, λd +⃗q/2, λ′ +d +z +⃗pe, λe +⃗p ′ +e, λ′ +e +FIG. 3. Kinematics for elastic electron-deuteron scattering in the Breit frame. The photon travels +along the positive z direction, and the deuteron comes from the right, along the negative z direction. +The three form factors are then extracted as [42, 43] +GC = +−1 +2md +√1 + η +G+ +00 − 2G+ ++− +3 +, GM = +2 +2md +√1 + η +Gx ++0 +√2η, GQ = +−1 +2md +√1 + η +G+ +00 + G+ ++− +2η +, +(3.3) +with η ≡ +Q2 +4m2 +d and the + superscript denoting the light-front sum of the 0 and z components. +For the helicity matrix elements, the model yields the following Q2 dependence: +G+ +00 = 0.5588N m +�m +Q +�9 � +1 + 129.1m2 +Q2 + O(m4 +Q4 ) +� +, +(3.4) +G+ ++− = −69.85N m +�m +Q +�11 � +1 + 4.8m2 +Q2 + O(m4 +Q4 ) +� +, +(3.5) +Gx ++0 = 8.851N m +�m +Q +�10 � +1 + 4.8m2 +Q2 + O(m4 +Q4 ) +� +, +(3.6) +with N the unknown normalization. The factor of m/Q associated with each helicity flip +[44] is clearly evident. For the form factors, we find +GC = − 0.5588 +√1 + η +N +12 +�m +Q +�9 � +1 + 379.1m2 +Q2 + O(m4 +Q4 ) +� +, +(3.7) +3 The difference between the proton and neutron masses is neglected in addition to the deuteron binding +energy, the two being of the same order. +7 + +GM = +8.851 +� +η(1 + η) +N +2 +√ +2 +�m +Q +�10 � +1 + 4.8m2 +Q2 + O(m4 +Q4 ) +� +, +(3.8) +GQ = − 0.5588 +η√1 + η +N +8 +�m +Q +�9 � +1 + 4.086m2 +Q2 + O(m4 +Q4 ) +� +. +(3.9) +The leading ± signs are as expected for large Q2. +We have left the kinematic factor η = Q2/16m2 without substitution, because there can +be three regimes for Q2. In addition to Q2 large or small, there can be an intermediate +region where Q2 is large but η is not. Such an intermediate regime does exist for GM and +GQ, where the coefficients of the nonleading terms are small enough for this correction to +be small while η is also small. For GC, this is not the case, because the coefficient of the +nonleading term is large enough to require a Q2 value for which η is also large. In the +intermediate regime, we obtain +GM ∼ +�m +Q +�11 +, GQ ∼ +�m +Q +�11 +, +(3.10) +and for the large-η regime +GC ∼ +�m +Q +�10 +, GM ∼ +�m +Q +�12 +, GQ ∼ +�m +Q +�12 +. +(3.11) +Ratios of these form factors at very large Q2 can be compared with the tree-level ratios +for a point-like spin-one particle, such as the W +, which are [43] +GC +GQ += 2 +3η − 1, +GM +GQ += −2. +(3.12) +Such behavior is immediately reproduced for form factors separated according to a Drell– +Yan frame [45], with the assumption of strict G+ +00 dominance [43]. In terms of our hadronic +matrix elements, we have +GC +GQ += 2 +3η − 2η +G+ ++− +G+ +00 + G+ ++− +, +GM +GQ += −2 +� +2η +Gx ++0 +G+ +00 + G+ ++− +. +(3.13) +As already observed in [43], these Breit-frame ratios cannot both be resolved by simply +assuming G+ +00 dominance. From our model, we obtain +GC +GQ += 2 +3η + 15.6 + O(m2 +Q2 ), +GM +GQ += −11.2 + O(m2 +Q2 ). +(3.14) +The leading 2 +3η is just kinematic. The deviations of 15.6 and -11.2 from -1 and -2, respec- +tively, are due to nonleading contributions multiplied by powers of η. Similar deviations will +arise for calculations done in the Drell–Yan frame, because η factors again interfere with +strict G+ +00 dominance. Plots of these ratios are shown in Fig. 4. +8 + +�����V����e/����� +V�� +V�� +� +�� +�� +�� +��/M����0 +� +�� +��� +FIG. 4. Ratios GC/GQ − 2η/3 (dashed) and GM/GQ (solid) for the model deuteron form factors. +B. +Structure functions +Experiments designed to extract these form factors measure cross sections and polariza- +tion observables in elastic electron-deuteron scattering. The unpolarized cross section +dσ +dΩ ∝ S, S ≡ A(Q2) + B(Q2) tan2(θe/2) +(3.15) +depends on the electron scattering angle θe and two structure functions +A(Q2) ≡ G2 +C + 8 +9η2G2 +Q + 2 +3ηG2 +M, +(3.16) +B(Q2) ≡ 4 +3η(1 + η)G2 +M. +(3.17) +These have been measured at the highest Q2 yet attained at JLab [22–24], and A has been +measured at comparable Q2 at SLAC [21]. However, these do not yet reach the Q2 values +needed for a definitive comparison. Figures 5 and 6 show plots of the data divided by the +model, including an arbitrary normalization factor. +In our model, expansions of these functions in inverse powers of Q2 are +A(Q2) = 0.1041N 2 +�m +Q +�20 � +1 + 1246m2 +Q2 + O(m4 +Q4 ) +� +, +(3.18) +B(Q2) = 13.06N 2 +�m +Q +�20 � +1 + 9.6m2 +Q2 + O(m4 +Q4 ) +� +. +(3.19) +Because the expansion for GC is valid only for large η, we have used the explicit form of η in +constructing the expansion for A. The function B is independent of η; the leading factor of +9 + +� +�l�70V����� +5 +6 +7 +� +� +�7Ql���70 +5 +6 +7 +� +� +� +� +� +FIG. 5. Data for the deuteron structure function A(Q2) divided by the model function, including +an arbitrary normalization. Experimental values are taken from [23] (circles) and [24] (squares). +η(1+η) in its definition exactly cancels against factors in the relationship of GM to hadronic +matrix elements. The expansion for B converges much faster than the expansion for A, and +the leading Q2 behavior is dominant for Q2 ≫ 10 GeV2 only for B. For A, one must wait +until impossibly large Q2, which enters a regime where the collective quark substructure is +important, including hidden-color effects [34], and the point-like approximation used in our +model is invalid. +In [22] the large Q2 behavior of B is quoted as being Q−24 from perturbative QCD. +This faster fall off compared to A is attributed to the extra suppression of the helicity flip +involved in GM. However, there are other compensating factors, and, just as in our model, +the behavior of B should be Q−20, which is the same as A. In Fig. 7 we plot the ratio of +B to A for a large range of Q2. This ratio becomes constant at very large Q2. Although +the plots begin at low Q2, there is nothing in the model that could reproduce diffractive +minima, hence the smooth appearance. +C. +Tensor polarization observables +Experiments can also extract tensor polarization observables [20, 25] +t20 ≡ − +1 +√ +2S +�8 +3ηGCGQ + 8 +9η2G2 +Q + 1 +3η +� +1 + 2(1 + η) tan2(θe/2) +� +G2 +M +� +, +(3.20) +t21 ≡ +2η +√ +3S cos(θe/2) +� +η + η2 sin2(θe/2)GMGQ, +(3.21) +t22 ≡ − +η +2 +√ +3S G2 +M. +(3.22) +10 + +� +�l��-G����� +1V � +V +V � +3 +3 � +� +� � +��Ql����- +3 +3 � +� +� � +� +FIG. 6. Data for the deuteron structure function B(Q2) divided by the model function, including +an arbitrary normalization. Experimental values are taken from [22]. +� +�8��15�8��1 +� +�� +�� +�� +�� +��� +��� +��� +��Q8����1 +��� +��� +��� +��� +��� +��� +��� +FIG. 7. Ratio of B to A for the model deuteron structure functions. +The highest Q2 measurements of these were also done at JLab [25]. When η is held explicit, +expansions in m/Q are +t20 = − +√ +2 + 1064[1 + 2(1 + η) tan2(θe/2)] +�m +Q +�2 ++ O(m4 +Q4 ), +(3.23) +11 + +t21 = 38.8 sec(θe/2) +� +η + sin2(θe/2)m +Q +(3.24) ++ sec(θe/2) +� +η + sin2(θe/2)[48606 + 77869(1 + η) tan2(θe/2)] +�m +Q +�3 ++ O(m4 +Q4 ), +t22 = −434.5 +�m +Q +�2 ++ [544703 + 872133(1 + η) tan2(θe/2)] +�m +Q +�4 ++ O(m6 +Q6 ). +(3.25) +While at very large Q2, they are +t20 = − +√ +2 + 133 tan2(θe/2) + 1064[1 + 2 tan2(θe/2)] +�m +Q +�2 ++ O(m4 +Q4 ), +(3.26) +t21 = 1217 sec(θe/2) sin(θe/2)[tan2(θe/2) − 0.007972] + [] +�m +Q +�2 ++ O(m4 +Q4 ), +(3.27) +t22 = −[434.5 − 54508 tan2(θe/2)] +�m +Q +�2 +(3.28) ++[544703 + 872133 tan2(θe/2)] +�m +Q +�4 ++ O(m6 +Q6 ). +The coefficients of nonleading terms are quite large. Thus, very large Q2 is required for +the leading term to be dominant, well beyond any available data. The limit of − +√ +2 for +t20 at θe = 0◦ was an early prediction of perturbative QCD [44, 46]. However, as argued +elsewhere [43], this value is obtained only at very large Q2, and the value is quite different for +small η. Figures 8 and 9 show plots of these observables at angles of 0◦ and 30◦, respectively. +We also compare with data [25] in Figs. 10, 11, and 12. At these ‘small’ values of Q2, only +t22 is consistent with data, something which is likely accidental with both data and model +values near zero. +IV. +PHOTODISINTEGRATION +In the photodisintegration of a deuteron, a real photon is absorbed and the two constituent +nucleons emitted. This process is depicted in Fig. 13. The initial deuteron and photon four- +momenta in the center-of-mass (c.m.) frame are p = (Ed, 0, 0, −qz) and q = (qz, 0, 0, qz), +where the incident photon is taken along the positive z axis. The final proton and neutron +four-momenta are p′ +p = (E′ +p, ⃗p ′ +p) and p′ +n = (E′ +n, ⃗p ′ +n), with θp and φp the polar and azimuthal +angles of the final proton. +By ignoring the nucleon mass difference, we have E′ +p = E′ +n, +because momentum conservation guarantees ⃗p ′ +n = −⃗p ′ +p in the c.m. frame. +In terms of the Mandelstam variable s, the c.m. energies and momenta are +Ed = (s + 4m2)/(2√s), qz = (s − 4m2)/(2√s), E′ +p = √s/2, |⃗p ′ +p| = +√ +s − 4m2/2. +(4.1) +The photon energy in the lab frame is Eγ = (s − 4m2)/4m. We will work at large s, so that +momentum transfers are large. +The standard definition of helicity amplitudes for photodisintegration is [47] +Fi± ≡ ǫν(λγ)Mν(λ′ +p, λ′ +n, λd) +(4.2) +12 + +� +��5e0���e0��� +V�)� +V�)� +V�)� +V� +V5)� +V5)� +V5)� +V5)� +5 +��08����Q +�55 +�5� +�5� +�5� +�5� +�5� +�5� +FIG. 8. +Deuteron tensor polarization observables t20 (solid), t21 (dashed), and t22 (dotted) as +computed in the model at an angle of θe = 0◦. +The asymptotic value of t20(0◦) is − +√ +2, as +predicted by perturbative QCD [44, 46]. +with ǫ the polarization vector for a photon with helicity λγ and Mν given in (2.8). The +index i is associated with particular helicity combinations as follows: +F1± = ǫν(1)Mν(±1 +2, ±1 +2, 1), F2± = ǫν(1)Mν(±1 +2, ±1 +2, 0), +(4.3) +F3± = ǫν(1)Mν(±1 +2, ±1 +2, −1), F4± = ǫν(1)Mν(±1 +2, ∓1 +2, 1), +(4.4) +F5± = ǫν(1)Mν(±1 +2, ∓1 +2, 0), F6± = ǫν(1)Mν(±1 +2, ∓1 +2, −1). +(4.5) +The other helicity combinations are related to these by parity. +The helicity amplitudes can be used to compute various polarization observables. The +recoil-proton polarization Py measures the asymmetry parallel/antiparallel to the normal +ˆy ∝ ⃗q × ⃗p ′ +p to the scattering plane: +Py = 2Im +3 +� +i=1 +[F † +i+Fi+3,− + F † +i+3,+Fi−]/f(θ), +(4.6) +where f(θ) = +�6 +i=1[|Fi+|2 + |Fi−|2] is the sum of all the helicity amplitudes squared. The +transferred polarizations Cx′ and Cz′ measure asymmetries parallel/antiparallel to the ˆx′ ∝ +⃗p ′ +p × ˆy and ˆz′ = ˆp ′ +p directions: +Cx′ = 2Re +3 +� +i=1 +[F † +i+Fi+3,− + F † +i+3,+Fi−]/f(θ), +(4.7) +Cz′ = +6 +� +i=1 +[|Fi+|2 − |Fi−|2]/f(θ). +(4.8) +13 + +� +���e0���e0��� +V�)� +V�)� +V�)� +V�)� +V�)� +� +��03����Q +��� +��� +��� +��� +��� +��� +��� +FIG. 9. Same as Fig. 8 but for an angle of θe = 30◦. +The asymmetry Σ for linearly polarized photons is given by +Σ = −2Re +�� +± +(F † +1±F3∓ − F † +4±F6∓) − F † +2+F2− + F † +5+F5− +� +/f(θ). +(4.9) +Each observable is formed as a ratio, which sets aside questions of normalization. +Because we only need to consider photons with helicity +1, the polarization vector is +always ǫ = − 1 +√ +2(0, 1, i, 0), relative to the momentum in the positive z direction. The final +Dirac spinors are +u′ +N = ̸p′ +N + m +� +E′ +N + m +� +φ(λ′ +N)(ˆp′ +N) +0 +� +, +(4.10) +with θn = π − θp, φn = φp + π = π, and +φ(1/2)(ˆp′ +N) = +� +cos(θN/2) +eiφN sin(θN/2) +� +, φ(−1/2)(ˆp′ +N) = +� +−e−iφN sin(θN/2) +cos(θN/2) +� +. +(4.11) +With these spinors as input, the amplitudes ǫνMν +X can be evaluated in terms of Dirac +matrix and spinor products and then combined to construct the predefined amplitudes Fi±. +At large s, these RNHA predictions for the helicity amplitudes reduce to +F1+ ∼ 4 +√ +2 +√scsc2(θp +2 )GEn(θp)GEp(θp), F1− ∼ 0, +(4.12) +F2+ ∼ 2m +s cot3(θp +2 )[GEn(θp)GMp(θp) − GMn(θp)GEp(θp)], +(4.13) +F2− ∼ 2m +s cot(θp +2 )[GMn(θp)GEp(θp) − GEn(θp)GMp(θp)], +F3+ ∼ 0, F3− ∼ 0, +(4.14) +14 + +� +��� +1��� +1��� +1��� +1��� +� +��� +��� +��� +��� +��0(����G +��� +��� +��� +��� +��� +��� +��� +FIG. 10. Plots of the tensor polarization observable t20 of the deuteron from both data [25] (circles) +and the model (squares) considered in the text. The angle θe varies and is as follows in order of +increasing Q2: 35.6◦, 33.4◦, 29.8◦, 27.3◦, 23.0◦, and 19.8◦. +F4+ ∼ −4 +√ +2m +s cot(θp +2 )GMn(θp)GEp(θp), F4− ∼ 4 +√ +2m +s cot(θp +2 )GEn(θp)GMp(θp), (4.15) +F5+ ∼ 2 +√s cot2(θp +2 )GEn(θp)GEp(θp), F5− ∼ 2 +√sGEn(θp)GEp(θp), +(4.16) +F6+ ∼ 4 +√ +2m +s cot3(θp +2 )GEn(θp)GMp(θp), F6− ∼ −4 +√ +2m +s cot(θp +2 )GMn(θp)GEp(θp).(4.17) +From these we can calculate the various observables. Plots of the results and recent data [48– +50] are given in Figs. 14, 15, 16, and 17. Because the tree-level amplitudes are real, Py is +automatically zero. That Cx′ is of order m/√s, rather than zero, is a correction to hadron +helicity conservation [51]. Also, we find the asymmetry Σ(90◦) to be approximately -0.06, +rather than the nominal expectation [52] of -1. In general, the trends with photon energy +seem to be modestly consistent with data. +V. +ELECTRODISINTEGRATION +The kinematics of the electrodisintegration process are shown in Fig. 18. The initial (final) +momentum and helicity of the electron are pe (p′ +e) and λe (λ′ +e). The intermediate photon +carries four-momentum q. The azimuthal angle φp of the proton measures the rotation of +the hadronic reaction plane relative to the electron scattering plane. +In the lab frame, with the z axis taken along the photon three-momentum and the electron +mass neglected, the initial and final electron four-momenta are +pe = (Ee, Ee sin θe, 0, Ee cos θe), p′ +e = (E′ +e, E′ +e sin θ′ +e, 0, E′ +e cos θ′ +e), +(5.1) +15 + +� +��� +V�)� +V�)� +� +�)� +�)� +�)� +�)� +�)� +�)� +��1Q���� +�)� +�)� +� +�)� +�)� +�)� +�)� +FIG. 11. Same as Fig. 10 but for t21. +� +��� +8�4�� +8�4� +8�4�� +8�4� +8�4�� +� +�4�� +�4� +�4�� +��-G����e +�4� +�4� +� +�4� +�4� +�4� +�4� +FIG. 12. Same as Fig. 10 but for t22. +with E′ +e and ˜θ = θ′ +e − θe, the angle of the scattered electron to the beam direction, being +measured. The photon four-momentum q = (Eγ, 0, 0, qz) is just pe − p′ +e, which yields +Q2 ≡ −q2 = 2EeE′ +e(1 − cos ˜θ), Eγ = Ee − E′ +e, qz = +� +E2γ + Q2. +(5.2) +The deuteron four-momentum is p = (md = 2m, 0, 0, 0), and in the zero-binding limit, the +initial proton and neutron four-momenta are pp = pn = (m, 0, 0, 0). +The final nucleon +16 + +⃗q, λγ +⃗p, λd +⃗p ′ +p, λ′ +p +⃗p ′ +n, λ′ +n +θn = π − θp +θp +z +FIG. 13. Kinematics for deuteron photodisintegration in the c.m. frame, with ⃗q the photon mo- +mentum and ⃗p = −⃗q the deuteron momentum. The final proton and neutron momenta are ⃗p ′ +p and +⃗p ′ +n. The λ’s are helicities. Coordinates are chosen such that the photon enters along the positive +z direction and the azimuthal angle φp of the proton is zero. +Eγ (GeV) +0 +1 +2 +3 +4 +5 +6 +Py (90 deg) +-1.0 +-0.8 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +θ (deg) +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +Py (Eγ=2 GeV) +-1.0 +-0.8 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(a) +(b) +FIG. 14. Recoil proton polarization Py as a function of (a) photon energy Eγ and (b) proton angle +θ. For the latter, the photon energy is 2 GeV. The solid line is the RNHA prediction; the data +points are from [48, 49]. +four-momenta are +p′ +p = (E′ +p = +� +⃗p ′2 +p + m2, |⃗p ′ +p| sin θp cos φp, |⃗p ′ +p| sin θp sin φp, |⃗p ′ +p| cos θp), +(5.3) +p′ +n = (E′ +n = +� +⃗p ′2 +n + m2, −|⃗p ′ +n| sin θn cos φp, −|⃗p ′ +n| sin θn sin φp, |⃗p ′ +n| cos θn). +(5.4) +Within the one-photon-exchange approximation, the scattering amplitude is proportional +17 + +Eγ (GeV) +0 +1 +2 +3 +4 +5 +6 +Cx' (90 deg) +-1.0 +-0.8 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +θ (deg) +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +Cx' (Eγ=2 GeV) +-1.0 +-0.8 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(a) +(b) +FIG. 15. Same as Fig. 14 but for the transferred polarization Cx′. +Eγ (GeV) +0 +1 +2 +3 +4 +5 +6 +Cz' (90 deg) +-1.0 +-0.8 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +θ (deg) +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +Cz' (Eγ=2 GeV) +-1.0 +-0.8 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(a) +(b) +FIG. 16. Same as Fig. 15 but for Cz′. +to +Med(λ′ +p, λ′ +n, λ′ +e; λd, λe) = ¯u′ +eγµue +Dµν +q2 Mν(λ′ +p, λ′ +n, λd), +(5.5) +with ue (u′ +e) the initial (final) spinor of the electron and Mν given in (2.8). The numerator +of the photon progator is the sum over photon polarizations +Dµν = +1 +� +λ=−1 +(−1)λǫ∗ +µ(λ)ǫν(λ). +(5.6) +18 + +Eγ (GeV) +0 +1 +2 +3 +4 +5 +6 +Σ (90 deg) +-1.0 +-0.8 +-0.6 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +θ (deg) +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +Σ (Eγ=2 GeV) +-0.10 +-0.08 +-0.06 +-0.04 +-0.02 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +(a) +(b) +FIG. 17. Same as Fig. 14 but for the asymmetry Σ. The data points are from [50]. +The polarization four-vectors are4 +ǫ(±1) = ∓ 1 +√ +2(0, 1, ±i, 0), ǫ(0) = (qz/Q, 0, 0, Eγ/Q) +(5.7) +relative to the photon four-momentum q = (Eγ, 0, 0, qz). Polarization observables [5–15] can +then be computed from these helicity amplitudes. +⃗pe, λe +⃗p ′ +e, λ′ +e +x +⃗q +⃗p ′ +p, λ′ +p +⃗p ′ +n, λ′ +n +y′ +x′ +z, z′ +y +θp +φp +θe +θ′ +e +z +θn +˜θ +FIG. 18. Kinematics for deuteron electrodisintegration. The unprimed axes are defined relative to +the electron scattering plane, and the primed axes relative to the final nucleon momenta. The final +proton momentum has polar angle θp and azimuthal angle φp relative to the unprimed frame. +4 In the hadronic c.m. frame, the longitudinal polarization vector is ǫ(0) = (q′ +z/Q, 0, 0, E′ +γ/Q). +19 + +In keeping with the notation of [6, 7] and [15], the differential cross section for elec- +trodisintegration, summed over the final electron and neutron helicities in the lab frame, +is [14, 15]5 +dσ ≡ +dσ5 +dE′dΩ′ +edΩ′ +p +(5.8) += mpmn|⃗p ′ +p| +16π3md +σMott +frec +[νLRL + νT RT + νTTRTT + νLTRLT + 2λeνLT ′TLT ′ + 2λeνT ′RT ′] , +where Ω′ +e (Ω′ +p) is the solid angle of the scattered electron (proton), σMott is the Mott cross +section, frec = |1 + (Eγ|⃗p ′ +p| − E′ +pqz cos θp)/(md|⃗p ′ +p|)| is the lab recoil factor, +νL = Q4 +q4z +, νT = Q2 +2q2z ++ tan2 ˜θ +2, νTT = Q2 +2q2z +, νLT = +Q2 +√ +2q2 +z +� +� +� +�Q2 +q2z ++ tan2 ˜θ +2, +(5.9) +νLT ′ = − Q2 +√ +2q2z +tan +˜θ +2, νT ′ = tan +˜θ +2 +� +� +� +�Q2 +q2 +z ++ tan2 ˜θ +2, +and ˜θ = θ′ +e − θe is the angle between the incoming and outgoing electron. The response +functions RX depend upon the hadronic helicity amplitudes and the azimuthal angle φp of +the hadronic scattering plane. The subscripts refer to the polarization of the intermediate +photon, which enters on substitution of the polarization expansion (5.6) for the numerator +of the photon propagator in the hadronic amplitude (5.5). The amplitude then decomposes +into separate leptonic and hadronic factors +Med(λ′ +p, λ′ +n, λ′ +e; λd, λe) = − +1 +� +λ=−1 +¯u′ +e̸ ǫ∗(λ)ue +(−1)λ +Q2 +ǫν(λ)Mν(λ′ +p, λ′ +n, λd). +(5.10) +The leptonic factors give rise to the νX coefficients, and the hadronic factors to the response +functions in the square of the amplitude used to construct the cross section [15]. +The +subscript L(T) indicates a purely longitudinal (transverse) contribution, while LT is a cross +term between longitudinal and transverse photon helicities. The TT subscript marks a cross +term between different transverse helicities. A prime indicates a different combination of +transverse helicities. +The response functions are computed from components of the hadronic tensor +wλ′,λ = 2 +3 +� +λ′′p,λ′p,λ′n,λ′′ +d,λd +ǫ∗ +ν(λ′)Mν∗(λ′′ +p, λ′ +n, λ′′ +d)ρp +λ′′p,λ′pǫµ(λ)Mµ(λ′ +p, λ′ +n, λd)ρd +λ′′ +d,λd, +(5.11) +with ρp(ρd) the density matrix for the proton (deuteron) helicity state. We construct these +in the xyz coordinate system of the electron scattering plane. The particular components +are [6] +RL = w0,0, RT = w1,1 + w−1,−1, RT ′ = w1,1 − w−1,−1, +(5.12) +RTT = 2Rew1,−1, RLT = −2Re [w0,1 − w0,−1] , RLT ′ = −2Re [w0,1 + w0,−1] . +5 In [6], h is 2λe but in [15], h is just λe, which leads to additional factors of 2. +20 + +For an unpolarized target, the deuteron density matrix is proportional to the identity, +ρd = +1 +3I; similarly, if the proton helicity is not detected, ρp = +1 +2I. +We then have the +unpolarized cross section [6] +dσunpol = mpmn|⃗p ′ +p| +16π3md +σMott +frec +σ0, σ0 ≡ νLRU +L + νTRU +T + νTTRU +TT + νLTRU +LT , +(5.13) +where the RU +X are computed with the simple density matrices. These are then computable +in our model, with the basic computation being the evaluation of ǫ(λγ)µMµ, which differs +from the photodisintegration calculation in only two ways: Q2 is not zero and λγ ranges +over all three possibilities. +The unpolarized response functions RU +LT ′ and RU +T ′ are identically zero. With ρd replaced +by 1 +3I and the form (5.7) of the polarization vectors taken into account, Rew(0, 1) is just +the negative of Rew(0, −1), and w1,1 is equal to w−1,−1. Thus, the inputs to RU +LT ′ and RU +T ′, +as given in (5.12), immediately cancel. +The recent ed → e′pn experiment at JLab [17] does not include polarization but does +begin to reach momentum transfers sufficient to consider the RNHA approach. Once po- +larization data is available, the expressions developed here and in the Appendix can be +compared. +VI. +SUMMARY +We have extended the reduced nuclear amplitude approach [1, 2] to helicity amplitudes +and applied this model to analysis of elastic electron-deuteron scattering, deuteron photo- +disintegration, and deuteron electrodisintegration. These are just examples of the approach, +which is generally applicable to exclusive nuclear processes. The primary limitation is that, +for any process, the net momentum transfer to every nucleon must be large; therefore, +as the number of nucleons increases, the required beam energy can increase dramatically. +The primary gain is precocious scaling in the dependence on momentum transfer. What the +model (or the original RNA approach) does not provide, though, is an overall normalization; +comparisons must be made in terms of ratios. +By considering helicity amplitudes, many more quantities can be studied, including po- +larization dependence. All three of the deuteron’s electromagnetic form factors can be cal- +culated and from there various elastic scattering observables can be constructed. In Sec. III +we considered the standard structure functions A and B as well as the tensor polarizations +t2m. Generally, the model implies the need for momentum transfers larger than one would +have hoped for seeing simple perturbative QCD scaling. However, our results do imply that +the deuteron structure function B is a good place to look, above a transfer of 10 GeV2. +The RNHA results for polarization observables in deuteron photodisintegration, consid- +ered in Sec. IV, are somewhat consistent with experiment. In particular, our result for the +asymmetry Σ, with a value of Σ(90◦) ≃ −0.06, is much better than the value of -1 originally +expected [52]. Higher photon energies would, of course, be useful. +We have also constructed the RNHA framework for analysis of deuteron electrodisinte- +gration, in Sec. V. This stands ready for comparison with experiment when data is available +at sufficient energies. One aspect that does remain is to consider polarization of the outgoing +proton, in addition to polarization of the beam and target. +Other processes that one might consider include deeply virtual Compton scattering on +the deuteron, pion photoproduction on the deuteron [3], and photodisintegration of 3He [4]. +21 + +In each case, our approach can provide not only information about helicity amplitudes but +also an analysis of nonleading momentum transfer dependence with respect to the onset of +perturbative QCD scaling. We look forward to experiments at larger momentum transfers +for all of these processes. +ACKNOWLEDGMENTS +This work began in conversations with S.J. Brodsky and D.-S. Hwang. Some calculations +were checked by W. Miller and C. Salveson. Diagrams were drawn with use of JaxoDraw [53]. +Appendix A: Electrodisintegration with polarization +If we consider polarization for the beam and the target,6 the proton density matrix is +still just ρp = 1 +2I, but the deuteron density matrix in the xyz frame is [6] +ρd = 1 +3 + + + + + +1 + +� +3 +2T10 + +1 +√ +2T20 − +� +3 +2(T ∗ +11 + T ∗ +21) +√ +3T ∗ +22 +− +� +3 +2(T11 + T21) +1 − +√ +2T20 +− +� +3 +2(T ∗ +11 − T ∗ +21) +√ +3T22 +− +� +3 +2(T11 − T21) 1 − +� +3 +2T10 + +1 +√ +2T20 + + + + + . +(A1) +For a target polarization defined relative to the beam direction, rather than the xyz system +used above, the tensor polarization coefficients TJM are related to the coefficients ˜TJM defined +relative to the beam [6]. If only ˜T10 and ˜T20 are nonzero,7 the nonzero TJM are +T10 = cos ˜θ ˜T10, T11 = − 1 +√ +2 sin ˜θ ˜T10, +(A2) +T20 = 1 +4(1 + 3 cos 2˜θ) ˜T20, T21 = − +� +3 +8 sin 2˜θ ˜T20, T22 = +� +3 +32(1 − cos 2˜θ) ˜T20. +The density matrix can then be written as +ρd = +�1 +3I + ˜T10ρdV + ˜T20ρdT +� +, +(A3) +where +ρdV = 1 +3 + + + + + +� +3 +2 cos ˜θ +√ +3 +2 sin ˜θ +0 +√ +3 +2 sin ˜θ +0 +√ +3 +2 sin ˜θ +0 +√ +3 +2 sin ˜θ − +� +3 +2 cos ˜θ + + + + + +(A4) +and +ρdT = 1 +3 + + + + +1 +4 +√ +2(1 + 3 cos 2˜θ) +3 +4 sin 2˜θ +3 +√ +32(1 − cos 2˜θ) +3 +4 sin 2˜θ +− +1 +2 +√ +2(1 + 3 cos 2˜θ) +−3 +4 sin 2˜θ +3 +√ +32(1 − cos 2˜θ) +−3 +4 sin 2˜θ +1 +4 +√ +2(1 + 3 cos 2˜θ) + + + + . +(A5) +6 For discussion of a polarized outgoing proton, see [7] and [15]. +7 The spherical tensor moments are related to the Cartesian tensor moments as ˜T10 = +� +3 +2Pz and ˜T20 = +1 +√ +2Pzz. +22 + +The response functions can then be separated into unpolarized, vector, and tensor contri- +butions as RX = RU +X + ˜T10RV +X + ˜T20RT +X, with RU +X, RV +X, and RT +X computed with ρd replaced +by 1 +3I, ρdV , and ρdT , respectively. +With dσunpol defined as the unpolarized cross section, given in (5.13), the full cross section +can be written as +dσ = +� +1 + ˜T10 +� +AV +d + 2λeAV +ed +� ++ ˜T20 +� +AT +d + 2λeAT +ed +�� +dσunpol, +(A6) +in terms of the single and double asymmetries +AV +d = +� +νLRV +L + νTRV +T + νTTRV +TT + νLT RV +LT +� +/σ0, +(A7) +AV +ed = +� +νLT ′RV +LT ′ + νT ′RV +T ′ +� +/σ0, +(A8) +AT +d = +� +νLRT +L + νTRT +T + νTTRT +TT + νLTRT +LT +� +/σ0, +(A9) +AT +ed = +� +νLT ′RT +LT ′ + νT ′RT +T ′ +� +/σ0. +(A10) +For a recent summary of data, see [54]. +[1] S.J. 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C 95, 024005 (2017). +26 + diff --git a/1tA0T4oBgHgl3EQfMv-f/content/tmp_files/load_file.txt b/1tA0T4oBgHgl3EQfMv-f/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e8bcf023ff839334a01b2b3bc5a0ce3c4297b90e --- /dev/null +++ b/1tA0T4oBgHgl3EQfMv-f/content/tmp_files/load_file.txt @@ -0,0 +1,993 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf,len=992 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='02137v1 [nucl-th] 5 Jan 2023 Reduced nuclear helicity amplitudes for deuteron electrodisintegration and other processes J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Flores and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Chabysheva Department of Physics, University of Idaho, Moscow ID 83844 USA J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Hiller Department of Physics, University of Idaho, Moscow ID 83844 USA and Department of Physics and Astronomy, University of Minnesota-Duluth, Duluth, Minnesota 55812 USA (Dated: January 6, 2023) 1 Abstract We extend the original idea of reduced nuclear amplitudes to capture individual helicity ampli- tudes and discuss various applications to exclusive processes involving the deuteron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Specifically, we consider deuteron form factors, structure functions, tensor polarization observables, photodisin- tegration, and electrodisintegration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The basic premise is that nuclear processes at high momentum transfer can be approximated by tree graphs for point-like nucleons supplemented by empirical form factors for each nucleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The latter represent the internal structure of the nucleon, and incorporate nonperturbative physics, which can allow for early onset of scaling behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The nucleon form factors are evaluated at the net momentum transfer experienced by the given nucleon, with use of GE for a no-flip contribution and GM for a helicity-flip contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Results are compared with data where available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The deuteron photodisintegration asymmetry Σ is obtained with a value of Σ(90◦) ≃ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='06, which is much closer to experiment than the value of -1 originally expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The method also provides an estimate of the momentum transfer values required for scaling onset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' We find that the deuteron structure function B is a good place to look, above momentum transfers of 10 GeV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' INTRODUCTION With the advent of the upgraded electron accelerator at the Thomas Jefferson National Accelerator Facility, scattering experiments with polarized beams and targets at high energy and high momentum transfer become possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' In the regime of high momentum transfer to all relevant nucleons, quantum chromodynamics (QCD) implies that the internal structure of every nucleon is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Until ab initio QCD (lattice) calculations for nuclear scattering processes are available for more than very simple processes, one is led to consider models that can represent the basic physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' One such approach is the reduced nuclear amplitude (RNA) analysis pioneered by Brod- sky and Chertok [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' In addition to their application to a generic deuteron form factor, the approach has been applied to deuteron disintegration [2], pion photoproduction [3], and photodisintegration of 3He [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' As originally developed, a nuclear process was modeled as a tree-level amplitude multiplied by a generic form factor for each nucleon, with each form factor evaluated at the net momentum transferred to that nucleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' In order to model the behavior of polarization observables [5–15], we extend this approach to a reduced nuclear helicity amplitude (RNHA) method to combine a tree-level helicity amplitude for point-like nucleons with the appropriate form factor for each nucleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' When the nucleon does (not) flip its helicity, we use the electric (magnetic) form factor GEN (GMN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' As a check on the procedure, virtual photon absorption by a single nucleon in the RNHA approach is consistent with the definitions of GEN and GMN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' A caveat in applications of the RNA approach is that the normalization is not determined by the model and is fixed to data at infinite momentum transfer by the coefficient of the leading power-law behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' This means that the normalization cannot be determined in practice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' fitting to a data point at some intermediate kinematics will give the wrong nor- malization and the wrong magnitude at higher momentum transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Instead, ratios need to be considered, so that the normalization becomes irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The primary criterion for the asymptotic region is in the momentum transfer to each nucleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' For every nucleon in the process, the momentum transfer must be above some common threshold, which is at least 1 GeV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' For example, for deuteron photodisintegration, 2 the momentum transfer to a nucleon is −tN = −(pN − p/2)2, where pN is the final four- momentum of the nucleon and p is the initial deuteron momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' When expressed in terms of the photon energy Eγ and the final nucleon angle θ, the constraint to be above 1 GeV2 becomes [16] mNEγ � 1 − � Eγ mN + Eγ | cos θ| � ≥ 1 GeV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='1) This relationship is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Notice that away from 90◦, the lower limit is quite high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' For electrodisintegration, only the most recent data [17, 18] begins to reach this θ (deg) 0 20 40 60 80 100 120 140 160 180 Eγ (GeV) 0 2 4 6 8 10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Angular dependence of the scale for large momentum transfer in deuteron photodisinte- gration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Here we will focus on deuteron processes, including photodisintegration and electrodisin- tegration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' For recent reviews of deuteron studies at high momentum transfer, see [16, 19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Elastic electron scattering data at high momentum transfer is presented in [21–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Recent photodisintegration data can be found in [26–29], and for electrodisintegration data, in [17, 18, 30–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Other analyses of deuteron processes include hidden-color contributions to deuteron form factors [34], the hard rescattering mechanism [35], quark-gluon strings [36], the Moscow NN potential [37], and AdS/QCD models [38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' One recent experiment [17] used the 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6 GeV electron beam at JLab and the Hall C spectrometers to measure electron scattering from a liquid deuterium target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The final electron and the proton were detected, with the kinematics restricted to the exclusive process ed → e′pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' One spectrometer measured the final electron at a nominal 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2◦ degrees from the beam direction, with a momentum of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='5-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='1 GeV such that the recorded events had a distribution of momentum transfer squared reaching 5 GeV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The events studied were taken from a bin of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='5 GeV2 in the tail of the distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' however, the nominal transfer was 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2 GeV2, because the majority of the events were in the lower half of the bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' A second spectrometer measured the proton momentum at a range of angles to the beam direction, tuned to select events where the (missing) neutron had an angle relative to the direction of the momentum transfer that fell within a chosen bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' In the one-photon exchange approximation, which we assume, the momentum transferred is, of course, the photon momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The published neutron angles are binned at 35◦, 45◦ and 75◦, with the first two selected to minimize final-state interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' For our purposes, the importance of 3 these two angles is that the momentum transferred to the neutron reaches 1 GeV in a zero- binding approximation, so that, rather than focus on the internal structure of the deuteron, we can consider the response to a large momentum transfer to all the nucleons involved and we can see that experiments may be approaching the threshold where our model can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The RNHA model is constructed in detail in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' II for two-nucleon processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' In the remainder of the paper, we consider various processes for the deuteron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' III, the form factors,1 structure functions, and tensor polarization observables of elastic electron scattering from the deuteron are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Photodisintegration and electrodisintegration are analyzed in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' IV and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Within the zero-binding approximation, elastic scattering and photodisintegration live at edges of the kinematic range of electrodisintegration and are essentially special cases that provide introductory examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Section VI contains a summary of the results and suggestions for additional applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Many details of the electrodisintegration helicity amplitudes are left to an appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' CONSTRUCTION OF THE MODEL The basic process for a two-nucleon system to absorb a photon and exchange momentum between the nucleons is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' These diagrams are modeled on the primitive process of γ∗ff → ff, with f representing a point-like nucleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2 The structure of each nucleon is then introduced by combining the Feynman amplitude for each diagram with the appropriate form factor for each nucleon, evaluated at the net momentum transfer for that nucleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' For a deuteron process in the zero-binding limit, the initial nucleons share the initial deuteron momentum p equally, so that pp = pn = p/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' We also neglect the nucleon mass difference, setting mp = mn ≡ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The distinction between different photon-absorption processes is then in the nature of the photon, being either real or virtual, and in the outcome for the final nucleons, bound as a deuteron or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The tree-level amplitudes for the four diagrams in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 2 are Mν a (λ′ p, λ′ n, λp, λn) = Aµν p (p/2 + q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' λ′ p, λp) 1 (p′ n − p/2)2Bnµ(λ′ n, λn), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='1) Mν b (λ′ p, λ′ n, λp, λn) = Aνµ p (p′ p − q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' λ′ p, λp) 1 (p′n − p/2)2Bnµ(λ′ n, λn), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2) Mν c (λ′ p, λ′ n, λp, λn) = Aµν n (p/2 + q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' λ′ n, λn) 1 (p′p − p/2)2Bpµ(λ′ p, λp), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='3) Mν d (λ′ p, λ′ n, λp, λn) = Aνµ n (p′ n − q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' λ′ n, λn) 1 (p′ p − p/2)2Bpµ(λ′ p, λp), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='4) where Aµν N (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' λ′ N, λN) = ¯u′ Nγµ ̸p + m p2 − m2γνuN, Bµ N(λ′ N, λN) = ¯u′ NγµuN, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='5) with uN (¯u′ N) the initial (final) spinor for the nucleon N with helicity λN (λ′ N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The sub- amplitude AN represents the fermion line that absorbs the photon, and BN represents the 1 For discussion specifically in terms of perturbative QCD, see [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 2 In [2], the primitive process was γ∗q¯q → q¯q, with q corresponding to a point-like proton and ¯q to a point- like neutron, and direct interaction of the photon with the neutron was neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Here we amend and extend this, to retain information about helicity states of the fermions and include photon absorption by the neutron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 4 pp, λp pn, λn p′ p, λ′ p p′ n, λ′ n q, λγ pp, λp pn, λn p′ p, λ′ p p′ n, λ′ n q, λγ (a) (b) pp, λp pn, λn p′ p, λ′ p p′ n, λ′ n q, λγ pp, λp pn, λn p′ p, λ′ p p′ n, λ′ n q, λγ (c) (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Tree graphs for deuteron processes that absorb a photon of momentum q and helicity λγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The initial (final) nucleon momentum and helicity are pN (p′ N) and λN (λ′ N), with N = p or n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The two nucleons exchange momentum via a vector particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The four diagrams differ in the nature of the photon-absorbing nucleon and the order of this absorption and momentum transfer between nucleons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' other fermion line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Calculation of these sub-amplitudes can be checked against the trace theorem for sums over helicities: � λN,λ′ N Aν′µ′∗ N (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' λ′ N, λN)Aµν N (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' λ′ N, λN) = Tr � γν′ ̸p + m p2 − m2γµ′(̸p′ N + m)γµ ̸p + m p2 − m2γν(̸pN + m) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6) � λN,λ′ N Bµ′∗ N (λ′ N, λN)Bµ N(λ′ N, λN) = Tr � γµ′(̸p′ N + m)γµ(̸pN + m) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='7) The full amplitude is constructed from the MX by combining them with form factors for each nucleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' For a deuteron with initial helicity λd, we have Mν(λ′ p, λ′ n, λd) = � λp,λn Cλd λpλn \uf8ee \uf8f0 � X=a,b,c,d Mν X(λ′ p, λ′ n, λp, λn) \uf8f9 \uf8fb Gpλ′pλp(Q2 p)Gnλ′nλn(Q2 n), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8) 5 where Q2 N = −(p′ N − pN)2, Cλd λpλn = \uf8f1 \uf8f2 \uf8f3 δλp± 1 2δλn± 1 2, λd = ±1 1 √ 2 � δλp 1 2δλn− 1 2 + δλp− 1 2δλn+ 1 2 � , λd = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='9) and GNλ′λ = � GEN, λ′ = λ GMN, λ′ = −λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='10) The form factors GEN and GMN represent the internal structure of the nucleons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' They can be represented by data or empirical fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' For simplicity, we use the fits [40] GEp ≃ � 1 + Q2 N m2 0 �−2 , GMp ≃ µpGEp, GMn ≃ µnGEp, GEn ≃ − µnτ 1 + 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6τ GEp, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='11) where m2 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='71 GeV2, τ = Q2 N 4m2 , µp = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='79, and µn = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' To limit the analysis to a single mass scale, we take the parameter m0 to be proportional to the nuclear mass, with m2 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='80 m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' We do not attempt to compute or assign an overall normalization to Mν, and the running of the strong coupling constant is not included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The initial nucleon spinor, for a deuteron traveling along the negative z direction, is [41] uN = ̸p/2 + m � Ed/2 + m � φ(λN)(−ˆz) 0 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='12) with φ(1/2)(−ˆz) = � 0 1 � , φ(−1/2)(−ˆz) = � 1 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='13) The final nucleon spinor is u′ N = ̸p′ N + m � E′ N + m � φ(λ′ N)(ˆp′ N) 0 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='14) with φ(1/2)(ˆp′ N) = � cos(θN/2) eiφN sin(θN/2) � , φ(−1/2)(ˆp′ N) = � −e−iφN sin(θN/2) cos(θN/2) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='15) where θN and φN are the polar and azimuthal angles of the outgoing momentum of the particular nucleon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' As discussed in the Introduction, the overall normalization of the RNHA amplitude is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' For comparison with data, we consider quantities which are themselves ratios or a ratio of the model to data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' ELASTIC ELECTRON SCATTERING A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Form factors The three deuteron form factors, GC, GM, and GQ, are readily obtained from the hadronic helicity amplitudes of elastic electron-deuteron scattering in the Breit frame [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The kine- matics are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The photon four-momentum is q = (0, 0, 0, qz) and the initial 6 (final) deuteron four-momentum is p = (Ed, 0, 0, −qz/2) (p′ = (Ed, 0, 0, qz/2)), with q2 z = Q2 and Ed = � Q2/4 + m2 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' In the zero-binding limit,3 md = 2m and the individual nucleon four-momenta are pp = pn = p/2 and p′ p = p′ n = p′/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The hadronic matrix elements are given by Gµ λ′ d,λd = � λ′p,λ′n C λ′ d λ′pλ′nMµ(λ′ p, λ′ n, λd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='1) The initial spinors are as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='12);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' the final spinors are specified by u′ N = ̸p′/2 + m � Ed/2 + m � φ(λ′ N)(ˆz) 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2) ⃗q, λγ −⃗q/2, λd ⃗q/2, λ′ d z ⃗pe, λe ⃗p ′ e, λ′ e FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Kinematics for elastic electron-deuteron scattering in the Breit frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The photon travels along the positive z direction, and the deuteron comes from the right, along the negative z direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The three form factors are then extracted as [42, 43] GC = −1 2md √1 + η G+ 00 − 2G+ +− 3 , GM = 2 2md √1 + η Gx +0 √2η, GQ = −1 2md √1 + η G+ 00 + G+ +− 2η , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='3) with η ≡ Q2 4m2 d and the + superscript denoting the light-front sum of the 0 and z components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' For the helicity matrix elements, the model yields the following Q2 dependence: G+ 00 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='5588N m �m Q �9 � 1 + 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='1m2 Q2 + O(m4 Q4 ) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='4) G+ +− = −69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='85N m �m Q �11 � 1 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8m2 Q2 + O(m4 Q4 ) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='5) Gx +0 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='851N m �m Q �10 � 1 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8m2 Q2 + O(m4 Q4 ) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6) with N the unknown normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The factor of m/Q associated with each helicity flip [44] is clearly evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' For the form factors, we find GC = − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='5588 √1 + η N 12 �m Q �9 � 1 + 379.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='1m2 Q2 + O(m4 Q4 ) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='7) 3 The difference between the proton and neutron masses is neglected in addition to the deuteron binding energy, the two being of the same order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 7 GM = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='851 � η(1 + η) N 2 √ 2 �m Q �10 � 1 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8m2 Q2 + O(m4 Q4 ) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8) GQ = − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='5588 η√1 + η N 8 �m Q �9 � 1 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='086m2 Q2 + O(m4 Q4 ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='9) The leading ± signs are as expected for large Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' We have left the kinematic factor η = Q2/16m2 without substitution, because there can be three regimes for Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' In addition to Q2 large or small, there can be an intermediate region where Q2 is large but η is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Such an intermediate regime does exist for GM and GQ, where the coefficients of the nonleading terms are small enough for this correction to be small while η is also small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' For GC, this is not the case, because the coefficient of the nonleading term is large enough to require a Q2 value for which η is also large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' In the intermediate regime, we obtain GM ∼ �m Q �11 , GQ ∼ �m Q �11 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='10) and for the large-η regime GC ∼ �m Q �10 , GM ∼ �m Q �12 , GQ ∼ �m Q �12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='11) Ratios of these form factors at very large Q2 can be compared with the tree-level ratios for a point-like spin-one particle, such as the W +, which are [43] GC GQ = 2 3η − 1, GM GQ = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='12) Such behavior is immediately reproduced for form factors separated according to a Drell– Yan frame [45], with the assumption of strict G+ 00 dominance [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' In terms of our hadronic matrix elements, we have GC GQ = 2 3η − 2η G+ +− G+ 00 + G+ +− , GM GQ = −2 � 2η Gx +0 G+ 00 + G+ +− .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='13) As already observed in [43], these Breit-frame ratios cannot both be resolved by simply assuming G+ 00 dominance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' From our model, we obtain GC GQ = 2 3η + 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6 + O(m2 Q2 ), GM GQ = −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2 + O(m2 Q2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='14) The leading 2 3η is just kinematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The deviations of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6 and -11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2 from -1 and -2, respec- tively, are due to nonleading contributions multiplied by powers of η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Similar deviations will arise for calculations done in the Drell–Yan frame, because η factors again interfere with strict G+ 00 dominance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Plots of these ratios are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 8 �����V����e/����� V�� V�� � �� �� �� ��/M����0 � �� ��� FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Ratios GC/GQ − 2η/3 (dashed) and GM/GQ (solid) for the model deuteron form factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Structure functions Experiments designed to extract these form factors measure cross sections and polariza- tion observables in elastic electron-deuteron scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The unpolarized cross section dσ dΩ ∝ S, S ≡ A(Q2) + B(Q2) tan2(θe/2) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='15) depends on the electron scattering angle θe and two structure functions A(Q2) ≡ G2 C + 8 9η2G2 Q + 2 3ηG2 M, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='16) B(Q2) ≡ 4 3η(1 + η)G2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='17) These have been measured at the highest Q2 yet attained at JLab [22–24], and A has been measured at comparable Q2 at SLAC [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' However, these do not yet reach the Q2 values needed for a definitive comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Figures 5 and 6 show plots of the data divided by the model, including an arbitrary normalization factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' In our model, expansions of these functions in inverse powers of Q2 are A(Q2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='1041N 2 �m Q �20 � 1 + 1246m2 Q2 + O(m4 Q4 ) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='18) B(Q2) = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='06N 2 �m Q �20 � 1 + 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6m2 Q2 + O(m4 Q4 ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='19) Because the expansion for GC is valid only for large η, we have used the explicit form of η in constructing the expansion for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The function B is independent of η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' the leading factor of 9 � �l�70V����� 5 6 7 � � �7Ql���70 5 6 7 � � � � � FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Data for the deuteron structure function A(Q2) divided by the model function, including an arbitrary normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Experimental values are taken from [23] (circles) and [24] (squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' η(1+η) in its definition exactly cancels against factors in the relationship of GM to hadronic matrix elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The expansion for B converges much faster than the expansion for A, and the leading Q2 behavior is dominant for Q2 ≫ 10 GeV2 only for B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' For A, one must wait until impossibly large Q2, which enters a regime where the collective quark substructure is important, including hidden-color effects [34], and the point-like approximation used in our model is invalid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' In [22] the large Q2 behavior of B is quoted as being Q−24 from perturbative QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' This faster fall off compared to A is attributed to the extra suppression of the helicity flip involved in GM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' However, there are other compensating factors, and, just as in our model, the behavior of B should be Q−20, which is the same as A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 7 we plot the ratio of B to A for a large range of Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' This ratio becomes constant at very large Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Although the plots begin at low Q2, there is nothing in the model that could reproduce diffractive minima, hence the smooth appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Tensor polarization observables Experiments can also extract tensor polarization observables [20, 25] t20 ≡ − 1 √ 2S �8 3ηGCGQ + 8 9η2G2 Q + 1 3η � 1 + 2(1 + η) tan2(θe/2) � G2 M � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='20) t21 ≡ 2η √ 3S cos(θe/2) � η + η2 sin2(θe/2)GMGQ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='21) t22 ≡ − η 2 √ 3S G2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='22) 10 � �l��-G����� 1V � V V � 3 3 � � � � ��Ql����- 3 3 � � � � � FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Data for the deuteron structure function B(Q2) divided by the model function, including an arbitrary normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Experimental values are taken from [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' � �8��15�8��1 � �� �� �� �� ��� ��� ��� ��Q8����1 ��� ��� ��� ��� ��� ��� ��� FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Ratio of B to A for the model deuteron structure functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The highest Q2 measurements of these were also done at JLab [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' When η is held explicit, expansions in m/Q are t20 = − √ 2 + 1064[1 + 2(1 + η) tan2(θe/2)] �m Q �2 + O(m4 Q4 ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='23) 11 t21 = 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8 sec(θe/2) � η + sin2(θe/2)m Q (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='24) + sec(θe/2) � η + sin2(θe/2)[48606 + 77869(1 + η) tan2(θe/2)] �m Q �3 + O(m4 Q4 ), t22 = −434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='5 �m Q �2 + [544703 + 872133(1 + η) tan2(θe/2)] �m Q �4 + O(m6 Q6 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='25) While at very large Q2, they are t20 = − √ 2 + 133 tan2(θe/2) + 1064[1 + 2 tan2(θe/2)] �m Q �2 + O(m4 Q4 ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='26) t21 = 1217 sec(θe/2) sin(θe/2)[tan2(θe/2) − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='007972] + [] �m Q �2 + O(m4 Q4 ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='27) t22 = −[434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='5 − 54508 tan2(θe/2)] �m Q �2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='28) +[544703 + 872133 tan2(θe/2)] �m Q �4 + O(m6 Q6 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The coefficients of nonleading terms are quite large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Thus, very large Q2 is required for the leading term to be dominant, well beyond any available data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The limit of − √ 2 for t20 at θe = 0◦ was an early prediction of perturbative QCD [44, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' However, as argued elsewhere [43], this value is obtained only at very large Q2, and the value is quite different for small η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Figures 8 and 9 show plots of these observables at angles of 0◦ and 30◦, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' We also compare with data [25] in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 10, 11, and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' At these ‘small’ values of Q2, only t22 is consistent with data, something which is likely accidental with both data and model values near zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' PHOTODISINTEGRATION In the photodisintegration of a deuteron, a real photon is absorbed and the two constituent nucleons emitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' This process is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The initial deuteron and photon four- momenta in the center-of-mass (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=') frame are p = (Ed, 0, 0, −qz) and q = (qz, 0, 0, qz), where the incident photon is taken along the positive z axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The final proton and neutron four-momenta are p′ p = (E′ p, ⃗p ′ p) and p′ n = (E′ n, ⃗p ′ n), with θp and φp the polar and azimuthal angles of the final proton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' By ignoring the nucleon mass difference, we have E′ p = E′ n, because momentum conservation guarantees ⃗p ′ n = −⃗p ′ p in the c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' In terms of the Mandelstam variable s, the c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' energies and momenta are Ed = (s + 4m2)/(2√s), qz = (s − 4m2)/(2√s), E′ p = √s/2, |⃗p ′ p| = √ s − 4m2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='1) The photon energy in the lab frame is Eγ = (s − 4m2)/4m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' We will work at large s, so that momentum transfers are large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The standard definition of helicity amplitudes for photodisintegration is [47] Fi± ≡ ǫν(λγ)Mν(λ′ p, λ′ n, λd) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2) 12 � ��5e0���e0��� V�)� V�)� V�)� V� V5)� V5)� V5)� V5)� 5 ��08����Q �55 �5� �5� �5� �5� �5� �5� FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Deuteron tensor polarization observables t20 (solid), t21 (dashed), and t22 (dotted) as computed in the model at an angle of θe = 0◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The asymptotic value of t20(0◦) is − √ 2, as predicted by perturbative QCD [44, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' with ǫ the polarization vector for a photon with helicity λγ and Mν given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The index i is associated with particular helicity combinations as follows: F1± = ǫν(1)Mν(±1 2, ±1 2, 1), F2± = ǫν(1)Mν(±1 2, ±1 2, 0), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='3) F3± = ǫν(1)Mν(±1 2, ±1 2, −1), F4± = ǫν(1)Mν(±1 2, ∓1 2, 1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='4) F5± = ǫν(1)Mν(±1 2, ∓1 2, 0), F6± = ǫν(1)Mν(±1 2, ∓1 2, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='5) The other helicity combinations are related to these by parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The helicity amplitudes can be used to compute various polarization observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The recoil-proton polarization Py measures the asymmetry parallel/antiparallel to the normal ˆy ∝ ⃗q × ⃗p ′ p to the scattering plane: Py = 2Im 3 � i=1 [F † i+Fi+3,− + F † i+3,+Fi−]/f(θ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6) where f(θ) = �6 i=1[|Fi+|2 + |Fi−|2] is the sum of all the helicity amplitudes squared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The transferred polarizations Cx′ and Cz′ measure asymmetries parallel/antiparallel to the ˆx′ ∝ ⃗p ′ p × ˆy and ˆz′ = ˆp ′ p directions: Cx′ = 2Re 3 � i=1 [F † i+Fi+3,− + F † i+3,+Fi−]/f(θ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='7) Cz′ = 6 � i=1 [|Fi+|2 − |Fi−|2]/f(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8) 13 � ���e0���e0��� V�)� V�)� V�)� V�)� V�)� � ��03����Q ��� ��� ��� ��� ��� ��� ��� FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 8 but for an angle of θe = 30◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The asymmetry Σ for linearly polarized photons is given by Σ = −2Re �� ± (F † 1±F3∓ − F † 4±F6∓) − F † 2+F2− + F † 5+F5− � /f(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='9) Each observable is formed as a ratio, which sets aside questions of normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Because we only need to consider photons with helicity +1, the polarization vector is always ǫ = − 1 √ 2(0, 1, i, 0), relative to the momentum in the positive z direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The final Dirac spinors are u′ N = ̸p′ N + m � E′ N + m � φ(λ′ N)(ˆp′ N) 0 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='10) with θn = π − θp, φn = φp + π = π, and φ(1/2)(ˆp′ N) = � cos(θN/2) eiφN sin(θN/2) � , φ(−1/2)(ˆp′ N) = � −e−iφN sin(θN/2) cos(θN/2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='11) With these spinors as input, the amplitudes ǫνMν X can be evaluated in terms of Dirac matrix and spinor products and then combined to construct the predefined amplitudes Fi±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' At large s, these RNHA predictions for the helicity amplitudes reduce to F1+ ∼ 4 √ 2 √scsc2(θp 2 )GEn(θp)GEp(θp), F1− ∼ 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='12) F2+ ∼ 2m s cot3(θp 2 )[GEn(θp)GMp(θp) − GMn(θp)GEp(θp)], (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='13) F2− ∼ 2m s cot(θp 2 )[GMn(θp)GEp(θp) − GEn(θp)GMp(θp)], F3+ ∼ 0, F3− ∼ 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='14) 14 � ��� 1��� 1��� 1��� 1��� � ��� ��� ��� ��� ��0(����G ��� ��� ��� ��� ��� ��� ��� FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Plots of the tensor polarization observable t20 of the deuteron from both data [25] (circles) and the model (squares) considered in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The angle θe varies and is as follows in order of increasing Q2: 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6◦, 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='4◦, 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8◦, 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='3◦, 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0◦, and 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' F4+ ∼ −4 √ 2m s cot(θp 2 )GMn(θp)GEp(θp), F4− ∼ 4 √ 2m s cot(θp 2 )GEn(θp)GMp(θp), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='15) F5+ ∼ 2 √s cot2(θp 2 )GEn(θp)GEp(θp), F5− ∼ 2 √sGEn(θp)GEp(θp), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='16) F6+ ∼ 4 √ 2m s cot3(θp 2 )GEn(θp)GMp(θp), F6− ∼ −4 √ 2m s cot(θp 2 )GMn(θp)GEp(θp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='17) From these we can calculate the various observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Plots of the results and recent data [48– 50] are given in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 14, 15, 16, and 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Because the tree-level amplitudes are real, Py is automatically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' That Cx′ is of order m/√s, rather than zero, is a correction to hadron helicity conservation [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Also, we find the asymmetry Σ(90◦) to be approximately -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='06, rather than the nominal expectation [52] of -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' In general, the trends with photon energy seem to be modestly consistent with data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' ELECTRODISINTEGRATION The kinematics of the electrodisintegration process are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The initial (final) momentum and helicity of the electron are pe (p′ e) and λe (λ′ e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The intermediate photon carries four-momentum q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The azimuthal angle φp of the proton measures the rotation of the hadronic reaction plane relative to the electron scattering plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' In the lab frame, with the z axis taken along the photon three-momentum and the electron mass neglected, the initial and final electron four-momenta are pe = (Ee, Ee sin θe, 0, Ee cos θe), p′ e = (E′ e, E′ e sin θ′ e, 0, E′ e cos θ′ e), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='1) 15 � ��� V�)� V�)� � �)� �)� �)� �)� �)� �)� ��1Q���� �)� �)� � �)� �)� �)� �)� FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 10 but for t21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' � ��� 8�4�� 8�4� 8�4�� 8�4� 8�4�� � �4�� �4� �4�� ��-G����e �4� �4� � �4� �4� �4� �4� FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 10 but for t22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' with E′ e and ˜θ = θ′ e − θe, the angle of the scattered electron to the beam direction, being measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The photon four-momentum q = (Eγ, 0, 0, qz) is just pe − p′ e, which yields Q2 ≡ −q2 = 2EeE′ e(1 − cos ˜θ), Eγ = Ee − E′ e, qz = � E2γ + Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2) The deuteron four-momentum is p = (md = 2m, 0, 0, 0), and in the zero-binding limit, the initial proton and neutron four-momenta are pp = pn = (m, 0, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The final nucleon 16 ⃗q, λγ ⃗p, λd ⃗p ′ p, λ′ p ⃗p ′ n, λ′ n θn = π − θp θp z FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Kinematics for deuteron photodisintegration in the c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' frame, with ⃗q the photon mo- mentum and ⃗p = −⃗q the deuteron momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The final proton and neutron momenta are ⃗p ′ p and ⃗p ′ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The λ’s are helicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Coordinates are chosen such that the photon enters along the positive z direction and the azimuthal angle φp of the proton is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Eγ (GeV) 0 1 2 3 4 5 6 Py (90 deg) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0 θ (deg) 0 20 40 60 80 100 120 140 160 180 Py (Eγ=2 GeV) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Recoil proton polarization Py as a function of (a) photon energy Eγ and (b) proton angle θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' For the latter, the photon energy is 2 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The solid line is the RNHA prediction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' the data points are from [48, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' four-momenta are p′ p = (E′ p = � ⃗p ′2 p + m2, |⃗p ′ p| sin θp cos φp, |⃗p ′ p| sin θp sin φp, |⃗p ′ p| cos θp), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='3) p′ n = (E′ n = � ⃗p ′2 n + m2, −|⃗p ′ n| sin θn cos φp, −|⃗p ′ n| sin θn sin φp, |⃗p ′ n| cos θn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content="4) Within the one-photon-exchange approximation, the scattering amplitude is proportional 17 Eγ (GeV) 0 1 2 3 4 5 6 Cx' (90 deg) 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content="0 θ (deg) 0 20 40 60 80 100 120 140 160 180 Cx' (Eγ=2 GeV) 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 14 but for the transferred polarization Cx′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=" Eγ (GeV) 0 1 2 3 4 5 6 Cz' (90 deg) 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content="0 θ (deg) 0 20 40 60 80 100 120 140 160 180 Cz' (Eγ=2 GeV) 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 15 but for Cz′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' to Med(λ′ p, λ′ n, λ′ e;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' λd, λe) = ¯u′ eγµue Dµν q2 Mν(λ′ p, λ′ n, λd), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='5) with ue (u′ e) the initial (final) spinor of the electron and Mν given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The numerator of the photon progator is the sum over photon polarizations Dµν = 1 � λ=−1 (−1)λǫ∗ µ(λ)ǫν(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6) 18 Eγ (GeV) 0 1 2 3 4 5 6 Σ (90 deg) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='0 θ (deg) 0 20 40 60 80 100 120 140 160 180 Σ (Eγ=2 GeV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='10 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 14 but for the asymmetry Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The data points are from [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The polarization four-vectors are4 ǫ(±1) = ∓ 1 √ 2(0, 1, ±i, 0), ǫ(0) = (qz/Q, 0, 0, Eγ/Q) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='7) relative to the photon four-momentum q = (Eγ, 0, 0, qz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Polarization observables [5–15] can then be computed from these helicity amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' ⃗pe, λe ⃗p ′ e, λ′ e x ⃗q ⃗p ′ p, λ′ p ⃗p ′ n, λ′ n y′ x′ z, z′ y θp φp θe θ′ e z θn ˜θ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Kinematics for deuteron electrodisintegration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The unprimed axes are defined relative to the electron scattering plane, and the primed axes relative to the final nucleon momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The final proton momentum has polar angle θp and azimuthal angle φp relative to the unprimed frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 4 In the hadronic c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' frame, the longitudinal polarization vector is ǫ(0) = (q′ z/Q, 0, 0, E′ γ/Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 19 In keeping with the notation of [6, 7] and [15], the differential cross section for elec- trodisintegration, summed over the final electron and neutron helicities in the lab frame, is [14, 15]5 dσ ≡ dσ5 dE′dΩ′ edΩ′ p (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='8) = mpmn|⃗p ′ p| 16π3md σMott frec [νLRL + νT RT + νTTRTT + νLTRLT + 2λeνLT ′TLT ′ + 2λeνT ′RT ′] , where Ω′ e (Ω′ p) is the solid angle of the scattered electron (proton), σMott is the Mott cross section, frec = |1 + (Eγ|⃗p ′ p| − E′ pqz cos θp)/(md|⃗p ′ p|)| is the lab recoil factor, νL = Q4 q4z , νT = Q2 2q2z + tan2 ˜θ 2, νTT = Q2 2q2z , νLT = Q2 √ 2q2 z � � � �Q2 q2z + tan2 ˜θ 2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='9) νLT ′ = − Q2 √ 2q2z tan ˜θ 2, νT ′ = tan ˜θ 2 � � � �Q2 q2 z + tan2 ˜θ 2, and ˜θ = θ′ e − θe is the angle between the incoming and outgoing electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The response functions RX depend upon the hadronic helicity amplitudes and the azimuthal angle φp of the hadronic scattering plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The subscripts refer to the polarization of the intermediate photon, which enters on substitution of the polarization expansion (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='6) for the numerator of the photon propagator in the hadronic amplitude (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The amplitude then decomposes into separate leptonic and hadronic factors Med(λ′ p, λ′ n, λ′ e;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' λd, λe) = − 1 � λ=−1 ¯u′ e̸ ǫ∗(λ)ue (−1)λ Q2 ǫν(λ)Mν(λ′ p, λ′ n, λd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='10) The leptonic factors give rise to the νX coefficients, and the hadronic factors to the response functions in the square of the amplitude used to construct the cross section [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The subscript L(T) indicates a purely longitudinal (transverse) contribution, while LT is a cross term between longitudinal and transverse photon helicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The TT subscript marks a cross term between different transverse helicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' A prime indicates a different combination of transverse helicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The response functions are computed from components of the hadronic tensor wλ′,λ = 2 3 � λ′′p,λ′p,λ′n,λ′′ d,λd ǫ∗ ν(λ′)Mν∗(λ′′ p, λ′ n, λ′′ d)ρp λ′′p,λ′pǫµ(λ)Mµ(λ′ p, λ′ n, λd)ρd λ′′ d,λd, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='11) with ρp(ρd) the density matrix for the proton (deuteron) helicity state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' We construct these in the xyz coordinate system of the electron scattering plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The particular components are [6] RL = w0,0, RT = w1,1 + w−1,−1, RT ′ = w1,1 − w−1,−1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='12) RTT = 2Rew1,−1, RLT = −2Re [w0,1 − w0,−1] , RLT ′ = −2Re [w0,1 + w0,−1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 5 In [6], h is 2λe but in [15], h is just λe, which leads to additional factors of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 20 For an unpolarized target, the deuteron density matrix is proportional to the identity, ρd = 1 3I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' similarly, if the proton helicity is not detected, ρp = 1 2I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' We then have the unpolarized cross section [6] dσunpol = mpmn|⃗p ′ p| 16π3md σMott frec σ0, σ0 ≡ νLRU L + νTRU T + νTTRU TT + νLTRU LT , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='13) where the RU X are computed with the simple density matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' These are then computable in our model, with the basic computation being the evaluation of ǫ(λγ)µMµ, which differs from the photodisintegration calculation in only two ways: Q2 is not zero and λγ ranges over all three possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The unpolarized response functions RU LT ′ and RU T ′ are identically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' With ρd replaced by 1 3I and the form (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='7) of the polarization vectors taken into account, Rew(0, 1) is just the negative of Rew(0, −1), and w1,1 is equal to w−1,−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Thus, the inputs to RU LT ′ and RU T ′, as given in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='12), immediately cancel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The recent ed → e′pn experiment at JLab [17] does not include polarization but does begin to reach momentum transfers sufficient to consider the RNHA approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Once po- larization data is available, the expressions developed here and in the Appendix can be compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' SUMMARY We have extended the reduced nuclear amplitude approach [1, 2] to helicity amplitudes and applied this model to analysis of elastic electron-deuteron scattering, deuteron photo- disintegration, and deuteron electrodisintegration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' These are just examples of the approach, which is generally applicable to exclusive nuclear processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The primary limitation is that, for any process, the net momentum transfer to every nucleon must be large;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' therefore, as the number of nucleons increases, the required beam energy can increase dramatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The primary gain is precocious scaling in the dependence on momentum transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' What the model (or the original RNA approach) does not provide, though, is an overall normalization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' comparisons must be made in terms of ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' By considering helicity amplitudes, many more quantities can be studied, including po- larization dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' All three of the deuteron’s electromagnetic form factors can be cal- culated and from there various elastic scattering observables can be constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' III we considered the standard structure functions A and B as well as the tensor polarizations t2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Generally, the model implies the need for momentum transfers larger than one would have hoped for seeing simple perturbative QCD scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' However, our results do imply that the deuteron structure function B is a good place to look, above a transfer of 10 GeV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The RNHA results for polarization observables in deuteron photodisintegration, consid- ered in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' IV, are somewhat consistent with experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' In particular, our result for the asymmetry Σ, with a value of Σ(90◦) ≃ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='06, is much better than the value of -1 originally expected [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Higher photon energies would, of course, be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' We have also constructed the RNHA framework for analysis of deuteron electrodisinte- gration, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' This stands ready for comparison with experiment when data is available at sufficient energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' One aspect that does remain is to consider polarization of the outgoing proton, in addition to polarization of the beam and target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Other processes that one might consider include deeply virtual Compton scattering on the deuteron, pion photoproduction on the deuteron [3], and photodisintegration of 3He [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 21 In each case, our approach can provide not only information about helicity amplitudes but also an analysis of nonleading momentum transfer dependence with respect to the onset of perturbative QCD scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' We look forward to experiments at larger momentum transfers for all of these processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work began in conversations with S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Brodsky and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Hwang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Some calculations were checked by W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Miller and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Salveson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Diagrams were drawn with use of JaxoDraw [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Appendix A: Electrodisintegration with polarization If we consider polarization for the beam and the target,6 the proton density matrix is still just ρp = 1 2I, but the deuteron density matrix in the xyz frame is [6] ρd = 1 3 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed 1 + � 3 2T10 + 1 √ 2T20 − � 3 2(T ∗ 11 + T ∗ 21) √ 3T ∗ 22 − � 3 2(T11 + T21) 1 − √ 2T20 − � 3 2(T ∗ 11 − T ∗ 21) √ 3T22 − � 3 2(T11 − T21) 1 − � 3 2T10 + 1 √ 2T20 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (A1) For a target polarization defined relative to the beam direction, rather than the xyz system used above, the tensor polarization coefficients TJM are related to the coefficients ˜TJM defined relative to the beam [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' If only ˜T10 and ˜T20 are nonzero,7 the nonzero TJM are T10 = cos ˜θ ˜T10, T11 = − 1 √ 2 sin ˜θ ˜T10, (A2) T20 = 1 4(1 + 3 cos 2˜θ) ˜T20, T21 = − � 3 8 sin 2˜θ ˜T20, T22 = � 3 32(1 − cos 2˜θ) ˜T20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' The density matrix can then be written as ρd = �1 3I + ˜T10ρdV + ˜T20ρdT � , (A3) where ρdV = 1 3 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed � 3 2 cos ˜θ √ 3 2 sin ˜θ 0 √ 3 2 sin ˜θ 0 √ 3 2 sin ˜θ 0 √ 3 2 sin ˜θ − � 3 2 cos ˜θ \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 (A4) and ρdT = 1 3 \uf8eb \uf8ec \uf8ec \uf8ed 1 4 √ 2(1 + 3 cos 2˜θ) 3 4 sin 2˜θ 3 √ 32(1 − cos 2˜θ) 3 4 sin 2˜θ − 1 2 √ 2(1 + 3 cos 2˜θ) −3 4 sin 2˜θ 3 √ 32(1 − cos 2˜θ) −3 4 sin 2˜θ 1 4 √ 2(1 + 3 cos 2˜θ) \uf8f6 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (A5) 6 For discussion of a polarized outgoing proton, see [7] and [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 7 The spherical tensor moments are related to the Cartesian tensor moments as ˜T10 = � 3 2Pz and ˜T20 = 1 √ 2Pzz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' 22 The response functions can then be separated into unpolarized, vector, and tensor contri- butions as RX = RU X + ˜T10RV X + ˜T20RT X, with RU X, RV X, and RT X computed with ρd replaced by 1 3I, ρdV , and ρdT , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' With dσunpol defined as the unpolarized cross section, given in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='13), the full cross section can be written as dσ = � 1 + ˜T10 � AV d + 2λeAV ed � + ˜T20 � AT d + 2λeAT ed �� dσunpol, (A6) in terms of the single and double asymmetries AV d = � νLRV L + νTRV T + νTTRV TT + νLT RV LT � /σ0, (A7) AV ed = � νLT ′RV LT ′ + νT ′RV T ′ � /σ0, (A8) AT d = � νLRT L + νTRT T + νTTRT TT + νLTRT LT � /σ0, (A9) AT ed = � νLT ′RT LT ′ + νT ′RT T ′ � /σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' (A10) For a recent summary of data, see [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content=' Brodsky and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tA0T4oBgHgl3EQfMv-f/content/2301.02137v1.pdf'} +page_content='T.' metadata={'source': 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+February 1, 2023 +ABSTRACT +We study the algorithmic problem faced by an information holder (seller) who wants to optimally +sell such information to a budged-constrained decision maker (buyer) that has to undertake some +action. Differently from previous works addressing this problem, we consider the case in which +the seller is an interested party, as the action chosen by the buyer does not only influence their +utility, but also seller’s one. This happens in many real-world settings, where the way in which +businesses use acquired information may positively or negatively affect the seller, due to the presence +of externalities on the information market. The utilities of both the seller and the buyer depend on +a random state of nature, which is revealed to the seller, but it is unknown to the buyer. Thus, the +seller’s goal is to (partially) sell their information about the state of nature to the buyer, so as to +concurrently maximize revenue and induce the buyer to take a desirable action. +We study settings in which buyer’s budget and utilities are determined by a random buyer’s type +that is unknown to the seller. In such settings, an optimal protocol for the seller must propose to +the buyer a menu of information-revelation policies to choose from, with the latter acquiring one +of them by paying its corresponding price. Moreover, since in our model the seller is an interested +party, an optimal protocol must also prescribe the seller to pay back the buyer contingently on their +action. +First, we show that the problem of computing a seller-optimal protocol can be solved in polynomial +time. This result relies on a quadratic formulation of the problem, which we solve by means of a +linear programming relaxation. Next, we switch the attention to the case in which a seller’s protocol +employs a single information-revelation policy, rather than proposing a menu. In such a setting, we +show that computing a seller-optimal protocol is APX-hard, even when either the number of actions +or that of states of nature is fixed. We complement such a negative result by providing a quasi- +polynomial-time approximation algorithm that, given any ρ > 0 and ǫ > 0 as input, provides a +multiplicative approximation ρ of the optimal seller’s expected utility, by only suffering a negligible +2−Ω(1/ρ) + ǫ additive loss. Such an algorithm runs in polynomial time whenever either the number +of buyer’s actions or that of states of nature is fixed. In order to derive our results, we draw a +connection between our information-selling problem and principal-agent problems with observable +actions. Finally, we complete the picture of the computational complexity of finding seller-optimal +protocols without menus by providing additional results for the specific setting in which the buyer +has limited liability, and by designing a polynomial-time algorithm for the case in which buyer’s +types are fixed. + +ARXIV PREPRINT - FEBRUARY 1, 2023 +1 +Introduction +Nowadays, there is a terrific amount of information being collected on the Web and other online platforms. Such infor- +mation ranges from consumer preferences, e.g., in e-commerce and streaming websites, to credit reports and location +histories. As a result, recent years have witnessed the born and exponential blowout of markets where specialized +companies sell information that is valuable to other businesses, such as advertisers, retailers, and loan providers. +Very recently, information markets have also received the attention of the algorithmic game theory research commu- +nity. However, while works addressing classical settings such as auctions (Daskalakis and Syrgkanis, 2022), signal- +ing (Dughmi and Xu, 2019), and contract design (Dütting et al., 2019) are now proliferating, only few papers studied +the problem of information selling, with (Babaioff et al., 2012) and (Chen et al., 2020) constituting two notable exam- +ples. +We study the algorithmic problem faced by an information holder (seller) who wants to optimally sell such information +to a budged-constrained decision maker (buyer) that has to undertake some action. Differently from previous works +addressing such a problem (see, e.g., (Chen et al., 2020)), we consider the case in which the seller is an interested +party, as the action chosen by the buyer does not only influence their utility, but also seller’s one. This happens in +many real-world settings, where the way in which businesses use acquired information may positively or negatively +affect the seller, due to the presence of externalities on the information market. The utilities of both the seller and the +buyer depend on a state of nature that is drawn according to a commonly-known probability distribution. The realized +state of nature is revealed to the seller, while it remains unknown to the buyer. Thus, the seller’s goal is to (partially) +sell their information about the state of nature to the buyer, so as to concurrently maximize revenue and induce the +buyer to take a desirable action. +We study settings in which buyer’s budget and utilities are determined by a random buyer’s type that is unknown to the +seller. In such settings, in order to optimally sell information, the seller has to commit upfront to a protocol working +as follows. First, the seller proposes to the buyer a menu of information-revelation policies to choose from, and the +latter acquires an expected-utility-maximizing one according to their (private) type, by paying its corresponding price. +By building on the Bayesian persuasion framework introduced by Kamenica and Gentzkow (2011), an information- +revelation policy is implemented as a signaling scheme, which is a randomized mapping from states of nature to signals +issued to the buyer. Then, the realized state of nature is disclosed to the seller, who reveals information about it to +the buyer according to the acquired signaling scheme. Finally, the buyer selects a best-response action according to +the just acquired information, and the seller pays back the buyer with a payment which depends on both the chosen +action and the signal that has been previously sent by the seller. Our protocol extends the one of Chen et al. (2020) by +adding a final payment from the seller to the buyer. As we show later, this is crucial in order to design seller-optimal +protocols in our setting where the seller is an interested party, since the latter is not only concerned with revenue, but +also with the buyer’s action. Moreover, the addition of payments from the buyer to the seller is also reasonable in many +real-world scenarios. For instance, think of a case in which the information holder asks the buyer to deposit additional +money, and this is given back to them only if the performed action respects some given rules on which the two parties +agreed upfront. +1.1 +Original Contributions +After introducing all the needed concepts in Section 2, we start providing our results in Section 3, where we analyze +the case of general protocols in which the seller proposes a menu of signaling schemes to the buyer. We show that +a seller-optimal protocol can be computed in polynomial time. In order to do that, we first formulate the problem of +finding a seller-optimal protocol as a quadratic problem. Then, we show that one can focus on direct and persuasive +signaling schemes, which are those that send signals corresponding to action recommendations for the buyer and +properly incentivize the latter to follow such recommendations. This in turn allows us to restrict the attention to +protocols that ask the buyer to pay their entire budget upfront and, then, pay back the buyer only if they take the +recommended action. These results allow us to formulate a suitable linear relaxation of the quadratic problem. A +similar technique has been employed in generalized principal-agent problems (Gan et al., 2022), where it is possible +to show that an optimal solution to the linear relaxation can be efficiently cast to an approximately-optimal solution to +the quadratic problem. Indeed, Castiglioni et al. (2022b) show that, even in the special case of hidden-action principal- +agent problems, obtaining an optimal solution to the quadratic problem is not possible in general, since the principal’s +optimization problem may not admit a maximum. Surprisingly, in our information-selling setting, we prove that an +optimal solution to our linear relaxation, which can be computed in polynomial time, can be used to recover a seller- +optimal protocol in polynomial time. As a byproduct, this also shows that, in our setting, the seller’s problem always +admits a maximum. +2 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +In the second part of the paper, we switch the attention to the case of protocols without menus, in which the seller does +not propose a menu of signaling schemes to the buyer, but they rather commit to a single signaling scheme. This is the +case in many real-world applications, where it is unreasonable that a buyer is asked to choose an information-revelation +policy among a range of options. Computing a seller-optimal protocol without menus begets considerable additional +computational challenges, since, intuitively, the seller has no way of extracting information about the buyer’s private +type, as instead it is the case when proposing a menu to choose from. +In Section 4, we draw a connection between the problem of computing a seller-optimal protocol without menus and +principal-agent problems with observable actions. These are problems in which a principal commits to an action- +dependent payment scheme in order to incentivize an agent to take some costly, observable action, in order to maximize +their expected utility. We prove that observable-action principal-agent problems are a special case of our information- +selling problem, and that, in such problems, computing an expected-utility-maximizing payment-scheme for the prin- +cipal is APX-hard. In particular, these results show that our information-selling problem is APX-hard even when the +number of states of nature is fixed and the buyer has limited liability, and, thus, the seller cannot charge a price for +a signaling scheme upfront. We also provide some preliminary technical results on observable-action principal-agent +problems, which are useful in order to prove some of our main claims in the paper, while also being of independent +interest. +In Section 5, we show how to circumvent the APX-hardness for settings in which the seller employs protocols without +menus and the buyer has limited liability. These special settings are of interested on their own, as a similar model has +been recently addressed by Dughmi et al. (2019). We focus on special cases where one of the parameters characterizing +a problem instance is fixed. In particular, we study what happens if we fix the number of buyer’s actions, showing +that the problem admits a PTAS. Moreover, we prove that, when instead the number of states of nature is fixed, there +exists a polynomial-time bi-criteria approximation algorithm that, given any ρ > 0 and ǫ > 0 as input, provides a +multiplicative approximation ρ of the optimal seller’s expected utility, by only suffering a 2−Ω(1/ρ) + ǫ additive loss. +Notice that such a loss is exponentially small in 1 +ρ, and, thus, it is negligible even for reasonably large values of ρ. As +shown by Castiglioni et al. (2022a), such an approximation result is tight for hidden-action principal-agent problems. +It remains an open problem to establish whether such an approximation guarantee is also tight for principal-agent +problems with observable actions, which are a special case of our information-selling problem. +Table 1: Summary of the results provided in the paper. Each cell specifies, on the first line, the computational com- +plexity of finding a seller-optimal protocol, while, additionally, on the second line, it specifies the approximation +guarantees that we can obtain in polynomial time, where OPT denotes the seller’s expected utility in an optimal proto- +col. The approximation guarantees that are shaded in gray can only be obtained by means of a quasi-polynomial-time +algorithm. +general +fixed # actions +fixed # states +fixed # types +Protocols with menus +P +P +P +P +Protocols w/o menus +Buyer w. limited liability +APX-hard +— +APX-hard +P +ρOPT−2−Ω(1/ρ)−ǫ +PTAS +ρOPT−2−Ω(1/ρ)−ǫ +Protocols w/o menus +Buyer w/o limited liability +APX-hard +APX-hard +APX-hard +P +ρOPT−2−Ω(1/ρ)−ǫ ρOPT−2−Ω(1/ρ)−ǫ ρOPT−2−Ω(1/ρ)−ǫ +In conclusion, in Section 6 we study the problem of computing seller-optimal protocols without menus in general +settings in which the buyer does not have limited liability, and, thus, the seller can charge a price for a signaling +scheme. We first prove a stronger negative result, by showing that, in such a setting, the problem of computing a seller- +optimal protocol is APX-hard even if the number of buyer’s actions is fixed. Then, we show how to circumvent such +a negative result by providing a quasi-polynomial-time bi-criteria approximation algorithm that, given any ρ > 0 and +ǫ > 0 as input, provides a multiplicative approximation ρ of the optimal seller’s expected utility, plus a2−Ω(1/ρ) + ǫ +additive loss. We prove that, when either the number of buyer’s action or that of states of nature is fixed, such an +algorithm runs in polynomial time. Finally, we show that, when the number of buyer’s types is fixed, the problem +admits a polynomial-time algorithm. This also implies that the seller’s optimization problem for protocols without +menus always admits a maximum. +We summarize the results provided in this paper in Table 1. All the proofs are in the Appendix. +3 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +1.2 +Related Works +The study of algorithmic ways of selling information to an imperfectly-informed buyer has received some attention +in the past. Babaioff et al. (2012) initiated the study by considering a buyer with an unlimited budget. They provide +an exponentially-sized linear program (LP) for computing an optimal mechanism for selling information, and they +efficiently solve it through the ellipsoid method. The main drawback of the approach presented by Babaioff et al. +(2012) is that an optimal mechanism may require a significant money transfer from the buyer to the seller and viceversa, +in order to only achieve a small, overall net transfer. Chen et al. (2020) complement the results in (Babaioff et al., 2012) +by studying the problem of selling information when both the buyer and the seller are budget-constrained. Moreover, +they also consider a setting in which the buyer’s budget is private, and the seller needs to elicit it in the mechanism. +Chen et al. (2020) show that the addition of budget constraints considerably simplifies the problem of computing an +optimal mechanism, since it can be formulated as a polynomially-sized LP. +The problem of selling information has also been addressed by Bergemann et al. (2018), who study the case of bi- +nary actions and states of nature, characterizing a revenue-maximizing mechanism in such a setting. Furthermore, +Bergemann et al. (2022) extend the analysis to the case in which there are more than two actions and binary states +of nature. In contrast, Liu et al. (2021) study a revenue-maximizing mechanism for selling information when the +stochasticity of the state of nature only affects a subset of the actions of the decision maker. +Our problem is also related to the Bayesian persuasion framework originally introduced by Kamenica and Gentzkow +(2011), where an informed sender wants to influence the behavior of a self-interested receiver via the strategic provision +of information. Dughmi et al. (2019) generalize the classical framework by considering the case in which there are +monetary transfers between the sender and the receiver. Our information-selling setting in which the buyer has limited +liability generalizes the model of Dughmi et al. (2019) by also introducing buyer’s types. +Finally, let us remark that our information-selling problem shares critical features with Bayesian principal-agent prob- +lems (see, e.g., (Castiglioni et al., 2022a; Alon et al., 2021, 2022; Guruganesh et al., 2021; Castiglioni et al., 2022c) +for some references). Indeed, as we show in Section 4, the problem of computing a seller-optimal protocol gener- +alizes particular principal-agent problems in which the agent’s action is observable. Such a connection between the +two settings is also demonstrated in terms of results. In particular, notice that Castiglioni et al. (2022a) design bi- +criteria approximation algorithms whose guarantees are similar to those provided in this paper. Moreover, Gan et al. +(2022) show how to find optimal protocols in generalized principal-agent problems by using a linear relaxation of the +principal’s optimization problem, which is quadratic. +2 +Preliminaries +We study the problem faced by an information holder (seller) selling information to a budget-constrained decision +maker (buyer). The information available to the seller is collectively termed state of nature and encoded as an element +of a finite set Θ := {θi}d +i=1 of d possible states, while the set of the m actions available to the buyer is A := {ai}m +i=1. +The buyer is also characterized by a private type, which is unknown to the seller and belongs to a finite set K := {ki}n +i=1 +of n possible types. Each buyer’s type k ∈ K is characterized by a utility function uk +θ : A → [0, 1] associated to each +state θ ∈ Θ and a budget bk ∈ R+ representing how much they can afford to pay. In our model, the seller’s utility +is not only determined by how much the buyer pays for acquiring information, but it also depends on the buyer’s +action. Specifically, for every state θ ∈ Θ, the sender gets an additional utility contribution determined by a function +us +θ : A → [0, 1]. We assume that both the seller and the buyer know the probability distribution µ ∈ ∆Θ according to +which the state of nature is drawn, as well as the probability distribution λ ∈ ∆K determining the buyer’s type.1 We +let µθ be the probability assigned to state θ ∈ Θ, while λk is the probability of type k ∈ K. +As in (Chen et al., 2020), we assume w.l.o.g. that information revelation happens only once during the seller-buyer +interaction. Thus, as it is the case in Bayesian persuasion Kamenica and Gentzkow (2011), the seller reveals infor- +mation to the buyer by committing to a signaling scheme φ, which is a randomized mapping from states of nature to +signals being issued to the buyer. Formally, φ : Θ → ∆S, where S is a finite set of signals. We denote by φθ ∈ ∆S +the probability distribution employed when the state of nature is θ ∈ Θ, with φθ(s) being the probability of sending +s ∈ S. +2.1 +Protocols with Menus +An information-selling protocol for the seller is defined as follows. The seller first proposes a menu of signaling +schemes to the buyer, with each signaling scheme being assigned with a price. Then, the buyer chooses a signaling +1In this work, given a finite set X, we let ∆X be the set of all the probability distributions defined over the elements of X. +4 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +scheme and pays its price upfront, before information is revealed.2 The seller also commits to action-dependent +payments, which are made by the seller in favor of the buyer after information is revealed and the latter has taken +an action. This is in contrast with what happens in the protocol introduced by Chen et al. (2020), where there are +no action-dependent money transfers. Intuitively, such payments are needed in order to incentivize the agent to play +an action that is profitable for the seller, and, thus, they are not needed in the setting of Chen et al. (2020) where the +seller’s utility function is only determined by how much the buyer pays for acquiring information. Formally, we define +a seller’s protocol as follows: +Definition 1 (Seller’s protocol). A protocol for the seller is a tuple {(φk, pk, πk)}k∈K, where: +• {φk}k∈K is a menu of signaling schemes φk : Θ → ∆S, one for each receiver’s type k ∈ K; +• {pk}k∈K is a menu of prices, with pk ∈ R+ representing how much the seller charges the buyer for selecting +the signaling scheme φk;3 +• {πk}k∈K is a menu of payment functions, which are defined as πk : S × A → R+ with πk(s, a) encoding +how much the seller pays the buyer whenever the latter plays action a ∈ A after selecting the signaling +scheme φk and receiving signal s ∈ S.4 +The seller and the buyer interact as follows: (i) the seller commits to a protocol {(φk, pk, πk)}k∈K; (ii) the buyer +selects a signaling scheme φk and pays pk to the seller (with k ∈ K possibly different from their true type); (iii) the +seller observes the realized state of nature θ ∼ µ, draws a signal s ∼ φk +θ according to the selected signaling scheme, +and communicates s to the buyer; (iv) given the signal s, the buyer infers a posterior distribution ξs ∈ ∆Θ over states +of nature, where the probability ξs +θ of state θ ∈ Θ is computed with the Bayes rule, as follows: +ξs +θ := +µθ φk +θ(s) +� +θ′∈Θ µθ′φk +θ′(s); +(v) given the posterior ξs, the buyer selects an action a ∈ A; and (vi) the seller pays πk(s, a) to the buyer. As in the +model by Chen et al. (2020), we assume that the seller is committed to following the protocol, while the buyer is not, +i.e., the buyer is free of leaving the interaction at any point. +In step (v), after observing a signal s ∈ S and computing the posterior ξs, the buyer plays a best response by choosing +an action a ∈ A maximizing their expected utility. Formally: +Definition 2 (ǫ-Best-response). Let ǫ ≥ 0. Given a signal s ∈ S, the induced posterior ξs ∈ ∆Θ, and a payment +function π : S × A → R+, the ǫ-best-response set of a buyer of type k ∈ K is: +Bk,ǫ +ξs,π := +� +a ∈ A : +� +θ∈Θ +ξs +θ uk +θ(a) + π(s, a) ≥ max +a′∈A +� +θ∈Θ +ξs +θ uk +θ(a′) + π(s, a′) − ǫ +� +. +We let bk,ǫ +ξs,π ∈ Bk,ǫ +ξs,π be an ǫ-best response played by the buyer. The best-response set Bk +ξs,π of a buyer of type k ∈ K +is defined for ǫ = 0, while bk +ξs,π ∈ Bk +ξs,π is a best response played by the buyer.5 +In the following, we will oftentimes work in the space of the distributions over posteriors. In that case, given a posterior +ξ ∈ ∆Θ, we abuse notation and write Bk +ξ,π, Bk,ǫ +ξ,π, bk,ǫ +ξ,π, and bk +ξ,π. +The seller’s goal is to implement an optimal (i.e., utility-maximizing) protocol {(φk, pk, πk)}k∈K. We focus on seller’s +protocols that are incentive compatible (IC) and individually rational (IR).6 Specifically, a seller’s protocol is IC if for +every pair of buyer’s types k, k′ ∈ K: +� +s∈S +� +θ∈Θ +µθφk +θ(s) +� +uk +θ(bk +ξs,πk) + πk(s, bk +ξs,πk) +� +− pk ≥ +� +s∈S +max +a∈A +� +θ∈Θ +µθφk′ +θ (s) +� +uk +θ(a) + πk′(s, a) +� +− pk′, +2Notice that proposing a menu of signaling schemes is equivalent to asking the buyer to report their type and then choosing a +signaling scheme based on that, as it is the case in (Chen et al., 2020). +3Assuming pk ≥ 0 is w.l.o.g., since, intuitively, the seller is never better off paying the buyer before they played any action. +4The assumption that πk(s, a) ≥ 0 is w.l.o.g., since the buyer does not commit to following the protocol, and, thus, πk(s, a) < 0 +would result in the buyer leaving the protocol without paying after taking an action. +5When the buyer is indifferent among multiple best responses (respectively, ǫ-best responses), we always assume that they break +ties in favor of the seller, choosing an action in Bk +ξs,π (respectively, Bk,ǫ +ξs,π) maximizing the seller’s expected utility. +6By a revelation-principle-style argument (see (Shoham and Leyton-Brown, 2008) for some examples), focusing on IC and IR +protocols is w.l.o.g. when looking for an optimal protocol. +5 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +while it is IR if for every buyers’ type k ∈ K: +� +s∈S +� +θ∈Θ +µθφk +θ(s) +� +uk +θ(bk +ξs,πk) + πk(s, bk +ξs,πk) +� +− pk ≥ max +a∈A +� +θ∈Θ +µθuk +θ(a). +Intuitively, an IC protocol incentivizes the buyer to select the signaling scheme φk corresponding ot their true type +k ∈ K, while an IR protocol ensures that the buyer gets more utility by acquiring information rather than leaving the +protocol before step (ii) and playing an action without information. Then, the seller’s expected utility is computed as +follows: +� +k∈K +λk +�� +s∈S +� +θ∈Θ +µθφk +θ(s) +� +us +θ(bk +ξs,πk) − πk(s, bk +ξs,πk) +� ++ pk +� +. +A crucial component of our results is that we can restrict the attention to protocols that are direct and persuasive. We +say that protocol is direct if it uses signaling schemes whose signals correspond to action recommendations for the +buyer, namely S = A, while a direct protocol is said to be persuasive whenever playing the recommended action is +always a best response for the buyer. +2.2 +Protocols without Menus +In the second part of the paper, we study the case of seller’s protocols without menus, in which the seller does not +propose a menu of signaling schemes to the buyer, but they rather commit to a single signaling scheme and a single +payment function.7 This allows us to simplify the definition of a protocol (see Definition 1), by denoting a seller’s +protocol without menus as a tuple (φ, p, π), where φ : Θ → ∆S is a signaling scheme, p ∈ R+ is a price for +such a signaling scheme, representing how much the seller charges the buyer to reveal information to them, and +π : S × A → R+ is a payment function. The seller-buyer interaction unfolds as in the general case with menus, but, +in this case, step (ii) only involves the payment of price p ∈ R+ on buyer’s part. +Some of our results on protocols without menus address the special case in which the buyer has limited liability, +which means that the buyer has no budget, and, thus, the seller cannot charge a price for a signaling scheme upfront. +Formally, this amounts to asking that bk = 0 for all k ∈ K. Notice that, while such a special case may seem of +scarce appeal for the problem of selling information, it is indeed interesting on its own, as it is similar to the model +studied by Dughmi et al. (2019). Indeed, our model can be seen as a generalization of the one in (Dughmi et al., 2019), +which adds buyer’s private types. Moreover, in the general case in which the buyer has no limited liability, our model +additionally builds on top of that of Dughmi et al. (2019) by adding the possibility for the seller to ask the buyer a +payments before information is revealed. +For protocols without menus, IC constraints are not needed anymore, while IR constraints are still required in order +to ensure that the buyer is incentivized to acquire information from the principal. Given a protocol without menus +(φ, p, π), only some of the buyer’s types are actually incentivized to participate in the protocol, i.e., all the types whose +corresponding IR constraint is satisfied. Formally, a protocol determines a subset Rφ,p,π ⊆ K of buyer’s types such +that, for every k ∈ Rφ,p,π, it holds that: (i) a buyer of type k has enough budget to buy information, namely bk ≥ p; +and (ii) the IR constraint is satisfied for a buyer of type k.8 In particular, point (ii) can be formally stated by saying +that the following condition is satisfied for every k ∈ Rφ,p,π: +� +s∈S +� +θ∈Θ +µθφθ(s) +� +uk +θ(bk +ξs,π) + π(s, bk +ξs,π) +� +− p ≥ max +a∈A +� +θ∈Θ +µθuk +θ(a). +Moreover, given a protocol without menus (φ, π, p), the seller’s expected utility is given by: +� +k∈Rφ,p,π +λk +�� +s∈S +� +θ∈Θ +µθφθ(s) +� +us +θ(bk +ξs,π) − π(s, bk +ξs,π) +� ++ p +� ++ +� +k̸∈Rφ,p,π +λk +� +θ∈Θ +µθuk +θ(bk +µ), +where bk +ξ ∈ arg maxa∈A +� +θ∈Θ ξθuk +θ(a) is a best response for a buyer’s type k ∈ K that only considers the posterior +ξ ∈ ∆Θ, where, as customary, ties are broken in favor of the seller. Notice that a buyer’s type k /∈ Rφ,p,π is among +those who decide to do not acquire information from the seller, and, thus, they play a best response to the probability +distribution µ (instead of a posterior). +7From the point of view of Chen et al. (2020), this is equivalent to assuming that there is no type reporting stage. +8Whenever the expected utility of a buyer’s type is the same by participating in the protocol as not doing that, we assume that +they take the option maximizing the seller’s expected utility. +6 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +Finally, when dealing with protocols without menus, it will be useful to directly work with distributions over posteriors +induced by signaling schemes, rather than with signaling schemes (Kamenica and Gentzkow, 2011). A signaling +scheme φ : Θ → ∆S induces a distribution γ over ∆Θ, which has a support supp(γ) := {ξs | s ∈ S} and satisfies the +following conditions: +� +ξ∈supp(γ) +γξ ξθ = µθ +∀θ ∈ Θ, +(1) +where γξ ∈ [0, 1] is the probability that γ assigns to the posterior ξ ∈ supp(γ). Thus, instead of working with signaling +schemes φ, one can w.l.o.g. work with distributions γ over ∆Θ that are consistent with the probability distribution µ, +i.e., they satisfy the condition in Equation (1). +When working with distributions over posteriors γ rather than with signaling schemes φ, with a slight abuse of notation, +we denote a seller’s protocol without menus as (γ, p, π), by identifying a signaling scheme with its induced distribution +over posteriors γ. Similarly, we slightly abuse notation in payment functions, by assuming that they are defined over +posteriors rather than signals. Formally, we let π : ∆Θ × A → R+, with π(ξ, a) denoting how much the buyer pays +back the seller when the induced posterior is ξ ∈ ∆Θ and they play action a ∈ A. +3 +Computing a Seller-optimal Protocol with Menus +We begin by studying the problem of computing a seller-optimal protocol in which the seller has the ability of propos- +ing a menu of signaling schemes and payment functions to the buyer. Formally, the problem of computing an optimal +IC and IR protocol with menus can be formulated as follows: +sup +φk +θ(s)≥0 +pk≥0 +πk(s,a)≥0 +� +k∈K +λk +�� +s∈S +� +θ∈Θ +µθφk +θ(s) +� +us +θ(bk +ξs,πk) − πk(s, bk +ξs,πk) +� ++ pk +� +s.t. +(2a) +� +s∈S +� +θ∈Θ +µθφk +θ(s) +� +uk +θ(bk +ξs,πk) + πk(s, bk +ξs,πk) +� +− pk +≥ +� +s∈S +max +a∈A +� +θ∈Θ +µθφk′ +θ (s) +� +uk +θ(a) + πk′(s, a) +� +− pk′ +∀k ∈ K, ∀k′ ∈ K +(2b) +� +s∈S +� +θ∈Θ +µθφk +θ(s) +� +uk +θ(bk +ξs,πk) + πk(s, bk +ξs,πk) +� +− pk ≥ max +a∈A +� +θ∈Θ +µθuk +θ(a) +∀k ∈ K +(2c) +� +s∈S +φk +θ(s) = 1 +∀k ∈ K, ∀θ ∈ Θ. +(2d) +Notice that Problem (2) is defined in terms of sup rather than max since, as it is the case in principal-agent problems +(see, e.g., (Castiglioni et al., 2022b; Gan et al., 2022)), it is not in general immediate to establish whether the seller’s +optimization problem always admits a maximum or not. Indeed, in the following we show that our problem always +admits a maximum. +As a first step, we prove that we can focus w.l.o.g on protocols which are direct and persuasive. +Lemma 1. Given any IC and IR seller’s protocol, it is always possible to recover an IC and IR seller’s protocol that +is direct and persuasive, and it provides the seller with the same expected utility. +Intuitively, Lemma 1 follows from the fact that, given any signaling scheme φk and price function πk corresponding +to some type k ∈ K, if two signals induce the same best response for a buyer of type k, then it is possible to merge +the two signals in a single one, recovering a new signaling scheme and a new price function for type k that achieve +the same seller’s expected utility. By doing such a procedure for every buyer’s type until there are no two signals +inducing the same best response for that type, we obtain a protocol that is direct and persuasive, and it has the same +seller’s expected utility as the original protocol. Notice that, since in direct protocols it holds § = A, whenever we +write πk(a, a′) for a, a′ ∈ A, the first action a is the seller’s recommendation (signal), while the second action a′ is +the one actually played by the buyer. +As a second crucial step, we exploit Lemma 1 in order to show that, given an IC and IR protocol that is direct and +persuasive, there exists another IC and IR protocol which is still direct and persuasive, it achieves the same seller’s +expected utility, and it is such that: (i) for every k ∈ K, the price pk of φk is equal to entire budget bk of a buyer +of type k, and (ii) the buyer is not paid back (i.e., they get a null payment) if they deviate from the seller’s action +recommendation. Formally: +7 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +Lemma 2. Given an IC and IR protocol {(φk, pk, πk)}k∈K that is direct and persuasive, it is always possible to +recover an IC and IR protocol {(φk, ˜pk, ˜πk)}k∈K such that: it is direct and persuasive, it provides the same seller’s +expected utility as the original protocol, and, for every buyers’ type k ∈ K, it satisfies ˜pk = bk and ˜πk(a, a′) = 0 for +all a ̸= a′ ∈ A. +As a direct consequence of Lemma 2, we can compactly denote πk(a, a) as πk(a) for every a ∈ A, since we can focus +w.l.o.g. on payment functions such that πk(a, a′) = 0 for all a ̸= a′. +We are now ready to introduce an LP with polynomially-many variables and constraints that is a linear relaxation of +Problem (2). In order to formulate the LP, we exploit Lemmas 1 and 2 to restrict the attention to direct and persuasive +protocols, prices such that pk = bk for every k ∈ K, and payments such that πk(a, a′) = 0 for every k ∈ K and +a ̸= a′ ∈ A. Moreover, we encode the terms � +θ∈Θ µθφk +θ(a)πk(a) as single variables lk(a). Then, the LP reads as +follows: +max +φk +θ (a)≥0 +lk(a)≥0 +yk,k′,a≥0 +� +k∈K +λk +� +a∈A +�� +θ∈Θ +µθφk +θ(a)us +θ(a) − lk(a) +� ++ bk +s.t. +(3a) +� +a∈A +�� +θ∈Θ +µθφk +θ(a)uk +θ(a) + lk(a) +� +− bk ≥ +� +a∈A +yk,k′,a − bk′ +∀k ∈ K, ∀k′ ∈ K +(3b) +yk,k′,a ≥ +� +θ∈Θ +µθφk′ +θ (a)uk +θ(a) + lk′(a) +∀k ∈ K, ∀k′ ∈ K, ∀a ∈ A +(3c) +yk,k′,a ≥ +� +θ∈Θ +µθφk′ +θ (a)uk +θ(a′) +∀k ∈ K, ∀k′ ∈ K, ∀a ̸= a′ ∈ A +(3d) +� +a∈A +�� +θ∈Θ +µθφk +θ(a)uk +θ(a) + lk(a) +� +− bk ≥ +� +θ∈Θ +µθuk +θ(a′) +∀k ∈ K, ∀a′ ∈ A +(3e) +� +θ∈Θ +µθφk +θ(a)uk +θ(a) + lk(a) ≥ +� +θ∈Θ +µθφk +θ(a)uk +θ(a′) +∀k ∈ K, ∀a ̸= a′ ∈ A +(3f) +� +a∈A +φk +θ(a) = 1 +∀k ∈ K, ∀θ ∈ Θ. +(3g) +In LP (3), Constraints (3b)–(3d) ensure that the protocol is IC, Constraints (3e) enforce that it is IR, while Con- +straints (3f) guarantee that the protocol is persuasive. +Given how LP (3) is obtained from Problem (2), it is not immediately clear how, given a feasible solution to LP (3), +one can recover a protocol that is a solution to Problem (2) with seller’s expected utility equal to the value of the +solution to LP (3). Indeed, in a solution to LP (3), a variable lk(a) could be strictly positive even when the variables +φk +θ(a) are equal to zero. In such a case, it is not possible to immediately recover a value for πk(a) starting from a +solution to LP (3), since lk(a) encodes � +θ∈Θ µθφk +θ(a)πk(a), from which computing πk(a) would require a division +by zero. +In the following, we show how, given an optimal solution to LP (3), it is indeed possible to build in polynomial time a +seller-optimal protocol with menus. First, we prove a preliminary result: +Lemma 3. The optimal value of LP (3) is at least as large as the supremum in Problem (2). +Then, we show that, given a solution to LP (3), it is possible to recover in polynomial time an IC and IR protocol with +at least the same value. Formally: +Lemma 4. Given a feasible solution to LP (3), it is possible to recover in polynomial time an IC and IR protocol +whose seller’s expected utility is greater than or equal to the value of the solution to LP (3). +Intuitively, Lemma 4 is proved by showing that, given a feasible solution to LP (3), it is possible to efficiently construct +a new solution in which, whenever some variable lk(a) > 0, then there exists at least one state of nature θ ∈ Θ for +which φk +θ(a) > 0, i.e., action a is recommended with strictly positive probability. Moreover, such a procedure does +not detriment the objective function value and retains the IC and IR conditions. Then, from the new solution, one can +recover a protocol that is a valid solution to Problem (2), by letting πk(a) = lk(a)/ � +θ∈Θ µθφk +θ(a) for all k ∈ K and +a ∈ A. +8 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +Finally, by exploiting Lemmas 3 and 4, we can design a polynomial-time algorithm that finds a seller-optimal protocol +with menus. Indeed, the algorithm can simply optimally solve LP (3) (in polynomial time), and use Lemma 4 to +recover an IC and IR protocol having at least the same value. Tanks to Lemma 3, such a protocol is optimal for the +seller. +Theorem 1. There exists a polynomial-time algorithm that computes a protocol with menus that maximizes the seller’s +expected utility. +Theorem 1 also shows as a byproduct that Problem (2) always admits a maximum. +Let us remark that the idea of formulating a linear relaxation of a quadratic problem by introducing a new variable has +already been used in generalized principal-agent problems by Gan et al. (2022). However, in such a setting, the linear +relaxation cannot be used to solve the principal’s optimization problem exactly, but only to recover a desirable approx- +imation of an optimal solution. This is because the problem may not admit a maximum, as shown by Castiglioni et al. +(2022b) even in the special case of hidden-action principal-agent problems. Surprisingly, in our information-selling +setting, the linear relaxation can be used to find an (exact) optimal solution. Intuitively, this is possible since, in our +setting, the seller observes the action undertaken by the buyer, while in hidden-action principal-agent problems the +principal does not directly observe the agent’s action. +4 +Drawing a Connection with Principal-agent Problems +In this section, we show that our information-selling problem is intimately related to a particular class of principal- +agent problems. Specifically, we show that the problem of computing a seller-optimal protocol without menus is a +generalization of the problem of computing an optimal contract in principal-agent problems in which the principal +observes the action undertaken by the agent. +In Section 4.1, we formally introduce principal-agent problems with observable actions. Then, in Section 4.2, we show +how such problems are related to our information-selling problem, and we prove an hardness result for them which +carries over to our problem Finally, in Section 4.3, we provide some preliminary technical results that will be useful +in the following sections. +4.1 +Principal-agent Problem with Observable Actions +We start by formally defining an instance of (Bayesian) observable-action principal-agent problem.9 For ease of ex- +position, we reuse some of the notation already introduced in Section 2, in order to denote elements that in observable- +action principal-agent problems have the same role as in our information-selling setting. The agent has a finite set K +of possible types, and a type k ∈ K is drawn with probability λk according to a known distribution λ ∈ ∆K. Each +agent’s type k ∈ K has a set A of actions, with each action having a type-dependent cost ck +a ∈ [0, 1]. The principal is +characterized by a reward ra ∈ [0, 1] for every agent’s action a ∈ A. Moreover, the principal can commit to a contract, +which can be encoded by a function π : A → R+ defining a payment π(a) from the principal to the agent for every +possible agent’s action a ∈ A. Given a contract, an agent of type k ∈ K plays a best response bk +π ∈ A, defined as +bk +π ∈ arg maxa∈A +� +π(a) − ck +a +� +, where, as usual, we assume that ties are broken in favor of the principal. Finally, the +principal’s goal is to commit to a contract maximizing their expected utility, which is defined as � +k∈K λk[rbkπ −π(bk +π)]. +4.2 +From Selling Information to Observable-action Principal-agent Problems +Next, we show that our information-selling problem in the case in which protocols are without menus and the buyer has +limited liability (i.e., bk = 0 for all k ∈ K) is strongly related to the problem of finding an optimal (i.e., expected-utility- +maximizing) contract in observable-action principal-agent problems. Specifically, we show that, given a posterior +ξ ∈ ∆Θ, designing a payment function π : ∆Θ × A → R+ that maximizes the seller’s expected utility conditioned on +the fact that the induced posterior is ξ is equivalent to finding an optimal contract in a suitably-defined principal-agent +problem with observable actions. Formally, for ease of presentation, we introduce the following notion of payment +function that is optimal for the seller in a given posterior: +9Notice that observable-action principal-agent problems are a special case of Bayesian hidden-action principal-agent problems. +Indeed, this can be easily seen by taking an instance of the hidden-action problem in which outcomes correspond one-to-one with +agent’s actions, and each action deterministically determines its corresponding outcome. +9 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +Definition 3 (Optimal payment function in a posterior). Given a posterior ξ ∈ ∆Θ, we say that a payment function +π : ∆Θ × A → R+ is optimal in ξ if the following holds: +π ∈ argmax +π′ +� +k∈K +λk +�� +θ∈Θ +ξθ us +θ(bk +ξ,π′) − π(ξ, bk +ξ,π′) +� +. +(4) +Notice that, in Problem (4), the price p of the signaling scheme φ does not appear in the seller’s expected utility, since +we are restricted to settings in which the buyer has limited liability, and, thus, it is always the case that p = 0. For the +same reason, we can safely assume that all the buyer’s types satisfy IR constraints. Then, we can state the following +crucial result: +Lemma 5. Given a posterior ξ ∈ ∆Θ, solving Problem (4) is equivalent to computing a contract maximizing the +principal’s expected utility in an instance of observable-action principal-agent problem such that, for every agent’s +type k ∈ K and action a ∈ A, the following holds: +ck +a = +� +θ∈Θ +ξθ +� +uk +θ(bk +ξ) − uk +θ(a) +� +and +ra = +� +θ∈Θ +ξθ us +θ(a). +Moreover, finding an optimal contract in any instance of observable-action principal-agent problem can be reduced in +polynomial time to computing a seller-optimal protocol without menus in a problem instance in which the buyer has +limited liability and there is only one state of nature. +The first statement in Lemma 5 implies that, given an instance of our information-selling problem in which the buyer +has limited liability and there is only one state of nature, it is possible to compute a seller-optimal protocol without +menus by finding an optimal contract in an instance of observable-action principal-agent problem defined as in the +lemma (notice that such an instance can be easily built in polynomial time). Thus, by Lemma 5, we can easily prove +the following: +Theorem 2. Restricted to instances in which the buyer has limited liability and there is only one state of nature, +computing a seller-optimal protocol without menus is equivalent to the problem of finding an optimal contract in +general instances of the observable-action principal-agent problem. +While the computational complexity of finding optimal contracts in hidden-action principal-agent problems is well +understood (see, e.g., (Castiglioni et al., 2022a)), to the best of our knowledge, there are no results on problems with +observable actions. In following theorem, we prove a strong hardness result for them: there exists a constant α < 1 +such that designing a contract which provides the principal with at least an α fraction of the expected utility in an +optimal contract is computationally intractable. Formally: +Theorem 3. In observable-action principal-agent problems, the problem of computing a contract maximizing the +principal’s expected utility is APX-hard. +Then, Theorem 2 immediately gives the following result: +Corollary 1. The problem of computing a seller-optimal protocol without menus is APX-hard, even when the buyer +has limited liability and the number of states of nature d is fixed. +As we show in the following sections (see Theorems 8 and 11), whenever the number of states of nature is fixed, +the problem of computing a seller-optimal protocol without menus admits a polynomial-time algorithm providing a +particular bi-criteria approximation of the seller’s expected utility in an optimal protocol. Such an approximation is +similar to the the bi-criteria guarantees provided by Castiglioni et al. (2022a) for Bayesian hidden-action principal- +agent problems. By Theorem 2, our polynomial-time bi-criteria approximation algorithm for the setting in which the +buyer has limited liability (Theorem 8) can be easily adapted to work with observable-action principal-agent problems. +Theorem 7 in (Castiglioni et al., 2022a) shows that, for hidden-action problems, such bi-criteria approximations are +tight. We leave as an open problem to establish whether these are also tight in our observable-action principal-agent +problems or one can obtain better guarantees in polynomial time for our specific case. +4.3 +Additional Preliminary Technical Results +We conclude the section by recalling two already-known results on hidden-action principal-agent problems. Clearly, +these also hold for the specific case of observable-action principal-agent problems. By Theorem 2, such results can be +easily cast to our information-selling problem. Indeed, we also show that one of them can be strengthen in our setting. +The first result that we are going to introduce makes use of linear contracts, which are payment schemes that pay the +agent a given fraction of the principal’s reward. Formally, in observable-action principal-agent problems, a contract +10 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +π : A → R+ is said to be linear if there exists a β ∈ [0, 1] such that π(a) = β ra for all a ∈ A. Despite their simplicity, +linear contracts provide good approximations with respect to general ones. In particular, the following holds: +Theorem 4 (Essentially Theorem 3 by Castiglioni et al. (2022a)). In an observable-action principal-agent problem, +for any ρ ∈ (0, 1/2], there exists a linear contract π : A → R+ such that: +� +k∈K +λk +� +rbkπ − π(bk +π) +� +≥ ρ max +π′ +� +k∈K +λk +� +rbk +π′ − π′(bk +π′) +� +− 2Ω(1/ρ). +Moreover, such a linear contract is defined by a parameter β = 1 − 2−i, for some i ∈ {1, . . ., ⌊1/2ρ⌋}. +We will make use of a stronger version of Theorem 4, which applies to our setting and directly follows from the +analysis of Castiglioni et al. (2022a) and Lemma 5. Formally: +Corollary 2. Given a posterior ξ ∈ ∆Θ, for any ρ ∈ (0, 1/2], there exists a payment function π : ∆Θ × A → R+ +such that π(ξ, a) = β � +θ∈Θ ξθ us +θ(a) for every a ∈ A, where β ∈ [0, 1] is an (action-independent) parameter, and, +additionally, the following holds: +� +k∈K +λk +� +θ∈Θ +ξθ +� +us +θ(bk +ξ,π) − π(s, bk +ξ,π) +� +≥ ρ +� +k∈K +λk max +a∈A +� +θ∈Θ +ξθ +� +us +θ(a) + uk +θ(a) − uk +θ(bk +ξ) +� +− 2Ω(1/ρ). +Moreover, such a parameter β is equal to 1 − 2−i for some i ∈ {1, . . . , ⌊1/2ρ⌋}. +Finally, we recall a useful result that establishes a connection between agent’s best responses and approximate best +responses in principal-agent problems. Intuitively, such a result states that, given a contract under which the agent is +allowed to play an ǫ-best response (for some ǫ ≥ 0), it is always possible to recover a new contract in which the agent +must play an (exact) best response, by only incurring in a small loss in the principal’s expected utility. Formally, given +ǫ ≥ 0 and a contract π : A → R+, for every k ∈ K, we let Bk,ǫ +π +⊆ A be the set of ǫ-best-response actions for an agent +of type k. Such a set is made by all the actions a ∈ A such that π(a) − ck +a ≥ maxa′∈A +� +π(a′) − ck +a′ +� +− ǫ. We denote +by bk,ǫ +π +∈ Bk,ǫ +π +an ǫ-best-response action that is actually played by an agent of type k, assuming that ties are broken in +favor of the principal, as usual. Then: +Theorem 5 (Essentially Proposition A.4 by Dutting et al. (2021)). Given ǫ ≥ 0, an instance of observable-action +principal-agent problem and, and a contract π : A → R+, there exists a contract π′ : A → R+ such that π′(a) = +(1 − √ǫ) π(a) + √ǫ ra for every a ∈ A, and the following holds: +� +k∈K +λk +� +rbk +π′ − π′(bk +π′) +� +≥ +� +k∈K +λk +� +rbk,ǫ +π +− π(bk,ǫ +π ) +� +− 2√ǫ. +Theorem 5 can be easily cast to our setting by means of Lemma 5. Formally: +Corollary 3. Given ǫ ≥ 0, a posterior ξ ∈ ∆Θ, and a payment a function π : ∆Θ × A → R+, there exists a payment +function π′ : ∆Θ × A → R+ such that π′(ξ, a) = (1 − √ǫ) π(ξ, a) + √ǫ � +θ∈Θ ξθus +θ(a) for every a ∈ A, and the +following holds: +� +k∈K +λk +�� +θ∈Θ +ξθ us +θ(bk +ξ,π′) − π′(ξ, bk +ξ,π′) +� +≥ +� +k∈K +λk +�� +θ∈Θ +ξθ us +θ(bk,ǫ +ξ,π) − π(ξ, bk,ǫ +ξ,π) +� +− 2√ǫ. +Corollary 3 will be crucial to provide our results in the following sections. +5 +Computing a Seller-optimal Protocol without Menus: +The Case of a Buyer with Limited Liability +In this section, we study the problem of computing a seller-optimal protocol without menus when the buyer has limited +liability, i.e., each buyer’s types k ∈ K has budget bk = 0. As remarked in Section 2.2, such a setting is of interest +on its own, since it is a generalization of the one addressed by Dughmi et al. (2019). Moreover, the technical results +derived in this section will be useful to deal with the general problem in which the buyer has no limited liability. We +show how to circumvent the APX-hardness result that we established in Corollary 1, first, in Section 5.1, by fixing the +number of buyer’s actions, and then, in Section 5.2, by fixing the number of states of nature. +In this section, since the buyer has limited liability, we can assume w.l.o.g. that p = 0, so that we can compactly +denote a protocol with a pair (γ, π), rather than with (γ, p, π). +11 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +5.1 +Fixing the Number of Buyer’s Actions +First, we show that, whenever the buyer has limited liability and the number of buyer’s actions m is fixed, the problem +of computing a seller-optimal protocol without menus admits a PTAS, i.e., we can design a protocol whose seller’s +expected utility is arbitrarily close to that of an optimal protocol in time polynomial in the instance size. +In order to design our PTAS, we start by observing that, since p = 0, the IR constraints are satisfied by all the protocols +(γ, π). This allows us to formulate the problem of computing a seller-optimal protocol without menus as the following +optimization problem:10 +max +γξ≥0 +π(ξ,a)≥0 +� +k∈K +λk +� +ξ∈supp(γ) +γξ +�� +θ∈Θ +ξθ us +θ(bk +ξ,π) − π(ξ, bk +ξ,π) +� +s.t. +(5a) +� +ξ∈supp(γ) +γξ ξθ = µθ +∀θ ∈ Θ. +(5b) +Notice that Problem (5) is defined over general distributions over posteriors γ, whose support supp(γ) may be not finite. +Thus, as we show in the following, the crucial result that we need to design a PTAS is the possibility of restricting the +attention to finite sets of posteriors. +We need to introduce a particular class of posteriors, which are called q-uniform posteriors. +Definition 4 (q-Uniform posterior). A posterior ξ ∈ ∆Θ is q-uniform if it can be obtained by averaging the elements +of a multi-set defined by q ∈ N>0 canonical basis vectors of Rd. +In the following, we denote by Ξq ⊆ ∆Θ (for a given q ∈ N>0) the finite set of all the q-uniform posteriors. As it is +easy to check, such a set satisfies |Ξq| ≤ min{dq, qd}. +In order to derive our PTAS, as a first preliminary result we show that, given any posterior ξ∗ ∈ ∆Θ, payment function +π : ∆Θ × A → R+, and ǫ > 0, there always exists a signaling scheme γ supported on Ξq which induces posterior +ξ∗ on average and guarantees a seller’s expected utility close to that provided by the posterior ξ∗ (assuming the buyer +plays an ǫ-best response). Formally: +Lemma 6. Given any ǫ, α > 0, a posterior ξ∗ ∈ ∆Θ, and a payment function π : ∆Θ × A → R+, there always exists +a signaling scheme γ ∈ ∆Ξq with q = 2 log(2m/α)/ǫ2 such that: +� +ξ∈Ξq +γξ +�� +θ∈Θ +ξθ us +θ(bk,ǫ +ξ,π) − π(ξ, bk,ǫ +ξ,π) +� +≥ +� +θ∈Θ +ξ∗ +θ us +θ(bk +ξ∗,π) − π(ξ∗, bk +ξ∗,π) − α, +for every buyer’s type k ∈ K, where we let π : ∆Θ ×A → R+ be a payment function that is optimal in every posterior +ξ ∈ Ξq when the buyer plays an ǫ-best response, i.e., π solves Problem (4) for every ξ ∈ Ξq with bk +ξ,π replaced by bk,ǫ +ξ,π. +Furthermore, the signaling scheme γ satisfies: +� +ξ∈Ξq +γξ ξθ = ξ∗ +θ +∀θ ∈ Θ. +Lemma 6 guarantees that, by decomposing each posterior ξ ∈ ∆Θ as a convex combination of the elements of Ξq, the +seller’s expected utility decreases by at most α. This implies that, assuming the buyer plays an ǫ-best response, it is +possible to work with signaling schemes (and thus payment functions) supported on Ξq, by only slightly degrading +the seller’s expected utility. +Another component that we need for our PTAS is an algorithm that, given a q-uniform posterior, computes an optimal +payment function in that posterior (i.e., a payment function solving Problem (4) for such a posterior). By Theorem 2, +it is easy to see that such an algorithm has to solve a problem that is equivalent to computing an optimal contract in +observable-action principal-agent problems. Thus, by Theorem 3, such a problem is APX-hard in general. Next, we +show that, whenever the number of buyer’s actions m is fixed, the APX-hardness result can be circumvented, and, thus, +we can provide an algorithm that solves the desired task and runs in polynomial time. Formally: +10Notice that, as it is the case for Problem (2) in Section 3, it is not immediately clear a priori whether the problem of computing +a seller-optimal protocol without menus admits a maximum or not. Thus, in principle we should start by defining the problem with +a sup rather than a max. However, in Section 6.2, we provide a (possibly exponential-time) algorithm which finds a seller-optimal +protocol without menus in general settings, and this implies that a maximum always exists. +12 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +Lemma 7. Restricted to instances where the buyer has limited liability and the number of buyer’s actions m is fixed, +there exists a polynomial-time algorithm that, given a posterior ξ ∈ ∆Θ as input, computes the payments π(ξ, a) for +a ∈ A of a payment function π : ∆Θ × A → R+ optimal in ξ. +Notice that, in order to get a payment function π : ∆Θ × A → R+ that is optimal in every posterior ξ ∈ Ξq, it is +sufficient to apply Lemma 7 for each ξ ∈ Ξq, then putting together all the computed payments π(ξ, a) in order to +obtain the overall payment function π. +The final piece that we need to complete the design of our PTAS is a way of coming back to work with buyer’s best +responses, rather than using ǫ-best responses. Indeed, this is possible thanks to Corollary 3, which allows us to modify +the payment function in all the induced posteriors, so as to achieve the desired result by only losing a small amount +2√ǫ of the seller’s expected utility. +Now, we are ready to design our PTAS that works whenever the buyer has limited liability and the number of buyer’s +actions m is fixed. By Lemma 6, we can focus on signaling schemes supported over q-uniform posteriors, for a +suitably-defined q ∈ N>0. Moreover, thanks to Lemma 7, we can compute a payment function that is optimal in all +the q-uniform posteriors, by running the polynomial-time algorithm in Lemma 7 for each q-uniform posterior in Ξq. +By Corollary 3, such an optimal payment function achieves a seller’s expected utility that is close to that obtained by +a payment function which is optimal in every q-uniform posterior when considering ǫ-best responses, thus allowing +for the application of the result in Lemma 6. In conclusion, our PTAS works by solving a modified version of LP (5), +where we set supp(γ) := Ξq in Equation (5a), and we take as payment function the one returned by applying Lemma 7 +in each posterior ξ ∈ Ξq. It is easy to see that the overall procedure requires time polynomial in the instance size when +the number of actions m is fixed, since |Ξq| ≤ dq and q = 2 log(2m/α)/ǫ2 as prescribed by Lemma 6. However, the +overall running time depends exponentially in α > 0, which the seller’s expected utility approximation provided by +the algorithm. This allows us to prove the following result: +Theorem 6. Restricted to instances where the buyer has limited liability and the number of buyer’s actions m is fixed, +the problem of computing a seller-optimal protocol without menus admits a PTAS. +Finally, we show that a similar approach can be employed to derive a quasi-polynomial time algorithm providing a +bi-criteria approximation of the seller’s expected utility in an optimal protocol, even when the number of buyer’s +actions m is arbitrary. Indeed, in our PTAS, the computation of an optimal payment function in a given q-uniform +posterior can be done in polynomial time only when the number of actions is fixed. While in general the problem +is APX-hard, an approximately-optimal price function can be computed in polynomial time by applying Corollary 2. +Moreover, since q = 2 log(2m/α)/ǫ2 and |Ξq| ≤ dq, the enumeration over the q-uniform posteriors can be performed +in time quasi polynomial in th number of actions m. This gives the following result: +Theorem 7. Restricted to instances in which the buyer has limited liability, there exists an algorithm that, given any +α, ǫ > 0 and ρ ∈ (0, 1/2] as input, returns a protocol without menus achieving a seller’s expected utility greater +than or equal to ρ OPT − 2−Ω(1/ρ) − (α + 2√ǫ), where OPT is the seller’s expected utility in an optimal protocol. +Moreover, the algorithm runs in time polynomial in Ilog m—where I is the size of the problem instance and m is +the number of buyer’s actions—, and the seller’s expected utility in the returned protocol is greater than or equal +to OPTLIN − (α + 2√ǫ), where OPTLIN is the best expected utility achieved by a protocol parametrized by β as in +Corollary 2. +The second part of the statement will be useful in deriving our results for the problem of computing seller-optimal +protocols without menus in the general case in which the buyer has no limited liability. Intuitively, it states that, even +if our approximation algorithm only provides a bi-criteria approximation of a seller-optimal protocol, the returned +protocol achieves a seller’s expected utility which is arbitrarily close to that achievable by using payment functions +that define the payments as a given fraction of the seller’s expected utility. +5.2 +Fixing the Number of States of Nature +Next, we study the case in which the buyer has limited liability and the number of states of nature d is fixed. We prove +that, in such a setting, it is possible to compute a bi-criteria approximation of an optimal protocol without menus +similar to that in Theorem 7, but in polynomial time. Notice that such a result circumvents the APX-hardness one +provided in Corollary 1, as the latter is based on a reduction working with instances with only one state of nature. +Similarly to Section 5.1, we first show that it is possible to employ signaling schemes supported on the set Ξq of +q-uniform posteriors (for a suitably-defined q ∈ N>0), by only suffering an arbitrarily small, additive loss in terms of +seller’s expected utility. While the following result is similar to the one obtained in Lemma 6, it is based on different +techniques and, in particular, on the fact that the seller’s expected utility is Lipschitz continuous in the buyers’ posterior. +Formally: +13 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +Lemma 8. Given any α > 0, a posterior ξ∗ ∈ ∆Θ, and a payment function π : ∆Θ × A → R+ that is optimal in +every posterior ξ ∈ Ξq with q = ⌈9d/α2⌉, there exists a signaling scheme γ ∈ ∆Ξq: +� +ξ∈Ξq +γξ +�� +θ∈Θ +ξθ us +θ(bk +ξ,π) − π(ξ, bk +ξ,π) +� +≥ +� +θ∈Θ +ξ∗ +θus +θ(bk +ξ∗,π) − π(ξ∗, bk +ξ∗,π) − α, +for every receiver’s type k ∈ K. Furthermore, the signaling scheme γ satisfies: +� +ξ∈Ξq +γξ ξθ = ξ∗ +θ +∀θ ∈ Θ. +Similarly to the case of a fixed number of actions, we employ Lemma 8 to restrict the attention to signaling schemes +(and thus payment functions) supported on Ξq. Moreover, in this case, we can apply Corollary 2 in each q-uniform +posterior in order to compute in polynomial time a payment function that provides a bi-criteria approximation of the +optimal seller’s expected utility in such a posterior. Finally, we design an algorithm that solves a modified version of +LP (5), where we set supp(γ) := Ξq in Equation (5a), and we take as payment function the one obtained by putting +together those computed by means of Corollary 2 for each ξ ∈ Ξq. Finally, the overall procedure requires polynomial +time, since |Ξq| ≤ qd and the number of states of nature d is fixed, and achieves a bi-criteria approximation of the +seller’s expected utility in an optimal protocol. Formally: +Theorem 8. Restricted to instances in which the buyer has limited liability and the number of states of nature d is fixed, +there exists an algorithm that, given α > 0 and ρ ∈ (0, 1/2] as input, returns in polynomial time a protocol without +menus achieving a seller’s expected utility greater than or equal to ρ OPT − 2−Ω(1/ρ) − α, where OPT is the seller’s +expected utility in an optimal protocol. Moreover, the seller’s expected utility in the returned protocol is greater than +or equal to ρ OPTLIN − 2−Ω(1/ρ) − α where OPTLIN is the best expected utility achieved by a protocol parametrized +by β as in Corollary 2. +Similarly to Theorem 7, the second part of the statement will be useful for deriving our results in the general case in +which the buyer has no limited liability. +6 +Computing a Seller-optimal Protocol without Menus: +The General Case +We conclude our analysis by considering the problem of computing a seller-optimal protocol without menus in general +instances in which the buyer has no limited liability. Thus, in such a setting, the seller also decides a price p ∈ R+ for +the signaling scheme proposed to the buyer. +First, we provide a negative result for general instances that is stronger than the one established in Corollary 1. In +particular, the latter result states that the seller’s optimization problem is APX-hard even in the special case in which +the buyer has limited liability and there is only one state of nature, relying on a reduction employing instances with +an arbitrary number of actions m. Indeed, for the specific case in which the buyer has limited liability and the number +of actions m is fixed, Theorem 6 provides a PTAS. Next, we show that, in general instances where the buyer may not +have limited liability, the problem is APX-hard even when the number of buyer’s actions m is fixed. To prove such an +hardness result, we employ a result by Guruswami and Raghavendra (2009) (see Theorem 9 below), which is about +the following promise problem related to the satisfiability of a fraction of linear equations with rational coefficients +and variables restricted to the hypercube.11 +Definition 5 (LINEQ-MA(1−ζ, δ) by Guruswami and Raghavendra (2009)). For any two constants ζ, δ ∈ R satisfying +0 ≤ δ ≤ 1 − ζ ≤ 1, LINEQ-MA(1 − ζ, δ) is the following promise problem: Given a set of linear equations Ax = c +over variables x ∈ Qnvar, with coefficients A ∈ Qneq×nvar and c ∈ Qneq, distinguish between the following two cases: +• there exists a vector ˆx ∈ {0, 1}nvar that satisfies at least a fraction 1 − ζ of the equations; +• every possible vector x ∈ Qnvar satisfies less than a fraction δ of the equations. +Theorem 9 (Guruswami and Raghavendra (2009)). For all the constants ζ, δ ∈ R which satisfy 0 ≤ δ ≤ 1 − ζ ≤ 1, +the problem LINEQ-MA(1 − ζ, δ) is NP-hard. +11In +the +definition +in +(Guruswami and Raghavendra, +2009), +the +vector +ˆx +can +be +non-binary. +How- +ever, Guruswami and Raghavendra (2009) use a binary vector ˆx in their proof and, thus, their hardness result also holds for +our definition. +14 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +Then, Theorem 5 allows us to prove the following hardness result: +Theorem 10. The problem of computing a seller-optimal protocol without menus is APX-hard, even when the number +of buyer’s actions m is fixed. +In the following, we show how to circumvent the hardness result in Theorem 10, by providing, in Section 6.1, a +quasi-polynomial-time bi-criteria approximation algorithm and, in Section 6.2, a polynomial-time (exact) algorithm +working when the number of buyer’s types is fixed. +6.1 +A General Quasi-polynomial-time Bi-criteria Approximation Algorithm +In order to circumvent the negative result presented in Theorem 10, we design a quasi-polynomial-time algorithm that +computes a protocol without menus providing a bi-criteria approximation of the seller’s expected utility in an optimal +protocol. Formally, our algorithm guarantees a multiplicative approximation ρ of the optimal utility, by only suffering +an additional 2−Ω(1/ρ) + α additive loss. Moreover, we show that our algorithm runs in polynomial time whenever +either the number of buyer’s actions m or that of states of nature d is fixed. +In order to prove the approximation guarantees of our algorithm, we rely on Theorems 7 and 8, and we decompose +the seller’s expected utility in an optimal protocol without menus into the sum of three different terms. Our algorithm +works by computing three protocols without menus, each one approximating one of the three terms. Choosing the best +protocol among the three provides the desired approximation guarantees. The following is an intuition of how each +term composing the optimal seller’s expected utility is approximated by our algorithm: +• The first term is related to the seller’s expected utility collected from buyer’s types for which the IR constraints +are not satisfied. Such a utility term can be trivially achieved by a protocol that charges no price, discloses no +information, and never pays back the buyer. +• The second term is related to the best seller’s expected utility which can be extracted from a buyer’s action. +This is related to the optimal seller’s expected utility in a setting with limited liability, since, in that case, +the seller’s expected utility is determined by the buyer’s action only. Thus, the second utility term can be +approximated by using either the algorithm provided in Theorem 7 or that given in Theorem 8.12 +• The third term is related to the seller’s expected utility obtained by the transfers between the seller and the +buyer, which include the charged price and the final payment. Such a utility term can be approximated by +using a protocol that reveals all the information to the buyer while charging a carefully-chosen price for that. +Formally, we prove the following main result: +Theorem 11. There exists an algorithm that, given any α > 0 and ρ ∈ (0, 1/6] as input, computes a protocol without +menus whose seller’s expected utility is greater than or equal to ρ OPT − 2−Ω(1/ρ) − α, where OPT is the seller’s +expected utility in an optimal protocol. Moreover, the algorithm runs in time polynomial in Ilog m—where I is the +size of the problem instance—when it is implemented with the algorithm in Theorem 7 as a subroutine, while it runs in +time polynomial in Id when it is implemented with the algorithm in Theorem 8 as a subroutine. +6.2 +Fixing the Number of Buyer’s Types +Next, we study the problem of computing a seller-optimal protocol without menus when the number of buyer’s types +is fixed, showing that it is possible to design a polynomial-time algorithm. As a byproduct, the existence of such an +algorithm shows that, for protocols without menus, the seller’s optimization problem always admits a maximum. +As a preliminary result, we show that it is always possible to focus on protocols without menus (φ, p, π) that employ +signals belonging to the set An, and define signaling schemes φ : Θ → ∆An and payment functions π : An × A → R+ +such that, for every signal a ∈ An and k ∈ K, it holds ak ∈ Bk +ξa,π, where ak ∈ A denotes the action corresponding +to type k in a. Intuitively, in such protocols, a signal specifies an action recommendation for each buyer’s type, so +that the buyer is always incentivized to follow such recommendations. With a slight abuse of notation, we say that +protocols without menus (φ, p, π) as described above are generalized-direct and generalized-persuasive. Formally, we +prove the following result: +Lemma 9. Given a seller’s protocol without menus, there always exists another protocol without menus which is +generalized-direct and generalized-persuasive, and achieves the same seller’s expected utility as the original protocol. +12Notice that we cannot employ Theorem 6 in place of Theorem 7, since the latter guarantees to achieve a seller’s expected utility +that is arbitrarily close to that of the best protocol employing payment functions parametrized by β, and this is needed in order to +derive the guarantees of our algorithm. Such a guarantee is not provided by Theorem 6, which only predicates on the quality of the +returned protocol with respect to an optimal protocol without menus. +15 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +In order to prove the lemma, we observe that, given a protocol, if two signals induce the same best response for every +buyer’s type, it is always possible to merge the two signals, retaining the same expected utility for both the seller and +the buyer. Then, by iterating such a process, we recover a signaling scheme and a payment function employing An as +set of signals . +As a second crucial step, we show that we can focus on protocols without menus (φ, p, π) whose price p is equal to +the budget bk of one buyer’s type k ∈ K. Formally: +Lemma 10. Given a protocol without menus, there always exists another protocol (φ, p, π) such that p = bk for some +k ∈ K, while achieving the same seller’s expected utility as the original protocol. +Finally, equipped with Lemma 9 and Lemma 10, we are ready to provide our polynomial-time algorithm. Intuitively, +since we can restrict the attention to protocols without menus (φ, p, π) that are generalized-direct and generalized- +persuasive, and whose prices p belong to the set {bk}k∈K, we can solve the seller’s problem by iterating over all the +possible price values p ∈ {bk}k∈K and, for each of them, over all the possible subsets R ⊆ K ∩ {k ∈ K : bk ≥ p} +of buyer’s types that satisfy the IR constraint. This can be done in polynomial time since the number of buyer’s types +n is fixed. Then, for every price value p ∈ {bk}k∈K and set R ⊆ K ∩ {k ∈ K : bk ≥ p}, it is sufficient to solve the +following optimization problem: +sup +φ≥0 +π≥0 +� +k∈R +λk +� +a∈An +� +θ∈Θ +µθφθ(a) [us +θ(ak) − π(a, ak)] + +� +k /∈R +λk +� +θ∈Θ +µθus +θ(bk +µ) +s.t. +(6a) +� +θ∈Θ +µθφθ(a) +� +uk +θ(ak) + π(a, ak) +� +≥ +� +θ∈Θ +µθφθ(a) +� +uk +θ(a′) + π(a, a′) +� +∀k ∈ R, ∀a ∈ An, ∀a′ ̸= ak ∈ A +(6b) +� +a∈An +� +θ∈Θ +µθφθ(a) +� +uk +θ(ak) + π(a, ak) +� +− bk ≥ +� +θ∈Θ +µθuk +θ(bk +µ) +∀k ∈ R +(6c) +� +a∈An +� +θ∈Θ +µθφθ(a) +� +uk +θ(ak) + π(a, ak) +� +− bk ≤ +� +θ∈Θ +µθuk +θ(bk +µ) +∀k ̸∈ R +(6d) +� +a∈An +φθ(a) = 1 +∀θ ∈ Θ. +(6e) +By using techniques similar to those used in Section 3 for protocols with menus, we can show that Problem (6) is +solvable in polynomial time by means of a suitable-defined LP. This allows us to state our last results: +Theorem 12. Restricted to instances in which the number of buyer’s types n is fixed, the problem of computing a +seller-optimal protocol without menus admits a polynomial-time algorithm. +Corollary 4. The problem of computing a seller-optimal protocol without menus always admits a maximum. +References +Paola Alimonti and Viggo Kann. 2000. Some APX-completeness results for cubic graphs. Theoretical Computer +Science 237 (04 2000), 123–134. https://doi.org/10.1016/S0304-3975(98)00158-3 +Tal Alon, Paul Dütting, and Inbal Talgam-Cohen. 2021. Contracts with Private Cost per Unit-of-Effort. In Proceedings +of the 22nd ACM Conference on Economics and Computation. 52–69. +Tal Alon, Paul Dütting, Yingkai Li, and Inbal Talgam-Cohen. 2022. +Bayesian Analysis of Linear Contracts. +arXiv:cs.GT/2211.06850 +Moshe Babaioff, Robert Kleinberg, and Renato Paes Leme. 2012. Optimal mechanisms for selling information. 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Proceedings of the 22nd ACM +Conference on Economics and Computation (2021). +Yoav Shoham and Kevin Leyton-Brown. 2008. Multiagent systems: Algorithmic, game-theoretic, and logical founda- +tions. Cambridge University Press. +17 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +A +Proofs Omitted from Section 3 +Lemma 1. Given any IC and IR seller’s protocol, it is always possible to recover an IC and IR seller’s protocol that +is direct and persuasive, and it provides the seller with the same expected utility. +Proof. Let { +� +φk, pk, πk +� +}k∈K be an IC and IR protocol such that there exist two signals s1, s2 ∈ S inducing the same +best response ¯a ∈ A for a given receiver’s type ¯k ∈ K, i.e., b¯k +ξs1 = b¯k +ξs2 = ¯a. +In the following, we show how to replace φ¯k and π¯k with a new signaling scheme ¯φ¯k and a new payment function ¯π¯k +by merging s1, s2 into a single signal ¯s. Formally we define: ¯φ¯k +θ(¯s) = φk +θ(s1) + φk +θ(s2) for each θ ∈ Θ. Similarly, we +define: +¯π +¯k(¯s, ¯a) = +� +θ∈Θ µθφ¯k +θ(s1)π¯k(s1, ¯a) + � +θ∈Θ µθφ¯k +θ(s2)π¯k(s2, ¯a) +� +θ∈Θ µθ(φ¯k +θ(s1) + φ¯k +θ(s2)) +, +Finally, we does not change all the other components of the protocol i.e., we leave these components of +{ +�¯φk, ¯pk, ¯πk +� +}k∈K equal to the one in { +� +φk, pk, πk +� +}k∈K. +To prove the lemma we show that the protocol +{ +�¯φk, ¯pk, ¯πk +� +}k∈K achieves the same seller’s expected utility of { +� +φk, pk, πk +� +}k∈K, while satisfying the IC and IR +constraints. As a first step, we observe that: +� +s∈S\{s1,s2} +� +θ∈Θ +µθ ¯φ +¯k +θ(s) +� +u +¯k +θ(b +¯k +ξs,¯π) + ¯π¯k(s, b +¯k +ξs,¯π) +� ++ +� +θ∈Θ +µθ ¯φ +¯k +θ(¯s)[u +¯k +θ(¯a) + ¯π¯k(¯s, ¯a)] − ¯p¯k = +� +s∈S\{s1,s2} +� +θ∈Θ +µθφ +¯k +θ(s) +� +u +¯k +θ(b +¯k +ξs,π) + π¯k(s, b +¯k +ξs,π) +� ++ +� +θ∈Θ +µθφ +¯k +θ(s1)[u +¯k +θ(¯a) + π¯k(s1, ¯a)]+ ++ +� +θ∈Θ +µθφ +¯k +θ(s2)[u +¯k +θ(¯a) + π¯k(s2, ¯a)] − p¯k. +The latter equality holds by linearity and proves that the protocol {(¯φk, ¯pk, ¯πk)}k∈K preserves the left-hand sides of +the IR and IC constraints. Moreover, thanks to the convexity of the max operator, we can show that: +max +a∈A +� +θ∈Θ +µθφ +¯k +θ(s1)[u +¯k +θ(a) + π¯k(s1, a)] + max +a∈A +� +θ∈Θ +µθφ +¯k +θ(s2)[u +¯k +θ(a) + π¯k(s2, a)] − p¯k ≥ +max +a∈A +� +θ∈Θ +µθ ¯φ +¯k +θ(¯s)[u +¯k +θ(a) + ¯π¯k(¯s, a)] − p¯k. +Then, by summing over the set (S ∪{¯s})\{s1, s2}, we notice that the value of the right-hand side of the IC constraints +achieved by the protocol {(¯φk, ¯pk, ¯πk)}k∈K is less or equal to the the value achieved by {(φk, pk, πk)}k∈K. Due to +that, we can easily conclude that the new protocol preserves the IC and the IR constraints. Finally, by observing that +the following equality holds: +� +θ∈Θ +µθ ¯φk +θ(¯s)[u +¯k +θ(¯a) + ¯π¯k(¯s, ¯a)] = +� +θ∈Θ +µθφ +¯k +θ(s1)[u +¯k +θ(¯a) + π¯k(s1, ¯a)] + +� +θ∈Θ +µθφ +¯k +θ(s2)[u +¯k +θ(¯a) + π¯k(s2, ¯a)], +we can easily prove that the two protocols achieve the same seller’s expected utility. Then, by iterating this procedure +for each buyer’s type and couple of signals until there are no two signals inducing the same best response for that type, +we get a protocol that employs direct and persuasive signals. +Lemma 2. Given an IC and IR protocol {(φk, pk, πk)}k∈K that is direct and persuasive, it is always possible to +recover an IC and IR protocol {(φk, ˜pk, ˜πk)}k∈K such that: it is direct and persuasive, it provides the same seller’s +expected utility as the original protocol, and, for every buyers’ type k ∈ K, it satisfies ˜pk = bk and ˜πk(a, a′) = 0 for +all a ̸= a′ ∈ A. +Proof. We first prove that by setting ˜πk(a, a′) = 0 for each a ̸= a′ ∈ A and k ∈ K, the seller’s expected utility +does not change and the IC and IR constraints are preserved. First, we show that by taking ˜πk(a, a′) = 0 for each +a ̸= a′ ∈ A and k ∈ K, the signaling schemes φk remain persuasive. Indeed, we have that: +� +θ∈Θ +µθφk +θ(a)uk +θ(a) + πk(a, a) ≥ +� +θ∈Θ +µθφk +θ(a′)uk +θ(a′) + πk(a, a′) +≥ +� +θ∈Θ +µθφk +θ(a′)uk +θ(a′). +18 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +Moreover, the left-hand side of all the equations defining the IR and IC constraints does not change since it depends +only from the payments ˜πk(a, a) = πk(a, a) for each a′ ∈ A that remains unchanged. Furthermore, by setting +˜πk(a, a′) = 0 for each a ̸= a′ ∈ A and k ∈ K, the right-hand sides of the IC constraints achieve smaller or equal +values, since, intuitively, we are setting to zero non-negative values. Hence, the IC constraints are satisfied. Finally, by +observing that the left-hand sides of the IR constraints and the seller’s expected utility do not embed terms ˜πk(a, a′) +with a ̸= a′ ∈ A and k ∈ K, we conclude the first part of the proof. +In the second part of the proof we show that it is always possible to define a protocol in which the seller asks +to each buyer’s type to deposit all their budget at the beginning of the interaction, achieving the same seller’s ex- +pected utility and satisfying the constraints. Formally, we show that given a protocol { +� +φk, pk, πk +� +}k∈K the protocol +{ +� +φk, ˜pk, ˜πk +� +}k∈K, with ˜pk = bk and ˜πk(a) = πk(a, a) + bk − pk for each k ∈ K and a ∈ A, achieves the same +seller’s expected utility. Indeed by linearity we have: +� +k∈K +λk +� � +a∈A +� +θ∈Θ +µθφk +θ(a)us +θ(a) − πk(a) + pk +� += +� +k∈K +λk +� � +a∈A +� +θ∈Θ +µθφk +θ(s)us +θ(a) − πk(a) + bk − bk + pk +� += +� +k∈K +λk +� � +a∈A +� +θ∈Θ +µθφk +θ(a)us +θ(a) − ˜πk(a) + bk +� +. +With similar arguments it is easy to check that the protocol { +� +φk, ˜pk, ˜πk +� +}k∈K satisfies the IC and IR constraints, +concluding the lemma. +Lemma 3. The optimal value of LP (3) is at least as large as the supremum in Problem (2). +Proof. We show that for each protocol { +� +φk, pk, πk +� +}k∈K that is feasible for Problem (2), we can derive a solution to +LP (3) with at least the same value. This is sufficient to prove the statement. +By Lemma 1 and Lemma 2, we focus without loss of generality on protocols { +� +φk, pk, πk +� +}k∈K that are direct and +persuasive. Then, we can build a solution (¯φ, ¯l, ¯y) to LP (3) letting +¯lk(a) = +� +θ∈Θ +µθφθ(a)πk(a) +for each k ∈ K and a ∈ A, and +¯yk,k′,a = max +�� +θ∈Θ +µθφk′ +θ (a) +� +uk +θ(a) + πk′(a, a) +� +, max +a′̸=a +� +θ∈Θ +µθφk′ +θ (a′)uk +θ(a′) +� +for each k, k′ ∈ K and a ∈ A. Moreover, we let ¯φ = φ. It is easy to verify that the solution (¯φ, ¯l, ¯y) results feasible +for LP (3). +Lemma 4. Given a feasible solution to LP (3), it is possible to recover in polynomial time an IC and IR protocol +whose seller’s expected utility is greater than or equal to the value of the solution to LP (3). +Proof. As a first step we show that from a feasible solution (φ, l, y) to LP (3) we can recover another solution with at +least the same value in which if πk(a) > 0 then there exists a θ ∈ Θ such that φk +θ(a) > 0. Specifically, given a k ∈ K +and an a ∈ A such that φk +θ(a) = 0 for all θ ∈ Θ and lk(a) > 0, let ¯a ∈ A be (¯a, ¯θ) be any couple of an action and a +state such that φk +¯θ(¯a) > 0. We now define a new feasible solution (¯φ, ¯l, ¯y) as follows: +• ¯lk(a) = 0 +• ¯lk(¯a) = lk(a) + lk(¯a) +• ¯yk′,k,a = 0 +∀k′ ∈ K +• ¯yk′,k,¯a = yk′,k,¯a + lk(a) +∀k′ ∈ K, +while we leave all the other terms variables equal to the ones in (φ, l, y). +It is easy to check that the solution +(¯φ, ¯l, ¯y)achieves the same seller’s expected utility while satisfying the constrains. Applying this procedure for each +couple (k, a) such that φk +θ(a) = 0 for all θ ∈ Θ and lk(a) > 0, we obtain a new solution (˜φ, ˜l, ˜y) such that if ˜lk(a) > 0 +then there exists a θ ∈ Θ such that ˜φk +θ(a) > 0. +19 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +To recover a feasible protocol we just need to set payments as follows. For each couple (k, a), if there exists a θ ∈ Θ +such that ˜φk +θ(a) > 0, we set ˜πk(a) = ˜lk(a)/(� +θ∈Θ µθ ˜φk +θ(a)). Otherwise, we set ˜πk(a) = 0. Notice that the ratio +˜lk(a)/(� +θ∈Θ µθ ˜φk +θ(a)) is always well defined. Moreover, it is easy to see that {˜φk, ˜pk, ˜πk}k∈K, with ˜pk = bk for +each k ∈ K is a feasible solution to Problem (2) with at least the same value as (φ, l, y) for LP (3) . This concludes +the proof. +Theorem 1. There exists a polynomial-time algorithm that computes a protocol with menus that maximizes the seller’s +expected utility. +Proof. The algorithm solves LP 3. By Lemma 3, this solution has value greater or equal to the supremum of Program 2. +Then, exploiting Lemma 4, we can recover in polynomial-time a protocol with at least the same utility, i.e., an optimal +one. +B +Proofs Omitted from Section 4 +Lemma 5. Given a posterior ξ ∈ ∆Θ, solving Problem (4) is equivalent to computing a contract maximizing the +principal’s expected utility in an instance of observable-action principal-agent problem such that, for every agent’s +type k ∈ K and action a ∈ A, the following holds: +ck +a = +� +θ∈Θ +ξθ +� +uk +θ(bk +ξ) − uk +θ(a) +� +and +ra = +� +θ∈Θ +ξθ us +θ(a). +Moreover, finding an optimal contract in any instance of observable-action principal-agent problem can be reduced in +polynomial time to computing a seller-optimal protocol without menus in a problem instance in which the buyer has +limited liability and there is only one state of nature. +Proof. We start proving the first part of the statement. Given an instance of Problem (4), we build an instance of the +observable-action principal-agent problem with ck +a = � +θ∈Θ ξθ +� +uk +θ(bk +ξ) − uk +θ(a) +� +and ra = � +θ∈Θ ξθ us +θ(a). To prove +the equivalence between the two settings, we first show that the set of best-responses for the two problems coincides. +Indeed, given a payment function π and a type k ∈ K, let a ∈ Bk +ξ,π. Then, for each a′ ∈ A +π(a) − ck +a = π(a) − +� +θ∈Θ +ξθ +� +uk +θ(bk +ξ) − uk +θ(a) +� += π(a) + +� +θ∈Θ +ξθuk +θ(a) − +� +θ∈Θ +ξθuk +θ(bk +ξ) +≥ π(a′) + +� +θ∈Θ +ξθuk +θ(a′) − +� +θ∈Θ +ξθuk +θ(bk +ξ) += π(a′) − ck +a′, +showing that a ∈ Bk +π. Similarly, we can prove that if a ∈ Bk +π, then a ∈ Bk +ξ,π This implies that the set of best responses +are equivalent for each payment function π, i.e., Bk +π = Bk +ξ,π. Hence, +argmax +π +� +k∈K +λk +� +rbkπ − π(bk +π) +� += argmax +π +� +k∈K +λk +�� +θ∈Θ +ξθ us +θ(bk +π) − +� +θ∈Θ +ξθ +� +uk +θ(bk +ξ) − uk +θ(bk +π) +� +� += argmax +π +� +k∈K +λk +�� +θ∈Θ +ξθ us +θ(bk +π) + +� +θ∈Θ +ξθuk +θ(bk +π) +� += argmax +π +� +k∈K +λk +�� +θ∈Θ +ξθ us +θ(bk +ξ,π) + +� +θ∈Θ +ξθuk +θ(bk +ξ,π) +� +, +showing the equivalence between the two problems. This proves the first part of the statement. +We now show that from an instance of the observable-action principal-agent problem we can always build an instance +of the selling-information problem without menus and with only a single state of nature θ. In particular, we set +us +θ(a) = ra for each a ∈ A, and uk +θ(a) = 1 − ck +a for each a ∈ A and k ∈ K. Following a analysis similar to the first +part of the proof, we can show that the two problems are equivalent. This concludes the proof. +20 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +Theorem 3. In observable-action principal-agent problems, the problem of computing a contract maximizing the +principal’s expected utility is APX-hard. +Proof. We reduce from vertex cover in cubic graphs. Formally, it is NP-hard to approximate the size of the minimum +vertex cover in cubic graphs with an approximation (1 + ε), for a given constant ε > 0 Alimonti and Kann (2000). Let +η = ε/7. We show that an (1 − η)-approximation to the principal-agent problem with observable actions can be used +to provide a (1 + ε) approximation to vertex cover, concluding the proof. +Consider an instance of vertex cover (V, E) with nodes V and edges E. Let ρ = |V | and ℓ = |E|. Given a vertex +v ∈ V , we let E(v) be the set of edges e such that v is one of the extreme of the edge e. Similarly, given an edge e ∈ E, +let V (e) be the set of vertexes v such that e is an edge with extreme v. We build an instance of the principal-agent +problem with observable actions as follows. For each vertex v ∈ V , there exists an agent’s type kv, while for each +e ∈ E, there exists a type ke. For each vertex v ∈ V , there exists an action av and an additional action a−. The cost +of a type ke, e ∈ E, is cke +a− = 0, cke +av = 1 +2 if e ∈ E(v) and 1 otherwise. The cost of a type kv, v ∈ V , is ckv +av = 0, and +1 otherwise. Finally, the principal’s utility is equal to 1 if the action is in {av}v∈V , while is equal to 0 otherwise, i.e., +us +av = 1 for each v ∈ V and us +a− = 0. All the types are equally probable, i.e., λk = +1 +ρ+ℓ for each k ∈ K. +First, we show that if there exists a vertex cover V ⋆ of size ν, the value of the problem is at least (ρ−ν)+ ν +2 + ℓ +2 +ρ+ℓ +. Consider +the payment function such that π(av) = 1 +2 if v ∈ V ⋆ and 0 otherwise. A type kv with v /∈ V ⋆ plays the action av and +receives a payment of 0. A type kv with v ∈ V ⋆ plays the action av and receives a payment of 1 +2. A type ke plays an +action av such that e ∈ E(v) (this action exists by construction) and receives a payment of 1 +2. It is easy to see that the +expected seller’s utility is (ρ−ν)+ ν +2 + ℓ +2 +ρ+ℓ +. +Suppose that there exists an algorithm that provides a 1 − η approximation. This implies that the algorithm returns a +solution, i.e., a payment function π, with value at least (1 − η) ρ− k +2 + ℓ +2 +ρ+ℓ +. We show how to exploit the payment function +π to build a vertex cover of size at most (1 + ε)ν in polynomial time. In particular, given π we recover a vertex +cover ¯V of the desired size as follows. First, it is easy to see that we can set the payment π(a−) = 0 and payments +π(av) ∈ {0, 1 +2} for each v ∈ V without decreasing the utility. Intuitively, payments are useful only to change the best +response of a type ke, e ∈ E, from a− to av, v ∈ V . To do so, it is sufficient a payment of 1 +2. Then, let ¯E be the set +of edges e ∈ E such that the best response of ke is a−, i.e., ¯E = {e ∈ E : bke +π = a−}. Consider a edge e ∈ ¯E and +a vertex v ∈ V (e). Since e ∈ ¯E the payment π(av) = 0 and no type ke plays action av. Hence, if we modify the +payments by letting π(av) = 1 +2 on the action av we have three effects: i) the type ke changes the best response to av, ii) +some other types e′ ∈ E could change from action a− to av,13 iii) the payment of type kv increases by 1 +2. Overall the +principal’s total utility increases by 1 +2λke since ke changes from action a− with payment 0 to action av with payment +1 +2, and it decreases by − 1 +2λkv as the payment to type kv increases by 1 +2. Moreover, if other types ke′, e′ ̸= e change +from a− with payment 0 to action av with payment 1 +2 the principal’s utility increases. This implies that the principal’s +utility does not decrease with this procedure. Hence, repeating this procedure we can build a payment function π with +the same utility such that all the agent’s type plays actions av, v ∈ V . Then, let ¯V be the set of vertexes with at least +one agent of type ke, e ∈ E that plays this action, i.e., ¯V = {v ∈ V : bke +π = av, e ∈ E}. We show that ¯V is an vertex +cover of size at most (1 + ǫ)ν, concluding the proof. Notice that we can set payment π(av) = 0 for each v ∈ V \ ¯V +without decreasing the seller’s utility. Removing the payment does not change any best response since the only type +playing the action av is the type kv, the utility ukv(av) = 1, and the utility of playing any other action a ̸= av is +ukv(a) + π(a) ≤ 1 +2. Then, the principal’s utility is +(ρ − | ¯V |) − 1 +2| ¯V | + 1 +2ℓ +ρ + ℓ += ρ − 1 +2| ¯V | + 1 +2ℓ +ρ + ℓ +, +since ρ − | ¯V | types kv, v ∈ V , play av and receive payment 0, | ¯V | types kv, v ∈ V , play av and receive payment 1 +2, +and all the types ke, e ∈ E, play an action av, v ∈ V and receive payment 1 +2. Then, since the principal’s utility is by +assumption ρ− 1 +2 | ¯V |+ 1 +2 ℓ +ρ+ℓ +≥ (1 − η) ρ− ν +2 + ℓ +2 +ρ+ℓ +, it holds +| ¯V | ≤ η(2ρ + ℓ) + ν ≤ (1 + 7η)ν = (1 + 7η)ν = (1 + ε)ν, +where the second inequality comes from ρ = 2 +3ℓ and ℓ ≤ 3ν. This concludes the proof. +13Notice that there could be other actions a′ +v with π(av′) = 1 +2 that provides the same utility of av for both the principal and the +agent. +21 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +Corollary 2. Given a posterior ξ ∈ ∆Θ, for any ρ ∈ (0, 1/2], there exists a payment function π : ∆Θ × A → R+ +such that π(ξ, a) = β � +θ∈Θ ξθ us +θ(a) for every a ∈ A, where β ∈ [0, 1] is an (action-independent) parameter, and, +additionally, the following holds: +� +k∈K +λk +� +θ∈Θ +ξθ +� +us +θ(bk +ξ,π) − π(s, bk +ξ,π) +� +≥ ρ +� +k∈K +λk max +a∈A +� +θ∈Θ +ξθ +� +us +θ(a) + uk +θ(a) − uk +θ(bk +ξ) +� +− 2Ω(1/ρ). +Moreover, such a parameter β is equal to 1 − 2−i for some i ∈ {1, . . . , ⌊1/2ρ⌋}. +Proof. As a first step, we notice that even if Castiglioni et al. (2022a) show that linear contracts provide the desired +approximation with respect to optimal contracts, their proof can be extended to show that linear contracts provide the +same approximation with respect to the optimal social welfare, i.e., � +k∈K λk maxa∈A[ra −ck +a]. To prove this result, it +is sufficient to follow all the steps of Theorem 3 of (Castiglioni et al., 2022a) except for the one in which Observation 1 +is employed to upperboud the value of the optimal contract with the social welfare. Finally, we can modify this result +to hold in our setting exploiting Lemma 5. +C +Proofs Omitted from Section 5 +Lemma 6. Given any ǫ, α > 0, a posterior ξ∗ ∈ ∆Θ, and a payment function π : ∆Θ × A → R+, there always exists +a signaling scheme γ ∈ ∆Ξq with q = 2 log(2m/α)/ǫ2 such that: +� +ξ∈Ξq +γξ +�� +θ∈Θ +ξθ us +θ(bk,ǫ +ξ,π) − π(ξ, bk,ǫ +ξ,π) +� +≥ +� +θ∈Θ +ξ∗ +θ us +θ(bk +ξ∗,π) − π(ξ∗, bk +ξ∗,π) − α, +for every buyer’s type k ∈ K, where we let π : ∆Θ ×A → R+ be a payment function that is optimal in every posterior +ξ ∈ Ξq when the buyer plays an ǫ-best response, i.e., π solves Problem (4) for every ξ ∈ Ξq with bk +ξ,π replaced by bk,ǫ +ξ,π. +Furthermore, the signaling scheme γ satisfies: +� +ξ∈Ξq +γξ ξθ = ξ∗ +θ +∀θ ∈ Θ. +Proof. Let ˜ξ ∈ Ξq be the empirical mean of q i.i.d. samples drawn according to ξ∗ ∈ ∆Θ, where each θ ∈ Θ +has probability ξ∗ +θ of being sampled. Therefore, ˜ξ ∈ Ξq is a random vector supported on q-uniform posteriors with +expectation ξ∗ ∈ ∆Θ. Moreover, let γ ∈ ∆Ξq be a probability distribution such as, for each ξ ∈ Ξq, it holds γξ := +Pr(˜ξ = ξ). We build a new payment function ˜π such that for each ξ ∈ ∆Θ and a ∈ A, we have ˜π(ξ, a) = π(ξ∗, a) +Moreover, we let Ξq,ǫ be the set of posteriors such that ξ ∈ Ξq,ǫ if and only if for each a ∈ A it holds: +����� +� +θ∈Θ +� +ξθuk +θ(a) − ξ∗ +θuk +θ(a) +� +����� ≤ ǫ +2. +(7) +Then, for each ξ ∈ Ξq,ǫ, we have that Bk +ξ∗,π ⊆ Bk,ǫ +ξ,π. In particular, for any a∗ ∈ Bk +ξ∗,π, ξ ∈ Ξq,ǫ and a ∈ A: +� +θ∈Θ +ξθuk +θ(a∗) + ˜π(ξ, a∗) ≥ +� +θ∈Θ +ξ∗ +θuk +θ(a∗) + ˜π(ξ∗, a∗) − ǫ +2 +(By Eq. (7) and the definition of Bk +ξ∗,π) +≥ +� +θ∈Θ +ξ∗ +θuk +θ(a) + ˜π(ξ∗, a) − ǫ +2 +≥ +� +θ∈Θ +ξθuk +θ(a) + ˜π(ξ, a) − ǫ +(By Equation (7)) +which is precisely the definition of Bk,ǫ +ξ∗,π. +For each a ∈ A, let ˜tk +a := � +θ∈Θ ˜ξθuk +θ(a)+˜π(˜ξ, a) and tk +a := � +θ∈Θ ξ∗ +θuk +θ(a)+˜π(ξ∗, a). By the Hoeffding’s inequality +we have that, for each a ∈ A, +Pr(|˜tk +a − E[˜tk +a]| ≥ ǫ +2) ≤ 2e−2q(ǫ/2)2 = 2e− log(2m/α) ≤ α +m. +(8) +22 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +Moreover, Equation (7) and the union bound yield the following: +� +ξ∈Ξq,ǫ +γξ = Pr(˜ξ ∈ Ξq,ǫ) += Pr( +� +a∈A +��˜tk +a − tk +a +�� ≤ ǫ +2) +≥ 1 − +� +a∈A +Pr( +��˜tk +a − tk +a +�� ≥ ǫ +2) +≥ 1 − α. +(By Equation (8)) +Let ¯ξ be a d-dimensional vector defined as ¯αθ := � +ξ∈Ξq\Ξq,ǫ γξξθ. By definition and for the previous result we have: +� +θ∈Θ ¯ξθ ≤ α. Finally, we can show: +� +ξ∈Ξq +γξ +� +θ∈Θ +ξθus +θ(bk,ǫ +ξ,π′) − π′(ξ, bk,ǫ +ξ,π′) +≥ +� +ξ∈Ξq +γξ +� +θ∈Θ +ξθus +θ(bk,ǫ +ξ,π) − π(ξ, bk,ǫ +ξ,π) +≥ +� +ξ∈Ξq,ǫ +γξ +� +θ∈Θ +ξθus +θ(bk,ǫ +ξ,π) − π(ξ, bk,ǫ +ξ,π) +≥ +� +ξ∈Ξq,ǫ +γξ +� +θ∈Θ +ξθus +θ(bk +ξ∗,π) − π(ξ∗, bk +ξ∗,π) +(Bk +ξ∗,π ⊆ Bk,ǫ +ξ,π for each ξ ∈ Ξq,ǫ) += +� +θ∈Θ +� +us +θ(bk +ξ∗,π) − π(ξ∗, bk +ξ∗,π) +�� � +ξ∈Ξq,ǫ +γξξθ +� += +� +θ∈Θ +� +us +θ(bk +ξ∗,π) − π(ξ∗, bk +ξ∗,π) +�� � +ξ∈Ξq +γξξθ − ¯ξθ +� +(By definition of ¯α) += +� +θ∈Θ +� +ξθus +θ(bk +ξ∗,π) − π(ξ∗, bk +ξ∗,π) +�� � +ξ∈Ξq +γξξθ +� +− +� +θ∈Θ +� +us +θ(bk +ξ∗,π) − π(ξ∗, bk +ξ∗,π) +� +¯ξθ +≥ +� +θ∈Θ +� +us +θ(bk +ξ∗,π) − π(ξ∗, bk +ξ∗,π) +�� � +ξ∈Ξq +γξξθ +� +− +� +θ∈Θ +¯ξθ +(Utilities in [0, 1]) +≥ +� +θ∈Θ +ξ∗ +θus +θ(bk +ξ∗,π) − π(ξ∗, bk +ξ∗,π) − α. +Finally, by definition of γ, we have that, for each θ ∈ Θ: +� +ξ∈Ξq +γξξθ = ξ∗ +θ. +This concludes the proof. +Lemma 7. Restricted to instances where the buyer has limited liability and the number of buyer’s actions m is fixed, +there exists a polynomial-time algorithm that, given a posterior ξ ∈ ∆Θ as input, computes the payments π(ξ, a) for +a ∈ A of a payment function π : ∆Θ × A → R+ optimal in ξ. +Proof. Given a posterior ξ ∈ ∆Θ and a tuple a ∈ A|K| we let Πa ⊆ Rm ++ be the set of payment functions π such +that for each k ∈ K it holds ak ∈ Bk +ξ,π. Given an a ∈ A|K|, the problem of computing an optimal payment function +restricted to payment functions in Πa can be formulated as follows: +min +π +� +k∈K +λkπ(ξ, ak) s.t. +� +θ∈Θ +ξθuk +θ(ak) + π(ξ, ak) ≥ +� +θ∈Θ +ξθuk +θ(a′) + π(ξ, a′) +∀a′ ∈ A, k ∈ K +π(ξ, a′) ≥ 0 +∀a′ ∈ A . +23 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +We observe that, for each tuple a ∈ A|K|, the vertexes of the regions Πa ⊆ Rm ++ are identified by m of the common +O(nm2 + m) constraints: +� +θ∈Θ +ξθuk +θ(a′) + π(ξ, a′) ≥ +� +θ∈Θ +ξθuk +θ(a′′) + π(ξ, a′′)∀a′ ̸= a′′ ∈ A, ∀k ∈ K +π(, ξ, a′) ≥ 0 +∀a′ ∈ A. +Hence, the total number of vertexes defining all the regions Πa, a ∈ A|K|, is at most +�nm2+m +m +� += O((nm2 + m)m). +Finally, since the objective function is linear in Πa for each tuple a ∈ A|K|, given the optimal tuple of induced actions +a∗ ∈ A|K| the optimum is attained in one of the vertexes of Πa∗. Moreover, there are overall +�� � +a∈A|K| V (Πa) +�� = +O((km2 + m)m) vertexes, where V (·) denotes the set of vertexes of the polytope. Hence, when m is fixed, it is +possible to enumerate in polynomial time over all the vertexes in � +a∈A|K| V (Πa) and compute the optimal payment +function. +Theorem 6. Restricted to instances where the buyer has limited liability and the number of buyer’s actions m is fixed, +the problem of computing a seller-optimal protocol without menus admits a PTAS. +Proof. Given two arbitrary constants α, ǫ > 0 we let (γ, π) be an optimal protocol. We show that an (α+2√ǫ)-optimal +protocol (γ∗, π∗) can be computed in polynomial time. As a first step we define a signaling scheme γ∗ supported in +Ξq as follows: +γ∗ +˜ξ = +� +ξ∈supp(γ) +γξγξ +˜ξ +∀˜ξ ∈ Ξq, +where γξ ∈ ∆Ξq is the signaling scheme satisfying Lemma 6 with q = 2 log(2m/α) +ǫ2 +. First we observe that γ∗ ∈ ∆Ξq +satisfies the consistency constraints, indeed we have: +� +˜ξ∈Ξq +γ∗ +˜ξ ˜ξθ = +� +ξ∈supp(γ) +γξ +� +˜ξ∈Ξq +γξ +˜ξ ˜ξθ = +� +ξ∈supp(γ) +γξξθ = µθ +∀θ ∈ Θ. +Moreover, let π∗ : ∆Θ × A → R+ be the optimal payment function in each ˜ξ ∈ Ξq. We show that the protocol +(γ∗, π∗, 0) is (α + 2√ǫ)-optimal. Let π′′ : ∆Θ × A → R+ be the optimal payment function in each ξ ∈ Ξq when the +buyer is playing an ǫ-best response, i.e., +π′′(ξ, ·) ∈ arg max +˜π(ξ,·) +� +k∈K +λk[ +� +θ +ξθus +θ(bk,ǫ +ξ,˜π) − ˜π(ξ)]. +Moreover, let π′ : ∆Θ ×A → R+ be the payment function such that π′(ξ, a) = (1−√ǫ)π′′(ξ, a)+√ǫ � +θ∈θ ξθus +θ(a) +for each ξ and A. Then, we have: +� +˜ξ∈Ξq +γ∗ +˜ξ +� � +θ∈Θ +˜ξθus +θ(bk +˜ξ,π∗) − π∗(˜ξ, bk +˜ξ,π∗) +� +≥ +� +˜ξ∈Ξq +γ∗ +˜ξ +� � +θ∈Θ +˜ξθus +θ(bk +˜ξ,π′) − π′(˜ξ, bk +˜ξ,π′) +� +(Optimality of π∗) +≥ +� +˜ξ∈Ξq +γ∗ +˜ξ +� � +θ∈Θ +˜ξθus +θ(bk,ǫ +˜ξ,π′′) − π′′(˜ξ, bk,ǫ +˜ξ,π′′) +� +− 2√ǫ +(By Proposition 3) +≥ +� +ξ∈supp(γ) +γξ +� � +˜ξ∈Ξq +γξ +˜ξ +� � +θ∈Θ +˜ξθus +θ(bk,ǫ +˜ξ,π′′) − π′′(˜ξ, bk,ǫ +˜ξ,π′′) +�� +− 2√ǫ +(By defintion of γ∗) +≥ +� +ξ∈supp(γ) +γξ +� � +θ∈Θ +ξθus +θ(bk +ξ,π) − π(ξ, bk +ξ,π) +� +− α − 2√ǫ +(By Lemma 6). +Notice that the optimal payment π∗ : ∆Θ × A → R+ in each ξ ∈ Ξq can be computed in polynomial time employing +Lemma 7. Hence, to compute the optimal signaling scheme γ∗ ∈ ∆Ξq we can solve the following LP: +� +k∈K +λk +� +ξ∈Ξq +γξ +� +θ∈Θ +ξθus +θ(bk +ξ,π∗) − π∗(ξ, bk +ξ,π∗) s.t. +24 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +� +ξ∈supp(γ) +γξξθ = µθ +∀θ ∈ Θ. +Note that since |Ξq| = O(dq), all the payment function π∗ : Ξq × A → R+ can be precomputed in polynomial time. +Moreover, the LP has polynomially many variables and constraints and can be solved efficiently. Finally, the solution +returned by the LP is α + 2√ǫ-optimal. This concludes the proof. +Theorem 7. Restricted to instances in which the buyer has limited liability, there exists an algorithm that, given any +α, ǫ > 0 and ρ ∈ (0, 1/2] as input, returns a protocol without menus achieving a seller’s expected utility greater +than or equal to ρ OPT − 2−Ω(1/ρ) − (α + 2√ǫ), where OPT is the seller’s expected utility in an optimal protocol. +Moreover, the algorithm runs in time polynomial in Ilog m—where I is the size of the problem instance and m is +the number of buyer’s actions—, and the seller’s expected utility in the returned protocol is greater than or equal +to OPTLIN − (α + 2√ǫ), where OPTLIN is the best expected utility achieved by a protocol parametrized by β as in +Corollary 2. +Proof. The proof follows the same steps of Theorem 6. However, it relies on Theorem 4 instead of Lemma 7 to +compute an approximate payment function for all q-uniform posteriors. +Lemma 8. Given any α > 0, a posterior ξ∗ ∈ ∆Θ, and a payment function π : ∆Θ × A → R+ that is optimal in +every posterior ξ ∈ Ξq with q = ⌈9d/α2⌉, there exists a signaling scheme γ ∈ ∆Ξq: +� +ξ∈Ξq +γξ +�� +θ∈Θ +ξθ us +θ(bk +ξ,π) − π(ξ, bk +ξ,π) +� +≥ +� +θ∈Θ +ξ∗ +θus +θ(bk +ξ∗,π) − π(ξ∗, bk +ξ∗,π) − α, +for every receiver’s type k ∈ K. Furthermore, the signaling scheme γ satisfies: +� +ξ∈Ξq +γξ ξθ = ξ∗ +θ +∀θ ∈ Θ. +Proof. We define a payment function π′ : ∆Θ ×A → R+ as follows: π′(ξ, a) = π(ξ∗, a) for each a ∈ A and ξ ∈ ∆Θ. +Furthermore, we define: +Iα(ξ∗) = +� +ξ ∈ ∆Θ : ∥ξ − ξ∗∥∞ ≤ α2 +18d +� +, +as the neighborhood of the given posterior ξ∗ ∈ ∆Θ and Ξ(ξ∗) = Iǫ(ξ∗) ∩ Ξq its intersection with the set Ξq. Notice +that if q ≥ 18d +α2 , it holds ξ∗ ∈ co(Ξ(ξ∗)). 14 +We show that for each ξ ∈ Iα(ξ∗), it holds bk +ξ∗,π′ ∈ Bk,ǫ +ξ,π′, where ǫ := α2/9. As a first step, by Hölder’s inequality we +have that +� +θ∈Θ +|(ξθ − ξ∗ +θ)us +θ(a)| ≤ d||ξ − ξ∗||∞ = ǫ/2 ∀a ∈ A, +Moreover, by the definition of best response and the previous inequality, we have that: +� +θ∈Θ +ξθuk +θ(bk +ξ∗,π′) + π′(ξ, bk +ξ∗,π′) ≥ +� +θ∈Θ +ξ∗ +θuk +θ(bk +ξ∗,π′) + π′(ξ∗, bk +ξ∗,π′) − ǫ/2 +≥ +� +θ∈Θ +ξ∗ +θuk +θ(bk +ξ,π′) + π′(ξ∗, bk +ξ,π′) − ǫ/2 +≥ +� +θ∈Θ +ξθuk +θ(bk +ξ,π′) + π′(ξ, bk +ξ,π′) − ǫ +≥ +� +θ∈Θ +ξθuk +θ(a′) + π′(ξ, a′) − ǫ, +for each a′ ∈ A. This shows that bk +ξ∗,π′ ∈ Bk,ǫ +ξ,π′. +Let π∗ : ∆Θ × A → R+ be the payment function prescribed by Proposition 3. Then, we have that: +14Given a finite set A we denote with co(A) the set containing all the convex combination of elements in A. +25 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +� +θ∈Θ +ξθus +θ(bk +ξ,π) − π(ξ, bk +ξ,π) ≥ +� +θ∈Θ +ξθus +θ(bk +ξ,π∗) − π∗(ξ, bk +ξ,π∗) +(Optimality of π′) +≥ +� +θ∈Θ +ξθus +θ(bk,ǫ +ξ,π′) − π′(ξ, bk,ǫ +ξ,π′) − 2√ǫ +(By Proposition 3 ) +≥ +� +θ∈Θ +ξθus +θ(bk +ξ∗,π′) − π′(ξ, bk +ξ∗,π′) − 2√ǫ +≥ +� +θ∈Θ +ξ∗ +θus +θ(bk +ξ∗,π′) − π′(ξ∗, bk +ξ∗,π′) − 2√ǫ − ǫ += +� +θ∈Θ +ξ∗ +θus +θ(bk +ξ∗,π′) − π′(ξ∗, bk +ξ∗,π′) − 3√ǫ += +� +θ∈Θ +ξ∗ +θus +θ(bk +ξ∗,π) − π(ξ∗, bk +ξ∗,π) − 3√ǫ. +This shows that the expected seller’s utility decreases of at most 3√ǫ when we consider sufficiently close posteriors. +Hence, by Caratheodory’s theorem we can decompose ξ∗ as follows: +� +ξ′∈Ξ(ξ) +γξ∗ +ξ′ ξ′ +θ = ξ∗ +θ +∀θ ∈ Θ +with γξ∗ ∈ ∆Ξ(ξ∗), where we recall that ξ∗ ∈ co(Ξ(ξ∗)). We show now that such a decomposition decreases the +expected seller’s utility only by the desired amount. Formally, we have that: +� +ξ′∈Ξ(ξ) +γξ∗ +ξ′ +� � +θ∈Θ +ξ′ +θus +θ(bk +ξ′,π) + π(ξ′, bk +ξ′,π) +� +≥ +� +ξ′∈Ξ(ξ) +γξ∗ +ξ′ +� � +θ∈Θ +ξ∗ +θus +θ(bk +ξ∗,π) + π(ξ, bk +ξ∗,π) − 3√ǫ +� += +� +θ∈Θ +ξ∗ +θus +θ(bk +ξ∗,π) + π(ξ∗, bk +ξ∗,π) − 3√ǫ. +Since 3√ǫ ≤ α, this concludes the proof. +Theorem 8. Restricted to instances in which the buyer has limited liability and the number of states of nature d is fixed, +there exists an algorithm that, given α > 0 and ρ ∈ (0, 1/2] as input, returns in polynomial time a protocol without +menus achieving a seller’s expected utility greater than or equal to ρ OPT − 2−Ω(1/ρ) − α, where OPT is the seller’s +expected utility in an optimal protocol. Moreover, the seller’s expected utility in the returned protocol is greater than +or equal to ρ OPTLIN − 2−Ω(1/ρ) − α where OPTLIN is the best expected utility achieved by a protocol parametrized +by β as in Corollary 2. +Proof. Given a constant α > 0 we let (γ, π) be an optimal protocol. As a first step, we show that there exists a protocol +(γ∗, π∗) achieving a seller’s expected utility of at least APX ≥ ρOPT−2−Ω(1/ρ)−α, where OPT is the utility achieved +with (γ, π). Moreover, the payment function π∗ is a linear function with parameter β ∈ {1 − 2−i}i∈{i,...,⌊ρ/2⌋}. We +define a signaling scheme γ∗ supported in Ξq as follows: +γ∗ +˜ξ = +� +ξ∈supp(γ) +γξγξ +˜ξ +∀˜ξ ∈ Ξq, +where γξ ∈ ∆Ξq is the signaling scheme satisfying Lemma 6 with q = 18d +α2 . First we observe that γ∗ ∈ ∆Ξq satisfies +the consistency constraints, indeed we have: +� +˜ξ∈Ξq +γ∗ +˜ξ ˜ξθ = +� +ξ∈supp(γ) +γξ +� +˜ξ∈Ξq +γξ +˜ξ ˜ξθ = +� +ξ∈supp(γ) +γξξθ = µθ +∀θ ∈ Θ. +Moreover, we can define as π∗ : ∆Θ × A → R+ as the payment function computed (in polynomial time) with +Corollary 2 in each q-uniform posterior. +Let π′ be the optimal payment function. We show that the protocol (γ∗, π∗) achieves the desired approximation. +Formally: +� +˜ξ∈Ξq +γ∗ +˜ξ +� � +θ∈Θ +˜ξθus +θ(bk +˜ξ,π∗) − π∗(˜ξ, bk +˜ξ,π∗) +� +26 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +≥ ρ + + � +˜ξ∈Ξq +γ∗ +˜ξ +� � +θ∈Θ +˜ξθus +θ(bk +˜ξ,π′) − π′(˜ξ, bk +˜ξ,π′) +� + + − 2Ω(1/ρ) +(Corollary 2) += +� +ξ∈supp(γ) +γξ +� � +˜ξ∈Ξq +γξ +˜ξ +� � +θ∈Θ +˜ξθus +θ(bk,ǫ +˜ξ,π) − π(˜ξ, bk,ǫ +˜ξ,π) +�� +(By defintion of γ∗) +≥ +� +ξ∈supp(γ) +γξ +� � +θ∈Θ +ξθus +θ(bk +ξ,π) − π(ξ, bk +ξ,π) +� +− α +(By Lemma 8). +This implies that since (γ∗, π∗) is feasible for the following LP, it has value at least ρOPT − 2Ω(1/ρ) − α. +max +γ≥0 +� +k∈K +λk +� +ξ∈Ξq +γξ +� +θ∈Θ +ξθus +θ(bk +ξ,π∗) − π∗(ξ, bk +ξ,π∗) s.t. +� +ξ∈supp(γ) +γξξθ = µθ +∀θ ∈ Θ. +Hence, to find the desired approximation it is sufficient to compute π∗ in each q-uniform posterior and solve the LP. +Note that since |Ξq| = O(qd), the computation of the payment function π∗ : ∆Θ × A → R+ and the computation of +the previous LP require polynomial time for each fixed α > 0. +Finally, to prove the second part of the statement it is sufficient to notice that π∗ is optimal with respect to the desired +set of linear payment functions. +D +Proofs Omitted from Section 6 +Theorem 10. The problem of computing a seller-optimal protocol without menus is APX-hard, even when the number +of buyer’s actions m is fixed. +Proof. We introduce a reduction from LINEQ-MA(1 − ζ, δ) to the design of the optimal protocol, showing that for ζ +and δ small enough, the following holds: +• Completeness: If an instance of LINEQ-MA(1 − ζ, δ) admits a 1 − ζ fraction of satisfiable equations when +variables are restricted to lie in the hypercube {0, 1}nvar, then there exists a protocol that provides to the +seller’s expected utility at least of η, where η will be defined in the following; +• Soundness: If at most a δ fraction of the equations can be satisfied, then every protocol provides to the seller’s +expected utility at most η − c, where c is a constant defined in the following. +In the rest of the proof, given a vector of variables x ∈ Qnvar, for i ∈ [nvar], we denote with xi the component +corresponding to the i-th variable. Similarly, for j ∈ [neq], cj is the j-th component of the vector c, whereas, for +i ∈ [nvar] and j ∈ [neq], the (j, i)-entry of A is denoted by Aji. +Reduction +As a preliminary step, we normalize the coefficients by letting ¯A := 1 +τ A and ¯c := +1 +τ 2 c, where we let +τ := 2M max +� +maxi∈[nvar],j∈[neq] Aji, maxj∈[neq] cj, n2 +var +� +and M will be defined in the following. It is easy to see +that the normalization preserves the number of satisfiable equations. Formally, the number of satisfied equations of +Ax = c is equal to the number of satisfied equations of ¯A¯x = ¯c, where ¯x = 1 +τ x. For every variable i ∈ [nvar], we +define a state of nature θi ∈ Θ. Moreover, we introduce three additional states θ0, θ1, θ2 ∈ Θ. The prior distribution +µ ∈ int(∆Θ) is defined in such a way that µθi = +1 +2n2var for every i ∈ [nvar], while µθ0 = +nvar−1 +2nvar , µθ1 = +1 +4, and +µθ2 = 1 +4 (notice that � +θ∈Θ µθ = 1). We define four buyer’s types k1 +j , k2 +j , k3 +j , k4 +j ∈ K for each equation j ∈ [neq], +where the probability of observing each buyer’s type is +1 +8neq . Moreover, we define an additional type k⋆. All the types +k ∈ K have budget bk = ν/2, where ν will be defined in the following. The buyer has 9 actions available, namely +A := {a0, a1, a2, a3, a4, a5, a6, a7, a8}. Then, we define the utilities of the players, where the utility is 0 when not +specified. For each k1 +j , j ∈ [neq], the utilities are: +• u +k1 +j +θi (a0) = 1 +2 for each i ∈ [nvar], +27 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +• u +k1 +j +θi (a1) = 1 +2 − ¯Aji + ¯cj for each i ∈ [nvar], +• u +k1 +j +θi (a2) = 1 +2 + ¯Aji − ¯cj for each i ∈ [nvar] +• u +k1 +j +θ0 (a0) = 1 +2, +• u +k1 +j +θ0 (a1) = 1 +2 + ¯cj, +• u +k1 +j +θ0 (a2) = 1 +2 − ¯cj. +• u +k1 +j +θ1 (a3) = 1 +2 + 2ν, +For each k2 +j , j ∈ [neq], the utilities are: +• u +k2 +j +θi (a0) = 1 +2 − ¯Aji + ¯cj for each i ∈ [nvar], +• u +k2 +j +θi (a7) = 1 +2 for each i ∈ [nvar] +• u +k2 +j +θ0 (a0) = 1 +2 + ¯cj, +• u +k2 +j +θ1 (a3) = 1 +2 + 2ν, +For each type k3 +j , j ∈ [neq] the utilities are: +• u +k3 +j +θi (a0) = 1 +2 + ¯Aji − ¯cj for each i ∈ [nvar], +• u +k3 +j +θi (a7) = 1 +2 for each i ∈ [nvar] +• u +k3 +j +θ0 (a1) = 1 +2 − ¯cj, +• u +k3 +j +θ1 (a3) = 1 +2 + 2ν, +For each type k4 +j , j ∈ [neq] the utilities are equivalent to the one of type k1 +j but with the following differences: +• ukj +θ (a5) = 1 +2 for each θ ∈ Θ, +• ukj +θi (a7) = 1 +2 for each i ∈ [nvar], +Finally, the utilities of type k⋆ are: +• uk⋆ +θ1 (a6) = 1, +• uk⋆ +θ (a1) = 1 for each θ ∈ Θ. +Moreover, we let uk +θ(a8) = 1 +2 for every k ∈ K and θ ∈ Θ. Finally, the utility of the seller is: +• us +θ(a6) = 1 +4 for each θ ∈ Θ, +• us +θ(a5) = 4ν for each θ ∈ Θ, +28 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +• us +θ(a0) = ν for each θ ∈ Θ, +• us +θ(a7) = 2ν for each θ ∈ Θ. +We recall that the utility is 0 when not defined explicitly. +Completeness. +Suppose that there exists a vector ˆx ∈ {0, 1}nvar such that at least a fraction 1 − ζ of the equations in +Aˆx = c are satisfied. Let X1 ⊆ [nvar] be the set of variables i ∈ [nvar] with ˆxi = 1, while X0 := [nvar] \ X1. Given +the definition of ¯A and ¯c, there exists a vector ¯x ∈ {0, 1 +τ }nvar such that at least a fraction 1 − ζ of the equations in +¯A¯x = ¯c are satisfied, and, additionally, ¯xi = 1 +τ for all the variables in i ∈ X1, while ¯xi = 0 whenever i ∈ X0. Let us +consider an (indirect) signaling scheme φ : Θ → ∆S where the set of signals is S := {s1, s2, s3}. Let q := nvar(nvar−1) +τ−|X1| . +For each i ∈ [nvar], let φθi(s1) = q and φθi(s2) = 1 − q if i ∈ X1, while φθi(s2) = 1 otherwise. Moreover, let +φθ0(s1) = 1, φθ1(s3) = 1 and φθ2(s2) = 1. Then, all the other probabilities φθ(s) are set to 0. It is easy to see that +the signaling scheme is feasible. Moreover, we set the price p = ν/2. Finally, we set π(s3, a6) = 2ν and all the other +payments π(s, a) = 0. +Now, we compute the expected seller’s utility due of each type of buyer. +• The buyer of type k⋆ in the posterior ξs3 plays the action a6 and gets utility � +θ∈θ ξs3 +θ uk⋆ +θ (a6) + π(s3, a6) = +1 + 2ν. +Moreover, in the other posteriors ξs1 and ξs2 the seller’s utility is at least 0. +Finally, the +protocol is IR for the buyer since the expected utility declining the protocol is 1 while accepting it is +−π/2 + 1 · 3 +4 + (1 + 2ν) 1 +4 = 1. Hence, the expected principal utility when the buyer’s type is k⋆ is at +least � +s∈S +� +θ∈θ ξs +θuk⋆ +θ (bk⋆ +ξs,π) = +1 +16. +• Consider a buyer k1 +j , j ∈ [neq], such that the j-th equality is satisfied by the vector ˆx. Now, let us take +the buyer’s posterior ξs1 ∈ ∆Θ induced by the signal s1. Let h := +q +n2var +� +i∈X1 +q +n2var + nvar−1 +nvar +. Then, using the +definition of ξs1, it is easy to check that ξ1 +θi = h for every i ∈ Xs1, ξs1 +θi = 0 for every i ∈ X0, while ξs1 +θ0 = +nvar−1 +nvar +� +i∈X1 +q +n2var + nvar−1 +nvar += 1 − h +��X1��. The buyer of type kj ∈ K experiences a utility of � +θ∈Θ ξs1 +θ ukj +θ (a0) = 1 +2 +by playing action a0. Instead, the utility she gets by playing a1 is defined as follows: +� +θ∈Θ +ξs1 +θ ukj +θ (a1) = +� +i∈X1 +h +�1 +2 − ¯Aji + ¯cj +� ++ ξ1 +θ0 +�1 +2 + ¯cj +� += += h +��X1�� +�1 +2 + ¯cj +� +− h +� +i∈X1 +¯Aji + +� +1 − h +��X1��� �1 +2 + ¯cj +� += += 1 +2 + ¯cj − h +� +i∈X1 +¯Aji = 1 +2 + ¯cj − 1 +τ +� +i∈X1 +¯Aji = 1 +2, +where the second to last equality holds since h = 1 +τ (by definition of h and q), while the last equality follows +from the fact that the j-th equation is satisfied, and, thus, 1 +τ +� +i∈X1 ¯Aji = ¯cj (recall that ¯xi = +1 +τ for all +i ∈ X1). Using similar arguments, we can write � +θ∈Θ ξs1 +θ ukj +θ (a2) = 1 +2. Moreover, all the other actions have +utility 0. Hence, the buyer plays a0 in the posterior ξs1. In posterior ξs2 induced by signal s2, the utility of +each action different from a8 is strictly smaller than 1 +2. Hence, the buyer will play a8, while in posterior ξs3 +induced by signal s3, the utility of action a3 is 1 +2 + 2ν and the buyer will play a3. Hence the expected utility +of the buyer is 3 +4 +1 +2 + 1 +4( 1 +2 + 2ν) − p = 1 +2. Moreover, the protocol is IR for the buyer since if she declines the +protocol the utility is 1 +2 while if she accepts the protocol the utility is 1 +2. Hence, when the buyer’s type is k1 +j +the expected seller’s utility is νµθ0 + ν/2. +• Consider a buyer k2 +j or k3 +j , j ∈ neq such that the j-th equality is satisfied. A similar argument as before +shows that in posterior ξs1 the buyer’s optimal action is a7, while in posterior ξs2, the optimal action is a8. +In posterior ξs3, the optimal action is a3. Hence, the expected buyer’s utility is 3 +4 +1 +2 + 1 +4( 1 +2 + 2ν) − p = 1 +2. +Hence, the protocol is IR for the buyer and provides expected seller’s utility at least 2νµθ0 + ν/2. +• Consider a buyer k4 +j , j ∈ [neq] such that the j-th equality is satisfied. The buyer has an utility similar to k1 +j +and plays the same best responses. Hence, it is indifferent in participating or not participating to the protocol. +29 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +We assume that they brake ties in favor of the seller and does not accept. She plays action a5 and the expected +seller’s utility is 2ν. +Since all the other buyer’s types provide positive utility —it never happens that the expected payment from the seller +to the buyer exceeds the payment from the buyer to the seller—, the expected seller’s utility is at least +η = 1 +32 + (1 − ζ)1 +8(µθ0ν + ν/2) + (1 − ζ)1 +4(µθ02ν + ν/2) + (1 − ζ)1 +82ν +Soundness +As a first step, we upperbound the expected seller’s utility from each type. It is easy to see that the +maximum expected utility that the seller can extract from the buyer’s type k⋆ is at most +1 +32. Moreover, the maximum +expected utility that the seller can extract from a buyer of type k1 +j , j ∈ [neq], is at most 1 +8(ν). The maximum expected +utility that the seller can extract from a buyer of type k2 +j or k3 +j , j ∈ [neq] is at most 1 +4 +3 +2ν. Finally, the maximum +expected utility that the seller can extract from a buyer of type k4 +j , j ∈ [neq], is 1 +82ν. +Using the previous upperbounds, we can bound the component of the utility due to each set of types. For each constant +t < 1, there exist constants c = c(t), ζ = ζ(t) such that if the expected utility is greater than η − c then the expected +utility from types k1 +j , j ∈ [neq], is at least t 1 +8(ν), the expected utility from types k2 +j , j ∈ [neq], and k3 +j , j ∈ [neq], is at +least t 1 +4 +3 +2ν, and the expected utility from types k4 +j , j ∈ [neq], is at least t 1 +82ν. To see that, consider for instance the +types k1 +j , j ∈ [neq]. It must hold: +1 +32 + ¯t1 +8ν + 1 +4 +3 +2ν + 1 +82ν ≥ 1 +32 + (1 − ζ)1 +8(µθ0ν + ν/2) + (1 − ζ)1 +4(µθ02ν + ν/2) + (1 − ζ)1 +82ν +Since for nvar large enough µθ0 is close to 1 +2, for c(t), ζ(t) small enough constant the equation is satisfied for ¯t ≥ t. A +similar result holds for every other set of types k2 +j with j ∈ [neq], k3 +j with j ∈ [neq], and k4 +j with j ∈ [neq]. +The next step is to show the existence of a posterior in which a t fraction of agent of types k1 +j , j ∈ [neq], play a0 and the +the same holds for each other set of types k2 +j ,k3 +j with action a7. Suppose by contradiction that there is no posterior in +which a t fraction of k1 +j , j ∈ [neq], plays a0. First, notice that the maximum payment is at most p = ν/2+ 1 +M , otherwise +all the buyer’s types k1 +j are not IR. Moreover, the seller’s utility minus payment is greater than 0 in a posterior only +if the agent plays a0. Finally, it is easy to see that it is sufficient to consider signaling schemes that induce posteriors +such that if ξθi > 0, then ξθ1 = 0 and ξθ2 = 0 since states ξθ1 and ξθ2 disincentivize the actions with high seller’s +utility. Hence, the maximal utility from agents of types k1 +j is at most +1 +8 +� +ν/2 + 1 +M + (t − 1/neq)1 +2ν +� +< t1 +8ν, +for M large enough, reaching a contradiction. A similar argument holds for the other types. This implies that there +exists a set Q ⊆ [neq] and a posterior ξ such that for each j ∈ Q all the buyers k1 +j , k2 +j , and k3 +j in the posterior play +a0,a7, and a7, respectively. Notice that |Q| ≥ 1 − 3(1 − t) and for t large enough |Q| > δ. +Suppose that there exists a signal inducing a posterior ξ ∈ ∆Θ in which all the buyer’s types k1 +j , j ∈ Q best respond by +playing action a0. We show that there exists at least one j ∈ Q such that it holds � +θ∈Θ ξθu +k1 +j +θ (a1) > � +θ∈Θ ξθu +k1 +j +θ (a0) +or � +θ∈Θ ξθu +k1 +j +θ (a2) > � +θ∈Θ ξθu +k1 +j +θ (a0). For every buyer’s type k1 +j ∈ K, it holds � +θ∈Θ ξθukj +θ (a0) = 1 +2. Moreover, it +is the case that: +� +θ∈Θ +ξθu +k1 +j +θ (a1) = +� +i∈[nvar] +ξθi +�1 +2 − ¯Aji + ¯cj +� ++ ξθ0 +�1 +2 + ¯cj +� += 1 +2 + ¯cj − +� +i∈[nvar] +ξθi ¯Aji. +Similarly, it holds: +� +θ∈Θ +ξθu +k1 +j +θ (a2) = 1 +2 − ¯cj + +� +i∈[nvar] +ξθi ¯Aji. +Suppose by contradiction that for every type k1 +j , j ∈ Q, it is the case that � +θ∈Θ ξθu +k1 +j +θ (a0) ≥ � +θ∈Θ ξθu +k1 +j +θ (a1), +which implies that ¯cj − � +i∈[nvar] ξθi ¯Aji ≤ 0, whereas it holds � +θ∈Θ ξθu +k1 +j +θ (a0) ≥ � +θ∈Θ ξθukj +θ (a2), implying +−¯cj + � +i∈[nvar] ξθi ¯Aji ≤ 0. +Thus, � +i∈[nvar] ξθi ¯Aji = ¯cj for every j ∈ Q and the vector ˆx ∈ Qnvar with +ˆxi = ξθi for all i ∈ [nvar] satisfies at least a fraction δ of the equations, reaching a contradiction. +Since we +30 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +have that t types k1 +j play a0, this implies that π(ξ, a0) > 0. However, at the same time we have that the buy- +ers of type k2 +j and k3 +j plays action a7. +Consider a j∗ ∈ Q such that � +θ∈Θ ξθu +k1 +j∗ +θ +(a1) > � +θ∈Θ ξθu +k1 +j∗ +θ +(a0) +or � +θ∈Θ ξθu +k1 +j∗ +θ +(a2) > � +θ∈Θ ξθu +k1 +j∗ +θ +(a0). +Recall that this buyer must play a7. +If the first inequality holds +then it must hold � +θ∈Θ ξθu +k2 +j∗ +θ +(a7) + π(ξ, a7) ≥ � +θ∈Θ ξθu +k2 +j∗ +θ +(a0) + π(ξ, a0). Moreover, � +θ∈Θ ξθu +k2 +j∗ +θ +(a7) = +� +θ∈Θ ξθu +k1 +j∗ +θ +(a0) < � +θ∈Θ ξθu +k1 +j∗ +θ +(a1) = � +θ∈Θ ξθu +k2 +j∗ +θ +(a0), implying π(ξ, a7) > π(ξ, a0). A similar argument +holds for the buyer k3 +j∗ if the second inequality is satisfied. This implies that type k4 +j∗ can play the same best responses +of player k1 +j in any posterior different from ξ and play action a7 in ξ. Hence, the expected utility of buyer k4 +j∗ is strictly +greater than the one of k1 +j∗ (that is IR), and hence it is strictly IR. +We conclude the proof showing that the utility of this buyer’s type is too small, reaching a contradiction. First, +notice that the seller must induce a posterior with ξθ1 ≥ +3 +4 with probability at least 1 +8. In all the other posteriors +the seller’s utility from type k∗ is 0. However, it must hold that the utility from type k⋆ is at least +1 +64 for ν small +enough. Hence, playing posteriors with ξθ1 ≥ +3 +4 with probability smaller than 1 +8 the seller’s utility form type k∗ +is at most 1 +2 +1 +4 +1 +8 < +1 +64. Now consider the type k4 +j∗ that is IR. In a posterior ξ with ξθ1 ≥ +3 +4, the seller’s utility +when the type is k4 +j∗ is at most 0. Hence, the total utility from this type is at most p + 7 +84ν ≤ ν + 1/M, where +the last inequality follows by the fact that the payment is at most ν +2 + 1/M. For |Q| large enough, we have that a +|Q|/neq − δ fraction of types k4 +j provide seller’s utility at most ν + 1/M. Hence, the total utility from type k4 +j is at +most 1 +8[(|Q|/neq − δ)(ν + 1/M) + (1 − (|Q|/neq − δ))2ν] ≤ t 1 +82ν. Thus, we reach a contradiction. +Theorem 11. There exists an algorithm that, given any α > 0 and ρ ∈ (0, 1/6] as input, computes a protocol without +menus whose seller’s expected utility is greater than or equal to ρ OPT − 2−Ω(1/ρ) − α, where OPT is the seller’s +expected utility in an optimal protocol. Moreover, the algorithm runs in time polynomial in Ilog m—where I is the +size of the problem instance—when it is implemented with the algorithm in Theorem 7 as a subroutine, while it runs in +time polynomial in Id when it is implemented with the algorithm in Theorem 8 as a subroutine. +Proof. Let (φ, p, π) be an optimal protocol. Then, the seller’s expected utility is given by: +� +k /∈Rφ,p,π +λk +� +θ∈Θ +µθus +θ(bk +µ) + +� +k∈Rφ,p,π +λk +�� +s∈S +� +θ∈Θ +µθφθ(s) +� +us +θ(bk +ξs,π) − π(s, bk +ξs,π) +� ++ p +� +, +where we recall that Rφ,p,π is the set of buyer’s types for which the IR constraint is satisfied under protocol (φ, p, π). +Given a signal s ∈ S and a type k ∈ K, let bk +ξs ∈ arg maxa∈A +� +θ∈Θ µθφθ(s)uk +θ(a). Intuitively, bk +ξs is an opti- +mal action for the buyer without considering the payment function. Then, the seller’s utility can be spitted in three +components: +(i) The utility from the buyer’s types that are not IR +U1 := +� +k /∈Rφ,p,π +λk +� +θ∈Θ +µθus +θ(bk +µ); +(ii) The maximum seller’s utility deriving from the buyer’s action +U2 := +� +k∈Rφ,p,π +λk +�� +s∈S +� +θ∈Θ +µθφθ(s) +� +us +θ(bk +ξs) + uk +θ(bk +ξs,π) − uk +θ(bk +ξs) +� +� +, +where we use the fact that to incentivize action bk +ξs,π over bk +ξs the payment must be at least +� +s∈S +� +θ∈Θ µθφθ(s)(uk +θ(bk +ξs)−uk +θ(bk +ξs,π)) +� +θ∈Θ µθφθ(s) +; +(iii) The utility related to the overall payment that the seller’s can extract from the buyer given the price function +π +U3 := +� +k∈Rφ,p,π +λk +� +p − +� +s +� +θ +µθφθ(s) +� +π(s, bk +s,π) + uk +θ(bk +s,π) − uk +θ(bk +ξs) +� +� +. +31 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +Notice that the term U2 + U3 is the utility deriving from buyer’s types for which the IR constraint is satisfied, where +we add, respectively subtract, the term +� +k∈Rφ,p,π +λk +�� +s∈S +� +θ∈Θ +µθφθ(s) +� +uk +θ(bk +ξs,π) − uk +θ(bk +ξs) +� +� +to U2, respectively U3. +In the following, we design three protocols (φ1, p1, π1), (φ2, p2, π2), and (φ3, p3, π3), each with seller’s utility that +approximates the corresponding utility terms U1, U2, and U3. We will show that this will implies that at least one +protocol provides a good approximation of the overall seller’s utility, i.e., of U1 + U2 + U3. +Approximate U1. +The protocol (φ1, p1, π1) that provides no information, charges no price, and does not provides +any payment has seller’s utility +� +k∈K +� +θ +µθus +θ(bk +µ) ≥ +� +k /∈Rφ,p,π +� +θ +µθus +θ(bk +µ) = U1 +Approximate U2. +By Corollary 2, we know that for each signal s ∈ S (inducing a posterior ξs) and ρ ∈ (0, 1/2], +there exists a linear contract π′(s, ·) such that π′(s, a) = β � +θ∈Θ ξs +θus +θ(a) with parameter β = 1 − 2−i, i ∈ +{1, . . . , ⌊ 1 +2ρ⌋} that guarantees: +� +k∈K +λk +� � +θ∈Θ +ξs +θ +� +us +θ(bk +ξs,π′) − π′(ξs, bk +ξs,π′) +� � +(11a) +≥ ρ +� +k∈K +λk +�� +θ +ξs +θ +� +us +θ(bk +ξs,π) + uk +θ(bk +ξs,π) − uk +θ(bk +ξs) +� +� +− 2−Ω(1/ρ) +(11b) +≥ ρ +� +k∈Rφ,p,π +λk +�� +θ +ξs +θ +� +us +θ(bk +ξs,π) + uk +θ(bk +ξs,π) − uk +θ(bk +ξs) +� +� +− 2−Ω(1/ρ) +(11c) +where the first inequality comes from Corollary 2, and the last one since we restrict the elements in the first summation. +Now, we need a protocol (φ2, p2, π2) that approximate the utility obtained by the optimal protocol that uses only linear +payment functions. When the number of states is fixed, we can approximate the optimal protocol that uses linear +payment functions using Theorem 8 with an additive loss α. Otherwise, we can use Theorem 7 that is polynomial time +when the number of actions is fixed, while it runs in quasi-polynomial time and provides a loss α when instantiated +with sufficiently small parameters. Hence, protocol (φ2, p2, π2) can be computed in time poly(min{Id, Ilog(m)}). +Notice that both the algorithms returns a protocol such that p = 0 and hence p2 = 0. Then, we can show that the +protocol (φ2, p2, π2) has seller’s utility +� +k∈K +λk +�� +s∈S +� +θ +µθφθ(s) +� +us +θ(bk +ξs,π2) − π2(s, bk +ξs,π2) +� +� +≥ +� +k∈K +λk +�� +s∈S +� +θ +µθφθ(s) +� +us +θ(bk +ξs,π′) − π′(s, bk +ξs,π′ +� +� +− α += +� +k∈K +λk +�� +s∈S +�� +θ +µθφθ(s) +� � +θ +ξs +θ +� +us +θ(bk +ξs,π′) − π′(s, bk +ξs,π′ +� +� +− α += +� +s∈S +�� +θ +µθφθ(s) +� � +k∈K +λk +�� +θ +ξs +θ +� +us +θ(bk +ξs,π′) − π′(s, bk +ξs,π′ +� +� +− α +≥ +� +s∈S +�� +θ +µθφθ(s) +� � +ρ +� +k∈R +λk +� +θ +ξs +θ +� +us +θ(bk +ξs,π) + uk +θ(bk +ξs,π) − uk +θ(bk +ξs) +� +− 2−Ω(1/ρ) +� +− α += +� +s∈S +�� +θ +µθφθ(s) +�  +ρ +� +k∈Rφ,p,π +λk +� +θ +ξs +θ +� +us +θ(bk +ξs,π) + uk +θ(bk +ξs,π) − uk +θ(bk +ξs) +� + + − 2−Ω(1/ρ) − α +32 + +ARXIV PREPRINT - FEBRUARY 1, 2023 += ρ +� +k∈Rφ,p,π +λk +� +s∈S +�� +θ +µθφθ(s) +� �� +θ +ξs +θ +� +us +θ(bk +ξs,π) + uk +θ(bk +ξs,π) − uk +θ(bk +ξs) +� +� +− 2−Ω(1/ρ) − α += ρ +� +k∈Rφ,p,π +λk +� +s∈S +� +θ +µθφθ(s) +� +us +θ(bk +ξs,π) + uk +θ(bk +ξs,π) − uk +θ(bk +ξs) +� +− 2−Ω(1/ρ) − α, +where the first inequality holds since π′ employs linear payments functions and (φ2, p2, π2) has an additive loss α +w.r.t. any protocol that employs linear payments functions, while the second inequality comes from Equation (11). +Approximate U3. +Let δk := � +θ µθuk +θ(bk +θ) − � +θ µθuk +θ(bk +µ) for each k ∈ K, where bk +θ is the best response of agent +of type k ∈ K when the state of nature is θ. For each k ∈ Rφ,p,π, by the definition of IR it holds +� +s∈S +� +θ∈Θ +µθφθ(s)[π(s, bk +ξs,π) + uk +θ(bk +ξs,π)] − p ≥ +� +θ +µθuk +θ(bk +µ), +(12) +Hence, +p − +� +s∈S +� +θ∈Θ +µθφθ(s) +� +π(s, bk +ξs,π) + uk +θ(bk +ξs,π) − uk +θ(bk +ξs) +� += p − +� +s∈S +� +θ∈Θ +µθφθ(s) +� +π(s, bk +ξs,π) + uk +θ(bk +ξs,π) +� ++ +� +s∈S +� +θ∈Θ +µθφθ(s)uk +θ(bk +ξs) +≤ p − +� +s∈S +� +θ∈Θ +µθφθ(s) +� +π(s, bk +ξs,π) + uk +θ(bk +ξs,π) +� ++ +� +s∈S +� +θ∈Θ +µθφθ(s)uk +θ(bk +θ) += p − +� +s∈S +� +θ∈Θ +µθφθ(s) +� +π(s, bk +ξs,π) + uk +θ(bk +ξs,π) +� ++ +� +θ∈Θ +µθuk +θ(bk +θ) +≤ − +� +θ∈Θ +µθuk +θ(bk +θ) + +� +θ∈Θ +µθuk +θ(bk +θ) +≤ δk, +where the first inequality follows by the optimality of action bk +θ in state θ, and the second one by Equation (12). +Next, we show that for each ζ ∈ [0, 1] we can design a protocol with seller’s utility of at least ζ +2 +� +k∈K δk − 2−1/ζ. +Let Pζ := {2−i}i∈{1,...,⌊1/ζ⌋} ∪ {0}, and for each k ∈ K let pk be the greatest p ∈ Pζ such that p ≤ δk. Then, +� +k∈K +λkpk ≥ +� +k∈K +λk +� +δk/2 − 2−⌊1/ζ⌋� += +� +k∈K +λkδk/2 − 2−⌊1/ζ⌋, +where the inequality holds since either pk ≥ δk/2 or pk ≤ 2−⌊1/ζ⌋ +Hence, � +p∈Pζ p � +k∈K:pk=p λk ≥ � +k∈K λkδk/2 − 2−⌊1/ζ⌋, implying +max +p∈Pζ p +� +k∈K:pk=p +λk ≥ +1 +2|Pζ| +� +k∈K +λkδk − 2−⌊1/ζ⌋ ≥ ζ +2 +� +k∈K +λkδk − 2−⌊1/ζ⌋. +Let p∗ = argmaxp∈Pζ p � +k∈K:pk=p λk. Consider the protocol (φ3, p3, π3) that charges payment p3 = p∗, reveals all +information with φ3 and set payment π3(s, a) = 0 for each s ∈ S and a ∈ A. We show that this protocol satisfies the +IR constraint for all the players such that pk = p∗. Indeed, for all these types it holds +� +θ∈Θ +µθuk +θ(bk +θ) − p∗ ≥ +� +θ∈Θ +µθuk +θ(bk +θ) − δk += +� +θ∈Θ +µθuk +θ(bk +θ) − +�� +θ +µθuk +θ(bk +θ) − +� +θ +µθuk +θ(bk +µ) +� +≥ 0. +(14) +Then, the utility of the protocol is at least the payment obtained by the buyers’ type in Rφ3,p3,π3 ⊇ {k ∈ K : pk = p∗}. +In particular, it is at least +p∗ +� +k∈K:pk=p∗ +λk ≥ ζ +2 +� +k∈K +λkδk − 2−⌊1/ζ⌋ +33 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +≥ ζ +2 +� +k∈Rφ,p,π +λkδk − 2−⌊1/ζ⌋ +≥ ζ +2 +� +k∈Rφ,p,π +λk +� +p − +� +s∈S +� +θ +µθφθ(s)[π(s, bk +ξs,π) + uk +θ(bk +ξs,π) − uk +θ(bk +ξs)] +� +− 2−⌊1/ζ⌋ += ζ +2U3 − 2−⌊1/ζ⌋, +where in the the first inequality we use Equation (14), and in the third inequality we use Equation (??). Equivalently, +setting ρ = ζ/2, we obtain that for each ρ ∈ [0, 1/2] there exists a protocol (φ3, p3, π3) that has seller’s utility at least +ρU3 − 2−Ω(1/ρ). +Wrapping up. +Let i = arg maxj∈{1,2,3} Uj and OPT be the seller’s utility with the optimal protocol (φ, p, π). Then, +since U1+U2+U3 = OPT, we have that Ui ≥ 1 +3OPT. Moreover, since for each ρ ∈ [0, 1/2] we can approximate each +utility Ui, i ∈ {1, 2, 3} with a protocol with utility at least ρUi −2−Ω(1/ρ) −α, the seller’s utility of our approximation +algorithm is at least ρUi − 2−Ω(1/ρ) − α ≥ ρOPT/3 − 2−Ω(1/ρ) − α. Finally, setting ρ′ = ρ/3, we obtain that for +each ρ′ ∈ [0, 1/6] the utility of the designed protocol is at least OPT − 2−Ω(1/ρ) − α. This concludes the proof. +Lemma 9. Given a seller’s protocol without menus, there always exists another protocol without menus which is +generalized-direct and generalized-persuasive, and achieves the same seller’s expected utility as the original protocol. +Proof. Let (φ, π, p) be a protocol and let be s1, s2 ∈ S be two signals such that bk +ξs1 = bk +ξs2 for each receiver’s type +k ∈ K. We show that it is always possible to define a new protocol (φ∗, π∗, p) that employs a single signal s∗ instead +of s1 and s2 achieving the same seller’s expected utility while satisfying the constraints. Formally, we define a new +signaling scheme φ∗ as follows: +�φ∗ +θ(s∗) = φθ(s1) + φθ(s2) ∀θ ∈ Θ +φ∗ +θ(s) = φθ(s) ∀θ ∈ Θ, ∀s ∈ S \ {s1, s2} +and a new payment function π∗ as follows: +�π∗(s∗, a) = zπ(s1, a) + (1 − z)π(s2, a) +∀a ∈ A +π∗(s, a) = π(s, a) ∀a ∈ A, +∀s ∈ S \ {s1, s2} +with z = � +θ∈Θ µθφθ(s1)/(� +θ∈Θ µθ(φθ(s1) + φθ(s2)). As a first step, we observe that for each k ∈ K it holds: +� +θ∈Θ +µθ +� +φθ(s1) +� +us +θ(bk +ξs1,π) − π(s1, bk +ξs1,π) +� ++ φθ(s2) +� +us +θ(bk +ξs2,π) − π(s2, bk +ξs2,π) +� � += +� +θ∈Θ +µθφ∗ +θ(s∗) +� +us +θ(bk +ξs∗,π∗) − π∗(s∗, bk +ξs∗,π∗) +� +. +Moreover, for each k ∈ K it holds: +� +θ∈Θ +µθ +� +φθ(s1) +� +uk +θ(bk +ξs1 ,π) + π(s1, bk +ξs1,π) +� ++ φθ(s2) +� +uk +θ(bk +ξs2,π) + π(s2, bk +ξs2,π) +� � += +� +θ∈Θ +µθφ∗ +θ(s∗) +� +uk +θ(bk +ξs∗,π∗) + π∗(s∗, bk +ξs∗,π∗) +� +. +Hence, noticing that for each signal s ∈ S \ {s1, s2} the seller’s utility and the buyer’s utility does not change from +(φ, π, p) to (φ∗, π∗, p), the set R of buyer’s type for which the IR is satisfied does not change. As a consequence, the +two protocols achieve the same seller’s expected utility. +Applying this procedure to all the couples of signals that induces the same vector of best responses, we obtain a +generalized-direct and generalized-persuasive protocol providing the same seller’s expected utility. +Lemma 10. Given a protocol without menus, there always exists another protocol (φ, p, π) such that p = bk for some +k ∈ K, while achieving the same seller’s expected utility as the original protocol. +34 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +Proof. Let (φ, π, p) be a protocol. We show that there exists a ˆk ∈ K and a payment function ˆπ such that the protocol +(φ, ˆπ, bˆk) provides the same seller’s expected utility. Let +ˆk ∈ arg +min +k∈Rφ,π,p:bk≥p{bk}. +We observe that all the buyer’s types k ∈ Rφ,π,p have enough budget to participate in the protocol,i.e., bk ≥ bˆk. +Furthermore, we define ˆπ(s, a) = π(s, a) + bˆk − p for each s ∈ S and a ∈ A. +Then, we show that the set of types Rφ,π,p = Rφ,ˆπ,ˆp. Indeed, for each type k ∈ Rφ,ˆπ,ˆp it holds +� +θ∈Θ +� +s∈S +µθφθ(s) +� +uk +θ(bk +ξs,ˆπ) + ˆπ(s, bk +ξs,ˆπ) +� +− bˆk += +� +θ∈Θ +� +s∈S +µθφθ(s) +� +uk +θ(bk +ξs,ˆπ) + π(s, bk +ξs,ˆπ) + bˆk − p +� +− bˆk += +� +θ∈Θ +� +s∈S +µθφθ(s) +� +uk +θ(bk +ξs,π) + π(s, bk +ξs,π) +� +− p, +and hence k ∈ Rφ,π,p. Similarly, we can prove that each buyer’s type k /∈ Rφ,ˆπ,ˆp does not belong to Rφ,π,p. It +follows that Rφ,π,p = Rφ,ˆπ,ˆp. +Finally, we can show that the seller’s utility results equal to the one in (φ, π, p). Indeed, we have: +� +k∈Rφ,p,π +λk +� � +θ∈Θ +� +s∈S +µθφθ(s) +� +us +θ(bk +ξs,π) − π(s, bk +ξs,π) +� ++ p +� ++ +� +k /∈Rφ,p,π +λk +� +θ∈Θ +µθus +θ(bk +µ) += +� +k∈Rφ,ˆ +p,ˆπ +λk +� � +θ∈Θ +� +s∈S +µθφθ(s) +� +us +θ(bk +ξs,ˆπ) − ˆπ(s, bk +ξs,ˆπ) +� ++ bˆk +� ++ +� +k /∈Rφ,ˆ +p,ˆπ +λk +� � +θ∈Θ +µθus +θ(bk +µ) +� +This concludes the proof. +Theorem 12. Restricted to instances in which the number of buyer’s types n is fixed, the problem of computing a +seller-optimal protocol without menus admits a polynomial-time algorithm. +Proof. In the following, we present an algorithm to compute an optimal protocol that works in polynomial time when +the number of buyer’s types is fixed. As a first step, we observe that, thanks to Lemma 10, the initial payment required +by the seller coincides with bk for some k ∈ K. Furthermore, we can focus on direct protocols by Lemma 9. Then, +given a price p ∈ {bk}k∈K and a set of buyer’s types R ⊆ K ∩ {k ∈ K : bk ≥ p} for which the IR constraint is +satisfied, the the problem of computing the optimal protocol can be formulated as Problem (6). Similarly to Section 3, +we can provide a linear relaxation of Problem (6) introducing a variable l(a, a′) that replaces � +θ∈Θ µθφθ(a)π(a, a′) +for each a ∈ An and a′ ∈ A. Then, we obtain the following LP. +max +φ≥0,l≥0 +� +k∈R +λk +� +a∈An +�� +θ∈Θ +µθφθ(a)us +θ(ak) − l(a, ak) +� ++ +� +k /∈R +λk +� +θ∈Θ +µθus +θ(bk +µ) +(15a) +� +θ∈Θ +µθφθ(a)uk +θ(ak) + l(a, ak) ≥ +� +θ∈Θ +µθφθ(a)uk +θ(a′) + l(a, a′) +∀k ∈ R, ∀a ∈ An, ∀a′ ̸= ak ∈ A +(15b) +� +a∈An +�� +θ∈Θ +µθφθ(a)uk +θ(ak) + l(a, ak) +� +− bk ≥ +� +θ∈Θ +µθuk +θ(bk +µ) +∀k ∈ R +(15c) +� +a∈An +�� +θ∈Θ +µθφθ(a)uk +θ(ak) + l(a, ak) +� +− bk ≤ +� +θ∈Θ +µθuk +θ(bk +µ) +∀k ̸∈ R +(15d) +� +a∈An +φθ(a) = 1 +∀θ ∈ Θ. +(15e) +Hence, once we fix bk and R, LP (15) returns a solution that has the same value of the optimal protocol. +35 + +ARXIV PREPRINT - FEBRUARY 1, 2023 +To compute the optimal protocol we can iterate over all the possible prices p ∈ {bk}k∈K and all the possible subsets +R ⊆ K ∩ {k ∈ K : bk ≥ p} of receivers types for which the IR constraint is satisfied. Notice that, given a price +p, the IR constraint can be satisfied only the buyer’s type k ∈ K with bk ≥ p. Then, we solve LP (15). Finally, we +return the solution with highest value. As we show in the first part of the proof, this solution has the same value of the +optimal protocol. Moreover, it is easy to check that the overall procedure requires to solve O(n2n) LPs, showing that +the algorithm runs in polynomial time. +To conclude the proof, we need to show how to modify the solution of LP 15 to obtain a protocol, i.e., a solution to +Problem (6), with at least the same value. To do so, we exploit a similar approach to the one presented in Section 3. +Let (φ, l) be the solution to LP (15) returned by the algorithm. Suppose that there exists a couple (¯a, ¯k) such that +l(¯a, a¯k) > 0 and � +θ∈Θ µθφθ(¯a) = 0. We show how to obtain a solution such that l(¯a, a) = 0 for each a ∈ A. Notice +that by Constraint (15b), it holds l(¯a, ¯ak) ≥ l(¯a, a) for each k ∈ K, a ∈ A. This implies that l(¯a, ¯ak) = l(¯a, ¯ak′) for +each k ̸= k′. We denote this value with l(¯a). Let ˆa ∈ An be any signal such that � +θ∈Θ µθφθ(ˆa) > 0. Consider a +assignment (φ, l′) to the variables such that +• l′(¯a, a) = 0 for each a ∈ A; +• l′(ˆa, a) = l(ˆa, a) + l(¯a) for each a ∈ A; +• l′(a) = l(a) for each a /∈ {¯a, ˆa}. +We show that this solution is feasible to LP (15) and has the same objective value of (φ, l). Indeed, it holds +� +k∈R +λk +� +a∈An +�� +θ∈Θ +µθφθ(a)us +θ(ak) − l′(a, ak) +� ++ +� +k /∈R +λk +� +θ∈Θ +µθus +θ(bk +µ) += +� +k∈R +λk +� +� +a∈An\{¯a,ˆa} +�� +θ∈Θ +µθφθ(a)us +θ(ak) − l′(a, ak) +� ++ +� +θ∈Θ +µθφθ(¯a)us +θ(¯ak) ++ +� +θ∈Θ +µθφθ(ˆa)us +θ(ˆak) − (l(ˆa, ˆak) − l(¯a)) +� ++ +� +k /∈R +λk +� +θ∈Θ +µθus +θ(bk +µ) += +� +k∈R +λk +� +� +a∈An\{¯a,ˆa} +�� +θ∈Θ +µθφθ(a)us +θ(ak) − l(a, ak) +� ++ +� +θ∈Θ +µθφθ(¯a)us +θ(¯ak) − l(¯a, ¯ak) ++ +� +θ∈Θ +µθφθ(ˆa)us +θ(ˆak) − l(ˆa, ˆak) +� ++ +� +k /∈R +λk +� +θ∈Θ +µθus +θ(bk +µ) += +� +k∈R +λk +� +a∈An +�� +θ∈Θ +µθφθ(a)us +θ(ak) − l(a, ak) +� ++ +� +k /∈R +λk +� +θ∈Θ +µθus +θ(bk +µ), +showing that the seller’s utility does not change. Moreover, Constraints (15b) relative to ¯a are satisfied since have the +form 0 ≥ 0. The Constraints (15b) relative to ˆa continue to be satisfied since we add a term l(¯a) on both sides of the +inequality. Finally, all the other Constraint (15b) are unchanged. Consider Constraint (15c) relative to a buyer’s type +k ∈ K. It holds +� +a∈An +�� +θ∈Θ +µθφθ(a)uk +θ(ak) + l′(a, ak) +� +− bk += +� +a∈An\{¯a,ˆa} +�� +θ∈Θ +µθφθ(a)uk +θ(ak) + l(a, ak) +� ++ +� +θ∈Θ +µθφθ(¯a)uk +θ(¯ak) ++ +� +θ∈Θ +µθφθ(ˆa)uk +θ(ˆak) + l(ˆa, ˆak) + l(¯a) − bk += +� +a∈An\{¯a,ˆa} +�� +θ∈Θ +µθφθ(a)uk +θ(ak) + l(a, ak) +� ++ +� +θ∈Θ +µθφθ(¯a)uk +θ(¯ak) ++ l(¯a, ¯ak) + +� +θ∈Θ +µθφθ(ˆa)uk +θ(ˆak) + l(ˆa, ˆak) − bk +36 + +ARXIV PREPRINT - FEBRUARY 1, 2023 += +� +a∈An +�� +θ∈Θ +µθφθ(a)uk +θ(ak) + l′(a, ak) +� +− bk +≥ +� +θ∈Θ +µθuk +θ(bk +µ) +Similarly, we can show that Constraints (15d) continue to hold. Hence, iteratively applying this procedure we ob- +tain a solution with the same value of the optimal protocol and such that for each tuple (a, k) if l(a, ak) > 0 and +� +θ∈Θ µθφθ(a) > 0. We can convert this solution into an optimal protocol, i.e., an optimal solution to Problem (6) +setting π(a, ak) = +l(a,ak) +� +θ∈Θ µθφθ(a) for each a ∈ An such that � +θ∈Θ µθφθ(a) = 0 and k ∈ K. Moreover, we set all +the other payments to 0. It is easy to see that the obtained protocol is a feasible optimal solution to Problem (6). This +concludes the proof. +37 + diff --git a/3tFST4oBgHgl3EQfZDim/content/tmp_files/load_file.txt b/3tFST4oBgHgl3EQfZDim/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1316ee6a2cf70e2ce38efccc13e7760095cbecde --- /dev/null +++ b/3tFST4oBgHgl3EQfZDim/content/tmp_files/load_file.txt @@ -0,0 +1,1340 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf,len=1339 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='13790v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='GT] 31 Jan 2023 SELLING INFORMATION WHILE BEING AN INTERESTED PARTY ARXIV PREPRINT Matteo Castiglioni Politecnico di Milano matteo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='castiglioni@polimi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='it Francesco Bacchiocchi Politecnico di Milano francesco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='bacchiocchi@polimi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='it Alberto Marchesi Politecnico di Milano alberto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='marchesi@polimi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='it Giulia Romano Politecnico di Milano giulia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='romano@polimi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='it Nicola Gatti Politecnico di Milano nicola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='gatti@polimi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='it February 1, 2023 ABSTRACT We study the algorithmic problem faced by an information holder (seller) who wants to optimally sell such information to a budged-constrained decision maker (buyer) that has to undertake some action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Differently from previous works addressing this problem, we consider the case in which the seller is an interested party, as the action chosen by the buyer does not only influence their utility, but also seller’s one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This happens in many real-world settings, where the way in which businesses use acquired information may positively or negatively affect the seller, due to the presence of externalities on the information market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The utilities of both the seller and the buyer depend on a random state of nature, which is revealed to the seller, but it is unknown to the buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Thus, the seller’s goal is to (partially) sell their information about the state of nature to the buyer, so as to concurrently maximize revenue and induce the buyer to take a desirable action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We study settings in which buyer’s budget and utilities are determined by a random buyer’s type that is unknown to the seller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In such settings, an optimal protocol for the seller must propose to the buyer a menu of information-revelation policies to choose from, with the latter acquiring one of them by paying its corresponding price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, since in our model the seller is an interested party, an optimal protocol must also prescribe the seller to pay back the buyer contingently on their action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' First, we show that the problem of computing a seller-optimal protocol can be solved in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This result relies on a quadratic formulation of the problem, which we solve by means of a linear programming relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Next, we switch the attention to the case in which a seller’s protocol employs a single information-revelation policy, rather than proposing a menu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In such a setting, we show that computing a seller-optimal protocol is APX-hard, even when either the number of actions or that of states of nature is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We complement such a negative result by providing a quasi- polynomial-time approximation algorithm that, given any ρ > 0 and ǫ > 0 as input, provides a multiplicative approximation ρ of the optimal seller’s expected utility, by only suffering a negligible 2−Ω(1/ρ) + ǫ additive loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Such an algorithm runs in polynomial time whenever either the number of buyer’s actions or that of states of nature is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In order to derive our results, we draw a connection between our information-selling problem and principal-agent problems with observable actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, we complete the picture of the computational complexity of finding seller-optimal protocols without menus by providing additional results for the specific setting in which the buyer has limited liability, and by designing a polynomial-time algorithm for the case in which buyer’s types are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ARXIV PREPRINT - FEBRUARY 1, 2023 1 Introduction Nowadays, there is a terrific amount of information being collected on the Web and other online platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Such infor- mation ranges from consumer preferences, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', in e-commerce and streaming websites, to credit reports and location histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As a result, recent years have witnessed the born and exponential blowout of markets where specialized companies sell information that is valuable to other businesses, such as advertisers, retailers, and loan providers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Very recently, information markets have also received the attention of the algorithmic game theory research commu- nity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' However, while works addressing classical settings such as auctions (Daskalakis and Syrgkanis, 2022), signal- ing (Dughmi and Xu, 2019), and contract design (Dütting et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', 2019) are now proliferating, only few papers studied the problem of information selling, with (Babaioff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', 2012) and (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', 2020) constituting two notable exam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We study the algorithmic problem faced by an information holder (seller) who wants to optimally sell such information to a budged-constrained decision maker (buyer) that has to undertake some action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Differently from previous works addressing such a problem (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', 2020)), we consider the case in which the seller is an interested party, as the action chosen by the buyer does not only influence their utility, but also seller’s one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This happens in many real-world settings, where the way in which businesses use acquired information may positively or negatively affect the seller, due to the presence of externalities on the information market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The utilities of both the seller and the buyer depend on a state of nature that is drawn according to a commonly-known probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The realized state of nature is revealed to the seller, while it remains unknown to the buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Thus, the seller’s goal is to (partially) sell their information about the state of nature to the buyer, so as to concurrently maximize revenue and induce the buyer to take a desirable action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We study settings in which buyer’s budget and utilities are determined by a random buyer’s type that is unknown to the seller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In such settings, in order to optimally sell information, the seller has to commit upfront to a protocol working as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' First, the seller proposes to the buyer a menu of information-revelation policies to choose from, and the latter acquires an expected-utility-maximizing one according to their (private) type, by paying its corresponding price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' By building on the Bayesian persuasion framework introduced by Kamenica and Gentzkow (2011), an information- revelation policy is implemented as a signaling scheme, which is a randomized mapping from states of nature to signals issued to the buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, the realized state of nature is disclosed to the seller, who reveals information about it to the buyer according to the acquired signaling scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, the buyer selects a best-response action according to the just acquired information, and the seller pays back the buyer with a payment which depends on both the chosen action and the signal that has been previously sent by the seller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Our protocol extends the one of Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2020) by adding a final payment from the seller to the buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As we show later, this is crucial in order to design seller-optimal protocols in our setting where the seller is an interested party, since the latter is not only concerned with revenue, but also with the buyer’s action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, the addition of payments from the buyer to the seller is also reasonable in many real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' For instance, think of a case in which the information holder asks the buyer to deposit additional money, and this is given back to them only if the performed action respects some given rules on which the two parties agreed upfront.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='1 Original Contributions After introducing all the needed concepts in Section 2, we start providing our results in Section 3, where we analyze the case of general protocols in which the seller proposes a menu of signaling schemes to the buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We show that a seller-optimal protocol can be computed in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In order to do that, we first formulate the problem of finding a seller-optimal protocol as a quadratic problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, we show that one can focus on direct and persuasive signaling schemes, which are those that send signals corresponding to action recommendations for the buyer and properly incentivize the latter to follow such recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This in turn allows us to restrict the attention to protocols that ask the buyer to pay their entire budget upfront and, then, pay back the buyer only if they take the recommended action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' These results allow us to formulate a suitable linear relaxation of the quadratic problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' A similar technique has been employed in generalized principal-agent problems (Gan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', 2022), where it is possible to show that an optimal solution to the linear relaxation can be efficiently cast to an approximately-optimal solution to the quadratic problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Indeed, Castiglioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2022b) show that, even in the special case of hidden-action principal- agent problems, obtaining an optimal solution to the quadratic problem is not possible in general, since the principal’s optimization problem may not admit a maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Surprisingly, in our information-selling setting, we prove that an optimal solution to our linear relaxation, which can be computed in polynomial time, can be used to recover a seller- optimal protocol in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As a byproduct, this also shows that, in our setting, the seller’s problem always admits a maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 2 ARXIV PREPRINT - FEBRUARY 1, 2023 In the second part of the paper, we switch the attention to the case of protocols without menus, in which the seller does not propose a menu of signaling schemes to the buyer, but they rather commit to a single signaling scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This is the case in many real-world applications, where it is unreasonable that a buyer is asked to choose an information-revelation policy among a range of options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Computing a seller-optimal protocol without menus begets considerable additional computational challenges, since, intuitively, the seller has no way of extracting information about the buyer’s private type, as instead it is the case when proposing a menu to choose from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In Section 4, we draw a connection between the problem of computing a seller-optimal protocol without menus and principal-agent problems with observable actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' These are problems in which a principal commits to an action- dependent payment scheme in order to incentivize an agent to take some costly, observable action, in order to maximize their expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We prove that observable-action principal-agent problems are a special case of our information- selling problem, and that, in such problems, computing an expected-utility-maximizing payment-scheme for the prin- cipal is APX-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In particular, these results show that our information-selling problem is APX-hard even when the number of states of nature is fixed and the buyer has limited liability, and, thus, the seller cannot charge a price for a signaling scheme upfront.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We also provide some preliminary technical results on observable-action principal-agent problems, which are useful in order to prove some of our main claims in the paper, while also being of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In Section 5, we show how to circumvent the APX-hardness for settings in which the seller employs protocols without menus and the buyer has limited liability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' These special settings are of interested on their own, as a similar model has been recently addressed by Dughmi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We focus on special cases where one of the parameters characterizing a problem instance is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In particular, we study what happens if we fix the number of buyer’s actions, showing that the problem admits a PTAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, we prove that, when instead the number of states of nature is fixed, there exists a polynomial-time bi-criteria approximation algorithm that, given any ρ > 0 and ǫ > 0 as input, provides a multiplicative approximation ρ of the optimal seller’s expected utility, by only suffering a 2−Ω(1/ρ) + ǫ additive loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Notice that such a loss is exponentially small in 1 ρ, and, thus, it is negligible even for reasonably large values of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As shown by Castiglioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2022a), such an approximation result is tight for hidden-action principal-agent problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' It remains an open problem to establish whether such an approximation guarantee is also tight for principal-agent problems with observable actions, which are a special case of our information-selling problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Table 1: Summary of the results provided in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Each cell specifies, on the first line, the computational com- plexity of finding a seller-optimal protocol, while, additionally, on the second line, it specifies the approximation guarantees that we can obtain in polynomial time, where OPT denotes the seller’s expected utility in an optimal proto- col.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The approximation guarantees that are shaded in gray can only be obtained by means of a quasi-polynomial-time algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' general fixed # actions fixed # states fixed # types Protocols with menus P P P P Protocols w/o menus Buyer w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' limited liability APX-hard — APX-hard P ρOPT−2−Ω(1/ρ)−ǫ PTAS ρOPT−2−Ω(1/ρ)−ǫ Protocols w/o menus Buyer w/o limited liability APX-hard APX-hard APX-hard P ρOPT−2−Ω(1/ρ)−ǫ ρOPT−2−Ω(1/ρ)−ǫ ρOPT−2−Ω(1/ρ)−ǫ In conclusion, in Section 6 we study the problem of computing seller-optimal protocols without menus in general settings in which the buyer does not have limited liability, and, thus, the seller can charge a price for a signaling scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We first prove a stronger negative result, by showing that, in such a setting, the problem of computing a seller- optimal protocol is APX-hard even if the number of buyer’s actions is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, we show how to circumvent such a negative result by providing a quasi-polynomial-time bi-criteria approximation algorithm that, given any ρ > 0 and ǫ > 0 as input, provides a multiplicative approximation ρ of the optimal seller’s expected utility, plus a2−Ω(1/ρ) + ǫ additive loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We prove that, when either the number of buyer’s action or that of states of nature is fixed, such an algorithm runs in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, we show that, when the number of buyer’s types is fixed, the problem admits a polynomial-time algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This also implies that the seller’s optimization problem for protocols without menus always admits a maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We summarize the results provided in this paper in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' All the proofs are in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 3 ARXIV PREPRINT - FEBRUARY 1, 2023 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='2 Related Works The study of algorithmic ways of selling information to an imperfectly-informed buyer has received some attention in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Babaioff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2012) initiated the study by considering a buyer with an unlimited budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' They provide an exponentially-sized linear program (LP) for computing an optimal mechanism for selling information, and they efficiently solve it through the ellipsoid method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The main drawback of the approach presented by Babaioff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2012) is that an optimal mechanism may require a significant money transfer from the buyer to the seller and viceversa, in order to only achieve a small, overall net transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2020) complement the results in (Babaioff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', 2012) by studying the problem of selling information when both the buyer and the seller are budget-constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, they also consider a setting in which the buyer’s budget is private, and the seller needs to elicit it in the mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2020) show that the addition of budget constraints considerably simplifies the problem of computing an optimal mechanism, since it can be formulated as a polynomially-sized LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The problem of selling information has also been addressed by Bergemann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2018), who study the case of bi- nary actions and states of nature, characterizing a revenue-maximizing mechanism in such a setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Furthermore, Bergemann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2022) extend the analysis to the case in which there are more than two actions and binary states of nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In contrast, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2021) study a revenue-maximizing mechanism for selling information when the stochasticity of the state of nature only affects a subset of the actions of the decision maker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Our problem is also related to the Bayesian persuasion framework originally introduced by Kamenica and Gentzkow (2011), where an informed sender wants to influence the behavior of a self-interested receiver via the strategic provision of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Dughmi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2019) generalize the classical framework by considering the case in which there are monetary transfers between the sender and the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Our information-selling setting in which the buyer has limited liability generalizes the model of Dughmi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2019) by also introducing buyer’s types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, let us remark that our information-selling problem shares critical features with Bayesian principal-agent prob- lems (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', (Castiglioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Alon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', 2021, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Guruganesh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Castiglioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', 2022c) for some references).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Indeed, as we show in Section 4, the problem of computing a seller-optimal protocol gener- alizes particular principal-agent problems in which the agent’s action is observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Such a connection between the two settings is also demonstrated in terms of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In particular, notice that Castiglioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2022a) design bi- criteria approximation algorithms whose guarantees are similar to those provided in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, Gan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2022) show how to find optimal protocols in generalized principal-agent problems by using a linear relaxation of the principal’s optimization problem, which is quadratic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 2 Preliminaries We study the problem faced by an information holder (seller) selling information to a budget-constrained decision maker (buyer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The information available to the seller is collectively termed state of nature and encoded as an element of a finite set Θ := {θi}d i=1 of d possible states, while the set of the m actions available to the buyer is A := {ai}m i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The buyer is also characterized by a private type, which is unknown to the seller and belongs to a finite set K := {ki}n i=1 of n possible types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Each buyer’s type k ∈ K is characterized by a utility function uk θ : A → [0, 1] associated to each state θ ∈ Θ and a budget bk ∈ R+ representing how much they can afford to pay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In our model, the seller’s utility is not only determined by how much the buyer pays for acquiring information, but it also depends on the buyer’s action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Specifically, for every state θ ∈ Θ, the sender gets an additional utility contribution determined by a function us θ : A → [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We assume that both the seller and the buyer know the probability distribution µ ∈ ∆Θ according to which the state of nature is drawn, as well as the probability distribution λ ∈ ∆K determining the buyer’s type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='1 We let µθ be the probability assigned to state θ ∈ Θ, while λk is the probability of type k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As in (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', 2020), we assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' that information revelation happens only once during the seller-buyer interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Thus, as it is the case in Bayesian persuasion Kamenica and Gentzkow (2011), the seller reveals infor- mation to the buyer by committing to a signaling scheme φ, which is a randomized mapping from states of nature to signals being issued to the buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally, φ : Θ → ∆S, where S is a finite set of signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We denote by φθ ∈ ∆S the probability distribution employed when the state of nature is θ ∈ Θ, with φθ(s) being the probability of sending s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='1 Protocols with Menus An information-selling protocol for the seller is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The seller first proposes a menu of signaling schemes to the buyer, with each signaling scheme being assigned with a price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, the buyer chooses a signaling 1In this work, given a finite set X, we let ∆X be the set of all the probability distributions defined over the elements of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 4 ARXIV PREPRINT - FEBRUARY 1, 2023 scheme and pays its price upfront, before information is revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='2 The seller also commits to action-dependent payments, which are made by the seller in favor of the buyer after information is revealed and the latter has taken an action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This is in contrast with what happens in the protocol introduced by Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2020), where there are no action-dependent money transfers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Intuitively, such payments are needed in order to incentivize the agent to play an action that is profitable for the seller, and, thus, they are not needed in the setting of Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2020) where the seller’s utility function is only determined by how much the buyer pays for acquiring information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally, we define a seller’s protocol as follows: Definition 1 (Seller’s protocol).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' A protocol for the seller is a tuple {(φk, pk, πk)}k∈K, where: {φk}k∈K is a menu of signaling schemes φk : Θ → ∆S, one for each receiver’s type k ∈ K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' {pk}k∈K is a menu of prices, with pk ∈ R+ representing how much the seller charges the buyer for selecting the signaling scheme φk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='3 {πk}k∈K is a menu of payment functions, which are defined as πk : S × A → R+ with πk(s, a) encoding how much the seller pays the buyer whenever the latter plays action a ∈ A after selecting the signaling scheme φk and receiving signal s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='4 The seller and the buyer interact as follows: (i) the seller commits to a protocol {(φk, pk, πk)}k∈K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (ii) the buyer selects a signaling scheme φk and pays pk to the seller (with k ∈ K possibly different from their true type);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (iii) the seller observes the realized state of nature θ ∼ µ, draws a signal s ∼ φk θ according to the selected signaling scheme, and communicates s to the buyer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (iv) given the signal s, the buyer infers a posterior distribution ξs ∈ ∆Θ over states of nature, where the probability ξs θ of state θ ∈ Θ is computed with the Bayes rule, as follows: ξs θ := µθ φk θ(s) � θ′∈Θ µθ′φk θ′(s);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (v) given the posterior ξs, the buyer selects an action a ∈ A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' and (vi) the seller pays πk(s, a) to the buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As in the model by Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2020), we assume that the seller is committed to following the protocol, while the buyer is not, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', the buyer is free of leaving the interaction at any point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In step (v), after observing a signal s ∈ S and computing the posterior ξs, the buyer plays a best response by choosing an action a ∈ A maximizing their expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally: Definition 2 (ǫ-Best-response).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let ǫ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given a signal s ∈ S, the induced posterior ξs ∈ ∆Θ, and a payment function π : S × A → R+, the ǫ-best-response set of a buyer of type k ∈ K is: Bk,ǫ ξs,π := � a ∈ A : � θ∈Θ ξs θ uk θ(a) + π(s, a) ≥ max a′∈A � θ∈Θ ξs θ uk θ(a′) + π(s, a′) − ǫ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We let bk,ǫ ξs,π ∈ Bk,ǫ ξs,π be an ǫ-best response played by the buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The best-response set Bk ξs,π of a buyer of type k ∈ K is defined for ǫ = 0, while bk ξs,π ∈ Bk ξs,π is a best response played by the buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='5 In the following, we will oftentimes work in the space of the distributions over posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In that case, given a posterior ξ ∈ ∆Θ, we abuse notation and write Bk ξ,π, Bk,ǫ ξ,π, bk,ǫ ξ,π, and bk ξ,π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The seller’s goal is to implement an optimal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', utility-maximizing) protocol {(φk, pk, πk)}k∈K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We focus on seller’s protocols that are incentive compatible (IC) and individually rational (IR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='6 Specifically, a seller’s protocol is IC if for every pair of buyer’s types k, k′ ∈ K: � s∈S � θ∈Θ µθφk θ(s) � uk θ(bk ξs,πk) + πk(s, bk ξs,πk) � − pk ≥ � s∈S max a∈A � θ∈Θ µθφk′ θ (s) � uk θ(a) + πk′(s, a) � − pk′, 2Notice that proposing a menu of signaling schemes is equivalent to asking the buyer to report their type and then choosing a signaling scheme based on that, as it is the case in (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 3Assuming pk ≥ 0 is w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', since, intuitively, the seller is never better off paying the buyer before they played any action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 4The assumption that πk(s, a) ≥ 0 is w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', since the buyer does not commit to following the protocol, and, thus, πk(s, a) < 0 would result in the buyer leaving the protocol without paying after taking an action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 5When the buyer is indifferent among multiple best responses (respectively, ǫ-best responses), we always assume that they break ties in favor of the seller, choosing an action in Bk ξs,π (respectively, Bk,ǫ ξs,π) maximizing the seller’s expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 6By a revelation-principle-style argument (see (Shoham and Leyton-Brown, 2008) for some examples), focusing on IC and IR protocols is w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' when looking for an optimal protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 5 ARXIV PREPRINT - FEBRUARY 1, 2023 while it is IR if for every buyers’ type k ∈ K: � s∈S � θ∈Θ µθφk θ(s) � uk θ(bk ξs,πk) + πk(s, bk ξs,πk) � − pk ≥ max a∈A � θ∈Θ µθuk θ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Intuitively, an IC protocol incentivizes the buyer to select the signaling scheme φk corresponding ot their true type k ∈ K, while an IR protocol ensures that the buyer gets more utility by acquiring information rather than leaving the protocol before step (ii) and playing an action without information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, the seller’s expected utility is computed as follows: � k∈K λk �� s∈S � θ∈Θ µθφk θ(s) � us θ(bk ξs,πk) − πk(s, bk ξs,πk) � + pk � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' A crucial component of our results is that we can restrict the attention to protocols that are direct and persuasive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We say that protocol is direct if it uses signaling schemes whose signals correspond to action recommendations for the buyer, namely S = A, while a direct protocol is said to be persuasive whenever playing the recommended action is always a best response for the buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='2 Protocols without Menus In the second part of the paper, we study the case of seller’s protocols without menus, in which the seller does not propose a menu of signaling schemes to the buyer, but they rather commit to a single signaling scheme and a single payment function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='7 This allows us to simplify the definition of a protocol (see Definition 1), by denoting a seller’s protocol without menus as a tuple (φ, p, π), where φ : Θ → ∆S is a signaling scheme, p ∈ R+ is a price for such a signaling scheme, representing how much the seller charges the buyer to reveal information to them, and π : S × A → R+ is a payment function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The seller-buyer interaction unfolds as in the general case with menus, but, in this case, step (ii) only involves the payment of price p ∈ R+ on buyer’s part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Some of our results on protocols without menus address the special case in which the buyer has limited liability, which means that the buyer has no budget, and, thus, the seller cannot charge a price for a signaling scheme upfront.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally, this amounts to asking that bk = 0 for all k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Notice that, while such a special case may seem of scarce appeal for the problem of selling information, it is indeed interesting on its own, as it is similar to the model studied by Dughmi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Indeed, our model can be seen as a generalization of the one in (Dughmi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', 2019), which adds buyer’s private types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, in the general case in which the buyer has no limited liability, our model additionally builds on top of that of Dughmi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2019) by adding the possibility for the seller to ask the buyer a payments before information is revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' For protocols without menus, IC constraints are not needed anymore, while IR constraints are still required in order to ensure that the buyer is incentivized to acquire information from the principal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given a protocol without menus (φ, p, π), only some of the buyer’s types are actually incentivized to participate in the protocol, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', all the types whose corresponding IR constraint is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally, a protocol determines a subset Rφ,p,π ⊆ K of buyer’s types such that, for every k ∈ Rφ,p,π, it holds that: (i) a buyer of type k has enough budget to buy information, namely bk ≥ p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' and (ii) the IR constraint is satisfied for a buyer of type k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='8 In particular, point (ii) can be formally stated by saying that the following condition is satisfied for every k ∈ Rφ,p,π: � s∈S � θ∈Θ µθφθ(s) � uk θ(bk ξs,π) + π(s, bk ξs,π) � − p ≥ max a∈A � θ∈Θ µθuk θ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, given a protocol without menus (φ, π, p), the seller’s expected utility is given by: � k∈Rφ,p,π λk �� s∈S � θ∈Θ µθφθ(s) � us θ(bk ξs,π) − π(s, bk ξs,π) � + p � + � k̸∈Rφ,p,π λk � θ∈Θ µθuk θ(bk µ), where bk ξ ∈ arg maxa∈A � θ∈Θ ξθuk θ(a) is a best response for a buyer’s type k ∈ K that only considers the posterior ξ ∈ ∆Θ, where, as customary, ties are broken in favor of the seller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Notice that a buyer’s type k /∈ Rφ,p,π is among those who decide to do not acquire information from the seller, and, thus, they play a best response to the probability distribution µ (instead of a posterior).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 7From the point of view of Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2020), this is equivalent to assuming that there is no type reporting stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 8Whenever the expected utility of a buyer’s type is the same by participating in the protocol as not doing that, we assume that they take the option maximizing the seller’s expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 6 ARXIV PREPRINT - FEBRUARY 1, 2023 Finally, when dealing with protocols without menus, it will be useful to directly work with distributions over posteriors induced by signaling schemes, rather than with signaling schemes (Kamenica and Gentzkow, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' A signaling scheme φ : Θ → ∆S induces a distribution γ over ∆Θ, which has a support supp(γ) := {ξs | s ∈ S} and satisfies the following conditions: � ξ∈supp(γ) γξ ξθ = µθ ∀θ ∈ Θ, (1) where γξ ∈ [0, 1] is the probability that γ assigns to the posterior ξ ∈ supp(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Thus, instead of working with signaling schemes φ, one can w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' work with distributions γ over ∆Θ that are consistent with the probability distribution µ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', they satisfy the condition in Equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' When working with distributions over posteriors γ rather than with signaling schemes φ, with a slight abuse of notation, we denote a seller’s protocol without menus as (γ, p, π), by identifying a signaling scheme with its induced distribution over posteriors γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Similarly, we slightly abuse notation in payment functions, by assuming that they are defined over posteriors rather than signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally, we let π : ∆Θ × A → R+, with π(ξ, a) denoting how much the buyer pays back the seller when the induced posterior is ξ ∈ ∆Θ and they play action a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 3 Computing a Seller-optimal Protocol with Menus We begin by studying the problem of computing a seller-optimal protocol in which the seller has the ability of propos- ing a menu of signaling schemes and payment functions to the buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally, the problem of computing an optimal IC and IR protocol with menus can be formulated as follows: sup φk θ(s)≥0 pk≥0 πk(s,a)≥0 � k∈K λk �� s∈S � θ∈Θ µθφk θ(s) � us θ(bk ξs,πk) − πk(s, bk ξs,πk) � + pk � s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2a) � s∈S � θ∈Θ µθφk θ(s) � uk θ(bk ξs,πk) + πk(s, bk ξs,πk) � − pk ≥ � s∈S max a∈A � θ∈Θ µθφk′ θ (s) � uk θ(a) + πk′(s, a) � − pk′ ∀k ∈ K, ∀k′ ∈ K (2b) � s∈S � θ∈Θ µθφk θ(s) � uk θ(bk ξs,πk) + πk(s, bk ξs,πk) � − pk ≥ max a∈A � θ∈Θ µθuk θ(a) ∀k ∈ K (2c) � s∈S φk θ(s) = 1 ∀k ∈ K, ∀θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2d) Notice that Problem (2) is defined in terms of sup rather than max since, as it is the case in principal-agent problems (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', (Castiglioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Gan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', 2022)), it is not in general immediate to establish whether the seller’s optimization problem always admits a maximum or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Indeed, in the following we show that our problem always admits a maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As a first step, we prove that we can focus w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='g on protocols which are direct and persuasive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given any IC and IR seller’s protocol, it is always possible to recover an IC and IR seller’s protocol that is direct and persuasive, and it provides the seller with the same expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Intuitively, Lemma 1 follows from the fact that, given any signaling scheme φk and price function πk corresponding to some type k ∈ K, if two signals induce the same best response for a buyer of type k, then it is possible to merge the two signals in a single one, recovering a new signaling scheme and a new price function for type k that achieve the same seller’s expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' By doing such a procedure for every buyer’s type until there are no two signals inducing the same best response for that type, we obtain a protocol that is direct and persuasive, and it has the same seller’s expected utility as the original protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Notice that, since in direct protocols it holds § = A, whenever we write πk(a, a′) for a, a′ ∈ A, the first action a is the seller’s recommendation (signal), while the second action a′ is the one actually played by the buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As a second crucial step, we exploit Lemma 1 in order to show that, given an IC and IR protocol that is direct and persuasive, there exists another IC and IR protocol which is still direct and persuasive, it achieves the same seller’s expected utility, and it is such that: (i) for every k ∈ K, the price pk of φk is equal to entire budget bk of a buyer of type k, and (ii) the buyer is not paid back (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', they get a null payment) if they deviate from the seller’s action recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally: 7 ARXIV PREPRINT - FEBRUARY 1, 2023 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given an IC and IR protocol {(φk, pk, πk)}k∈K that is direct and persuasive, it is always possible to recover an IC and IR protocol {(φk, ˜pk, ˜πk)}k∈K such that: it is direct and persuasive, it provides the same seller’s expected utility as the original protocol, and, for every buyers’ type k ∈ K, it satisfies ˜pk = bk and ˜πk(a, a′) = 0 for all a ̸= a′ ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As a direct consequence of Lemma 2, we can compactly denote πk(a, a) as πk(a) for every a ∈ A, since we can focus w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' on payment functions such that πk(a, a′) = 0 for all a ̸= a′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We are now ready to introduce an LP with polynomially-many variables and constraints that is a linear relaxation of Problem (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In order to formulate the LP, we exploit Lemmas 1 and 2 to restrict the attention to direct and persuasive protocols, prices such that pk = bk for every k ∈ K, and payments such that πk(a, a′) = 0 for every k ∈ K and a ̸= a′ ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, we encode the terms � θ∈Θ µθφk θ(a)πk(a) as single variables lk(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, the LP reads as follows: max φk θ (a)≥0 lk(a)≥0 yk,k′,a≥0 � k∈K λk � a∈A �� θ∈Θ µθφk θ(a)us θ(a) − lk(a) � + bk s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (3a) � a∈A �� θ∈Θ µθφk θ(a)uk θ(a) + lk(a) � − bk ≥ � a∈A yk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='a − bk′ ∀k ∈ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ∀k′ ∈ K (3b) yk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='a ≥ � θ∈Θ µθφk′ θ (a)uk θ(a) + lk′(a) ∀k ∈ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ∀k′ ∈ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ∀a ∈ A (3c) yk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='a ≥ � θ∈Θ µθφk′ θ (a)uk θ(a′) ∀k ∈ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ∀k′ ∈ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ∀a ̸= a′ ∈ A (3d) � a∈A �� θ∈Θ µθφk θ(a)uk θ(a) + lk(a) � − bk ≥ � θ∈Θ µθuk θ(a′) ∀k ∈ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ∀a′ ∈ A (3e) � θ∈Θ µθφk θ(a)uk θ(a) + lk(a) ≥ � θ∈Θ µθφk θ(a)uk θ(a′) ∀k ∈ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ∀a ̸= a′ ∈ A (3f) � a∈A φk θ(a) = 1 ∀k ∈ K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ∀θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (3g) In LP (3), Constraints (3b)–(3d) ensure that the protocol is IC, Constraints (3e) enforce that it is IR, while Con- straints (3f) guarantee that the protocol is persuasive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given how LP (3) is obtained from Problem (2), it is not immediately clear how, given a feasible solution to LP (3), one can recover a protocol that is a solution to Problem (2) with seller’s expected utility equal to the value of the solution to LP (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Indeed, in a solution to LP (3), a variable lk(a) could be strictly positive even when the variables φk θ(a) are equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In such a case, it is not possible to immediately recover a value for πk(a) starting from a solution to LP (3), since lk(a) encodes � θ∈Θ µθφk θ(a)πk(a), from which computing πk(a) would require a division by zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In the following, we show how, given an optimal solution to LP (3), it is indeed possible to build in polynomial time a seller-optimal protocol with menus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' First, we prove a preliminary result: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The optimal value of LP (3) is at least as large as the supremum in Problem (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, we show that, given a solution to LP (3), it is possible to recover in polynomial time an IC and IR protocol with at least the same value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given a feasible solution to LP (3), it is possible to recover in polynomial time an IC and IR protocol whose seller’s expected utility is greater than or equal to the value of the solution to LP (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Intuitively, Lemma 4 is proved by showing that, given a feasible solution to LP (3), it is possible to efficiently construct a new solution in which, whenever some variable lk(a) > 0, then there exists at least one state of nature θ ∈ Θ for which φk θ(a) > 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', action a is recommended with strictly positive probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, such a procedure does not detriment the objective function value and retains the IC and IR conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, from the new solution, one can recover a protocol that is a valid solution to Problem (2), by letting πk(a) = lk(a)/ � θ∈Θ µθφk θ(a) for all k ∈ K and a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 8 ARXIV PREPRINT - FEBRUARY 1, 2023 Finally, by exploiting Lemmas 3 and 4, we can design a polynomial-time algorithm that finds a seller-optimal protocol with menus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Indeed, the algorithm can simply optimally solve LP (3) (in polynomial time), and use Lemma 4 to recover an IC and IR protocol having at least the same value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Tanks to Lemma 3, such a protocol is optimal for the seller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' There exists a polynomial-time algorithm that computes a protocol with menus that maximizes the seller’s expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Theorem 1 also shows as a byproduct that Problem (2) always admits a maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let us remark that the idea of formulating a linear relaxation of a quadratic problem by introducing a new variable has already been used in generalized principal-agent problems by Gan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' However, in such a setting, the linear relaxation cannot be used to solve the principal’s optimization problem exactly, but only to recover a desirable approx- imation of an optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This is because the problem may not admit a maximum, as shown by Castiglioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2022b) even in the special case of hidden-action principal-agent problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Surprisingly, in our information-selling setting, the linear relaxation can be used to find an (exact) optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Intuitively, this is possible since, in our setting, the seller observes the action undertaken by the buyer, while in hidden-action principal-agent problems the principal does not directly observe the agent’s action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 4 Drawing a Connection with Principal-agent Problems In this section, we show that our information-selling problem is intimately related to a particular class of principal- agent problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Specifically, we show that the problem of computing a seller-optimal protocol without menus is a generalization of the problem of computing an optimal contract in principal-agent problems in which the principal observes the action undertaken by the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='1, we formally introduce principal-agent problems with observable actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='2, we show how such problems are related to our information-selling problem, and we prove an hardness result for them which carries over to our problem Finally, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='3, we provide some preliminary technical results that will be useful in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='1 Principal-agent Problem with Observable Actions We start by formally defining an instance of (Bayesian) observable-action principal-agent problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='9 For ease of ex- position, we reuse some of the notation already introduced in Section 2, in order to denote elements that in observable- action principal-agent problems have the same role as in our information-selling setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The agent has a finite set K of possible types, and a type k ∈ K is drawn with probability λk according to a known distribution λ ∈ ∆K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Each agent’s type k ∈ K has a set A of actions, with each action having a type-dependent cost ck a ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The principal is characterized by a reward ra ∈ [0, 1] for every agent’s action a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, the principal can commit to a contract, which can be encoded by a function π : A → R+ defining a payment π(a) from the principal to the agent for every possible agent’s action a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given a contract, an agent of type k ∈ K plays a best response bk π ∈ A, defined as bk π ∈ arg maxa∈A � π(a) − ck a � , where, as usual, we assume that ties are broken in favor of the principal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, the principal’s goal is to commit to a contract maximizing their expected utility, which is defined as � k∈K λk[rbkπ −π(bk π)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='2 From Selling Information to Observable-action Principal-agent Problems Next, we show that our information-selling problem in the case in which protocols are without menus and the buyer has limited liability (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', bk = 0 for all k ∈ K) is strongly related to the problem of finding an optimal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', expected-utility- maximizing) contract in observable-action principal-agent problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Specifically, we show that, given a posterior ξ ∈ ∆Θ, designing a payment function π : ∆Θ × A → R+ that maximizes the seller’s expected utility conditioned on the fact that the induced posterior is ξ is equivalent to finding an optimal contract in a suitably-defined principal-agent problem with observable actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally, for ease of presentation, we introduce the following notion of payment function that is optimal for the seller in a given posterior: 9Notice that observable-action principal-agent problems are a special case of Bayesian hidden-action principal-agent problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Indeed, this can be easily seen by taking an instance of the hidden-action problem in which outcomes correspond one-to-one with agent’s actions, and each action deterministically determines its corresponding outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 9 ARXIV PREPRINT - FEBRUARY 1, 2023 Definition 3 (Optimal payment function in a posterior).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given a posterior ξ ∈ ∆Θ, we say that a payment function π : ∆Θ × A → R+ is optimal in ξ if the following holds: π ∈ argmax π′ � k∈K λk �� θ∈Θ ξθ us θ(bk ξ,π′) − π(ξ, bk ξ,π′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (4) Notice that, in Problem (4), the price p of the signaling scheme φ does not appear in the seller’s expected utility, since we are restricted to settings in which the buyer has limited liability, and, thus, it is always the case that p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' For the same reason, we can safely assume that all the buyer’s types satisfy IR constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, we can state the following crucial result: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given a posterior ξ ∈ ∆Θ, solving Problem (4) is equivalent to computing a contract maximizing the principal’s expected utility in an instance of observable-action principal-agent problem such that, for every agent’s type k ∈ K and action a ∈ A, the following holds: ck a = � θ∈Θ ξθ � uk θ(bk ξ) − uk θ(a) � and ra = � θ∈Θ ξθ us θ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, finding an optimal contract in any instance of observable-action principal-agent problem can be reduced in polynomial time to computing a seller-optimal protocol without menus in a problem instance in which the buyer has limited liability and there is only one state of nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The first statement in Lemma 5 implies that, given an instance of our information-selling problem in which the buyer has limited liability and there is only one state of nature, it is possible to compute a seller-optimal protocol without menus by finding an optimal contract in an instance of observable-action principal-agent problem defined as in the lemma (notice that such an instance can be easily built in polynomial time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Thus, by Lemma 5, we can easily prove the following: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Restricted to instances in which the buyer has limited liability and there is only one state of nature, computing a seller-optimal protocol without menus is equivalent to the problem of finding an optimal contract in general instances of the observable-action principal-agent problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' While the computational complexity of finding optimal contracts in hidden-action principal-agent problems is well understood (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', (Castiglioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', 2022a)), to the best of our knowledge, there are no results on problems with observable actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In following theorem, we prove a strong hardness result for them: there exists a constant α < 1 such that designing a contract which provides the principal with at least an α fraction of the expected utility in an optimal contract is computationally intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In observable-action principal-agent problems, the problem of computing a contract maximizing the principal’s expected utility is APX-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, Theorem 2 immediately gives the following result: Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The problem of computing a seller-optimal protocol without menus is APX-hard, even when the buyer has limited liability and the number of states of nature d is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As we show in the following sections (see Theorems 8 and 11), whenever the number of states of nature is fixed, the problem of computing a seller-optimal protocol without menus admits a polynomial-time algorithm providing a particular bi-criteria approximation of the seller’s expected utility in an optimal protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Such an approximation is similar to the the bi-criteria guarantees provided by Castiglioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2022a) for Bayesian hidden-action principal- agent problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' By Theorem 2, our polynomial-time bi-criteria approximation algorithm for the setting in which the buyer has limited liability (Theorem 8) can be easily adapted to work with observable-action principal-agent problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Theorem 7 in (Castiglioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', 2022a) shows that, for hidden-action problems, such bi-criteria approximations are tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We leave as an open problem to establish whether these are also tight in our observable-action principal-agent problems or one can obtain better guarantees in polynomial time for our specific case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='3 Additional Preliminary Technical Results We conclude the section by recalling two already-known results on hidden-action principal-agent problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Clearly, these also hold for the specific case of observable-action principal-agent problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' By Theorem 2, such results can be easily cast to our information-selling problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Indeed, we also show that one of them can be strengthen in our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The first result that we are going to introduce makes use of linear contracts, which are payment schemes that pay the agent a given fraction of the principal’s reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally, in observable-action principal-agent problems, a contract 10 ARXIV PREPRINT - FEBRUARY 1, 2023 π : A → R+ is said to be linear if there exists a β ∈ [0, 1] such that π(a) = β ra for all a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Despite their simplicity, linear contracts provide good approximations with respect to general ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In particular, the following holds: Theorem 4 (Essentially Theorem 3 by Castiglioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2022a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In an observable-action principal-agent problem, for any ρ ∈ (0, 1/2], there exists a linear contract π : A → R+ such that: � k∈K λk � rbkπ − π(bk π) � ≥ ρ max π′ � k∈K λk � rbk π′ − π′(bk π′) � − 2Ω(1/ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, such a linear contract is defined by a parameter β = 1 − 2−i, for some i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', ⌊1/2ρ⌋}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We will make use of a stronger version of Theorem 4, which applies to our setting and directly follows from the analysis of Castiglioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2022a) and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally: Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given a posterior ξ ∈ ∆Θ, for any ρ ∈ (0, 1/2], there exists a payment function π : ∆Θ × A → R+ such that π(ξ, a) = β � θ∈Θ ξθ us θ(a) for every a ∈ A, where β ∈ [0, 1] is an (action-independent) parameter, and, additionally, the following holds: � k∈K λk � θ∈Θ ξθ � us θ(bk ξ,π) − π(s, bk ξ,π) � ≥ ρ � k∈K λk max a∈A � θ∈Θ ξθ � us θ(a) + uk θ(a) − uk θ(bk ξ) � − 2Ω(1/ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, such a parameter β is equal to 1 − 2−i for some i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' , ⌊1/2ρ⌋}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, we recall a useful result that establishes a connection between agent’s best responses and approximate best responses in principal-agent problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Intuitively, such a result states that, given a contract under which the agent is allowed to play an ǫ-best response (for some ǫ ≥ 0), it is always possible to recover a new contract in which the agent must play an (exact) best response, by only incurring in a small loss in the principal’s expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally, given ǫ ≥ 0 and a contract π : A → R+, for every k ∈ K, we let Bk,ǫ π ⊆ A be the set of ǫ-best-response actions for an agent of type k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Such a set is made by all the actions a ∈ A such that π(a) − ck a ≥ maxa′∈A � π(a′) − ck a′ � − ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We denote by bk,ǫ π ∈ Bk,ǫ π an ǫ-best-response action that is actually played by an agent of type k, assuming that ties are broken in favor of the principal, as usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then: Theorem 5 (Essentially Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='4 by Dutting et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given ǫ ≥ 0, an instance of observable-action principal-agent problem and, and a contract π : A → R+, there exists a contract π′ : A → R+ such that π′(a) = (1 − √ǫ) π(a) + √ǫ ra for every a ∈ A, and the following holds: � k∈K λk � rbk π′ − π′(bk π′) � ≥ � k∈K λk � rbk,ǫ π − π(bk,ǫ π ) � − 2√ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Theorem 5 can be easily cast to our setting by means of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally: Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given ǫ ≥ 0, a posterior ξ ∈ ∆Θ, and a payment a function π : ∆Θ × A → R+, there exists a payment function π′ : ∆Θ × A → R+ such that π′(ξ, a) = (1 − √ǫ) π(ξ, a) + √ǫ � θ∈Θ ξθus θ(a) for every a ∈ A, and the following holds: � k∈K λk �� θ∈Θ ξθ us θ(bk ξ,π′) − π′(ξ, bk ξ,π′) � ≥ � k∈K λk �� θ∈Θ ξθ us θ(bk,ǫ ξ,π) − π(ξ, bk,ǫ ξ,π) � − 2√ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Corollary 3 will be crucial to provide our results in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 5 Computing a Seller-optimal Protocol without Menus: The Case of a Buyer with Limited Liability In this section, we study the problem of computing a seller-optimal protocol without menus when the buyer has limited liability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', each buyer’s types k ∈ K has budget bk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As remarked in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='2, such a setting is of interest on its own, since it is a generalization of the one addressed by Dughmi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, the technical results derived in this section will be useful to deal with the general problem in which the buyer has no limited liability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We show how to circumvent the APX-hardness result that we established in Corollary 1, first, in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='1, by fixing the number of buyer’s actions, and then, in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='2, by fixing the number of states of nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In this section, since the buyer has limited liability, we can assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' that p = 0, so that we can compactly denote a protocol with a pair (γ, π), rather than with (γ, p, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 11 ARXIV PREPRINT - FEBRUARY 1, 2023 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='1 Fixing the Number of Buyer’s Actions First, we show that, whenever the buyer has limited liability and the number of buyer’s actions m is fixed, the problem of computing a seller-optimal protocol without menus admits a PTAS, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', we can design a protocol whose seller’s expected utility is arbitrarily close to that of an optimal protocol in time polynomial in the instance size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In order to design our PTAS, we start by observing that, since p = 0, the IR constraints are satisfied by all the protocols (γ, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This allows us to formulate the problem of computing a seller-optimal protocol without menus as the following optimization problem:10 max γξ≥0 π(ξ,a)≥0 � k∈K λk � ξ∈supp(γ) γξ �� θ∈Θ ξθ us θ(bk ξ,π) − π(ξ, bk ξ,π) � s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (5a) � ξ∈supp(γ) γξ ξθ = µθ ∀θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (5b) Notice that Problem (5) is defined over general distributions over posteriors γ, whose support supp(γ) may be not finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Thus, as we show in the following, the crucial result that we need to design a PTAS is the possibility of restricting the attention to finite sets of posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We need to introduce a particular class of posteriors, which are called q-uniform posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Definition 4 (q-Uniform posterior).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' A posterior ξ ∈ ∆Θ is q-uniform if it can be obtained by averaging the elements of a multi-set defined by q ∈ N>0 canonical basis vectors of Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In the following, we denote by Ξq ⊆ ∆Θ (for a given q ∈ N>0) the finite set of all the q-uniform posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As it is easy to check, such a set satisfies |Ξq| ≤ min{dq, qd}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In order to derive our PTAS, as a first preliminary result we show that, given any posterior ξ∗ ∈ ∆Θ, payment function π : ∆Θ × A → R+, and ǫ > 0, there always exists a signaling scheme γ supported on Ξq which induces posterior ξ∗ on average and guarantees a seller’s expected utility close to that provided by the posterior ξ∗ (assuming the buyer plays an ǫ-best response).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally: Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given any ǫ, α > 0, a posterior ξ∗ ∈ ∆Θ, and a payment function π : ∆Θ × A → R+, there always exists a signaling scheme γ ∈ ∆Ξq with q = 2 log(2m/α)/ǫ2 such that: � ξ∈Ξq γξ �� θ∈Θ ξθ us θ(bk,ǫ ξ,π) − π(ξ, bk,ǫ ξ,π) � ≥ � θ∈Θ ξ∗ θ us θ(bk ξ∗,π) − π(ξ∗, bk ξ∗,π) − α, for every buyer’s type k ∈ K, where we let π : ∆Θ ×A → R+ be a payment function that is optimal in every posterior ξ ∈ Ξq when the buyer plays an ǫ-best response, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', π solves Problem (4) for every ξ ∈ Ξq with bk ξ,π replaced by bk,ǫ ξ,π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Furthermore, the signaling scheme γ satisfies: � ξ∈Ξq γξ ξθ = ξ∗ θ ∀θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Lemma 6 guarantees that, by decomposing each posterior ξ ∈ ∆Θ as a convex combination of the elements of Ξq, the seller’s expected utility decreases by at most α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This implies that, assuming the buyer plays an ǫ-best response, it is possible to work with signaling schemes (and thus payment functions) supported on Ξq, by only slightly degrading the seller’s expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Another component that we need for our PTAS is an algorithm that, given a q-uniform posterior, computes an optimal payment function in that posterior (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', a payment function solving Problem (4) for such a posterior).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' By Theorem 2, it is easy to see that such an algorithm has to solve a problem that is equivalent to computing an optimal contract in observable-action principal-agent problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Thus, by Theorem 3, such a problem is APX-hard in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Next, we show that, whenever the number of buyer’s actions m is fixed, the APX-hardness result can be circumvented, and, thus, we can provide an algorithm that solves the desired task and runs in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally: 10Notice that, as it is the case for Problem (2) in Section 3, it is not immediately clear a priori whether the problem of computing a seller-optimal protocol without menus admits a maximum or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Thus, in principle we should start by defining the problem with a sup rather than a max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' However, in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='2, we provide a (possibly exponential-time) algorithm which finds a seller-optimal protocol without menus in general settings, and this implies that a maximum always exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 12 ARXIV PREPRINT - FEBRUARY 1, 2023 Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Restricted to instances where the buyer has limited liability and the number of buyer’s actions m is fixed, there exists a polynomial-time algorithm that, given a posterior ξ ∈ ∆Θ as input, computes the payments π(ξ, a) for a ∈ A of a payment function π : ∆Θ × A → R+ optimal in ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Notice that, in order to get a payment function π : ∆Θ × A → R+ that is optimal in every posterior ξ ∈ Ξq, it is sufficient to apply Lemma 7 for each ξ ∈ Ξq, then putting together all the computed payments π(ξ, a) in order to obtain the overall payment function π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The final piece that we need to complete the design of our PTAS is a way of coming back to work with buyer’s best responses, rather than using ǫ-best responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Indeed, this is possible thanks to Corollary 3, which allows us to modify the payment function in all the induced posteriors, so as to achieve the desired result by only losing a small amount 2√ǫ of the seller’s expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Now, we are ready to design our PTAS that works whenever the buyer has limited liability and the number of buyer’s actions m is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' By Lemma 6, we can focus on signaling schemes supported over q-uniform posteriors, for a suitably-defined q ∈ N>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, thanks to Lemma 7, we can compute a payment function that is optimal in all the q-uniform posteriors, by running the polynomial-time algorithm in Lemma 7 for each q-uniform posterior in Ξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' By Corollary 3, such an optimal payment function achieves a seller’s expected utility that is close to that obtained by a payment function which is optimal in every q-uniform posterior when considering ǫ-best responses, thus allowing for the application of the result in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In conclusion, our PTAS works by solving a modified version of LP (5), where we set supp(γ) := Ξq in Equation (5a), and we take as payment function the one returned by applying Lemma 7 in each posterior ξ ∈ Ξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' It is easy to see that the overall procedure requires time polynomial in the instance size when the number of actions m is fixed, since |Ξq| ≤ dq and q = 2 log(2m/α)/ǫ2 as prescribed by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' However, the overall running time depends exponentially in α > 0, which the seller’s expected utility approximation provided by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This allows us to prove the following result: Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Restricted to instances where the buyer has limited liability and the number of buyer’s actions m is fixed, the problem of computing a seller-optimal protocol without menus admits a PTAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, we show that a similar approach can be employed to derive a quasi-polynomial time algorithm providing a bi-criteria approximation of the seller’s expected utility in an optimal protocol, even when the number of buyer’s actions m is arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Indeed, in our PTAS, the computation of an optimal payment function in a given q-uniform posterior can be done in polynomial time only when the number of actions is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' While in general the problem is APX-hard, an approximately-optimal price function can be computed in polynomial time by applying Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, since q = 2 log(2m/α)/ǫ2 and |Ξq| ≤ dq, the enumeration over the q-uniform posteriors can be performed in time quasi polynomial in th number of actions m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This gives the following result: Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Restricted to instances in which the buyer has limited liability, there exists an algorithm that, given any α, ǫ > 0 and ρ ∈ (0, 1/2] as input, returns a protocol without menus achieving a seller’s expected utility greater than or equal to ρ OPT − 2−Ω(1/ρ) − (α + 2√ǫ), where OPT is the seller’s expected utility in an optimal protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, the algorithm runs in time polynomial in Ilog m—where I is the size of the problem instance and m is the number of buyer’s actions—, and the seller’s expected utility in the returned protocol is greater than or equal to OPTLIN − (α + 2√ǫ), where OPTLIN is the best expected utility achieved by a protocol parametrized by β as in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The second part of the statement will be useful in deriving our results for the problem of computing seller-optimal protocols without menus in the general case in which the buyer has no limited liability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Intuitively, it states that, even if our approximation algorithm only provides a bi-criteria approximation of a seller-optimal protocol, the returned protocol achieves a seller’s expected utility which is arbitrarily close to that achievable by using payment functions that define the payments as a given fraction of the seller’s expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='2 Fixing the Number of States of Nature Next, we study the case in which the buyer has limited liability and the number of states of nature d is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We prove that, in such a setting, it is possible to compute a bi-criteria approximation of an optimal protocol without menus similar to that in Theorem 7, but in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Notice that such a result circumvents the APX-hardness one provided in Corollary 1, as the latter is based on a reduction working with instances with only one state of nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Similarly to Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='1, we first show that it is possible to employ signaling schemes supported on the set Ξq of q-uniform posteriors (for a suitably-defined q ∈ N>0), by only suffering an arbitrarily small, additive loss in terms of seller’s expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' While the following result is similar to the one obtained in Lemma 6, it is based on different techniques and, in particular, on the fact that the seller’s expected utility is Lipschitz continuous in the buyers’ posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally: 13 ARXIV PREPRINT - FEBRUARY 1, 2023 Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given any α > 0, a posterior ξ∗ ∈ ∆Θ, and a payment function π : ∆Θ × A → R+ that is optimal in every posterior ξ ∈ Ξq with q = ⌈9d/α2⌉, there exists a signaling scheme γ ∈ ∆Ξq: � ξ∈Ξq γξ �� θ∈Θ ξθ us θ(bk ξ,π) − π(ξ, bk ξ,π) � ≥ � θ∈Θ ξ∗ θus θ(bk ξ∗,π) − π(ξ∗, bk ξ∗,π) − α, for every receiver’s type k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Furthermore, the signaling scheme γ satisfies: � ξ∈Ξq γξ ξθ = ξ∗ θ ∀θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Similarly to the case of a fixed number of actions, we employ Lemma 8 to restrict the attention to signaling schemes (and thus payment functions) supported on Ξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, in this case, we can apply Corollary 2 in each q-uniform posterior in order to compute in polynomial time a payment function that provides a bi-criteria approximation of the optimal seller’s expected utility in such a posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, we design an algorithm that solves a modified version of LP (5), where we set supp(γ) := Ξq in Equation (5a), and we take as payment function the one obtained by putting together those computed by means of Corollary 2 for each ξ ∈ Ξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, the overall procedure requires polynomial time, since |Ξq| ≤ qd and the number of states of nature d is fixed, and achieves a bi-criteria approximation of the seller’s expected utility in an optimal protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally: Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Restricted to instances in which the buyer has limited liability and the number of states of nature d is fixed, there exists an algorithm that, given α > 0 and ρ ∈ (0, 1/2] as input, returns in polynomial time a protocol without menus achieving a seller’s expected utility greater than or equal to ρ OPT − 2−Ω(1/ρ) − α, where OPT is the seller’s expected utility in an optimal protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, the seller’s expected utility in the returned protocol is greater than or equal to ρ OPTLIN − 2−Ω(1/ρ) − α where OPTLIN is the best expected utility achieved by a protocol parametrized by β as in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Similarly to Theorem 7, the second part of the statement will be useful for deriving our results in the general case in which the buyer has no limited liability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 6 Computing a Seller-optimal Protocol without Menus: The General Case We conclude our analysis by considering the problem of computing a seller-optimal protocol without menus in general instances in which the buyer has no limited liability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Thus, in such a setting, the seller also decides a price p ∈ R+ for the signaling scheme proposed to the buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' First, we provide a negative result for general instances that is stronger than the one established in Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In particular, the latter result states that the seller’s optimization problem is APX-hard even in the special case in which the buyer has limited liability and there is only one state of nature, relying on a reduction employing instances with an arbitrary number of actions m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Indeed, for the specific case in which the buyer has limited liability and the number of actions m is fixed, Theorem 6 provides a PTAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Next, we show that, in general instances where the buyer may not have limited liability, the problem is APX-hard even when the number of buyer’s actions m is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' To prove such an hardness result, we employ a result by Guruswami and Raghavendra (2009) (see Theorem 9 below), which is about the following promise problem related to the satisfiability of a fraction of linear equations with rational coefficients and variables restricted to the hypercube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='11 Definition 5 (LINEQ-MA(1−ζ, δ) by Guruswami and Raghavendra (2009)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' For any two constants ζ, δ ∈ R satisfying 0 ≤ δ ≤ 1 − ζ ≤ 1, LINEQ-MA(1 − ζ, δ) is the following promise problem: Given a set of linear equations Ax = c over variables x ∈ Qnvar, with coefficients A ∈ Qneq×nvar and c ∈ Qneq, distinguish between the following two cases: there exists a vector ˆx ∈ {0, 1}nvar that satisfies at least a fraction 1 − ζ of the equations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' every possible vector x ∈ Qnvar satisfies less than a fraction δ of the equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Theorem 9 (Guruswami and Raghavendra (2009)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' For all the constants ζ, δ ∈ R which satisfy 0 ≤ δ ≤ 1 − ζ ≤ 1, the problem LINEQ-MA(1 − ζ, δ) is NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 11In the definition in (Guruswami and Raghavendra, 2009), the vector ˆx can be non-binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' How- ever, Guruswami and Raghavendra (2009) use a binary vector ˆx in their proof and, thus, their hardness result also holds for our definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 14 ARXIV PREPRINT - FEBRUARY 1, 2023 Then, Theorem 5 allows us to prove the following hardness result: Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The problem of computing a seller-optimal protocol without menus is APX-hard, even when the number of buyer’s actions m is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In the following, we show how to circumvent the hardness result in Theorem 10, by providing, in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='1, a quasi-polynomial-time bi-criteria approximation algorithm and, in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='2, a polynomial-time (exact) algorithm working when the number of buyer’s types is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='1 A General Quasi-polynomial-time Bi-criteria Approximation Algorithm In order to circumvent the negative result presented in Theorem 10, we design a quasi-polynomial-time algorithm that computes a protocol without menus providing a bi-criteria approximation of the seller’s expected utility in an optimal protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally, our algorithm guarantees a multiplicative approximation ρ of the optimal utility, by only suffering an additional 2−Ω(1/ρ) + α additive loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, we show that our algorithm runs in polynomial time whenever either the number of buyer’s actions m or that of states of nature d is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In order to prove the approximation guarantees of our algorithm, we rely on Theorems 7 and 8, and we decompose the seller’s expected utility in an optimal protocol without menus into the sum of three different terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Our algorithm works by computing three protocols without menus, each one approximating one of the three terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Choosing the best protocol among the three provides the desired approximation guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The following is an intuition of how each term composing the optimal seller’s expected utility is approximated by our algorithm: The first term is related to the seller’s expected utility collected from buyer’s types for which the IR constraints are not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Such a utility term can be trivially achieved by a protocol that charges no price, discloses no information, and never pays back the buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The second term is related to the best seller’s expected utility which can be extracted from a buyer’s action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This is related to the optimal seller’s expected utility in a setting with limited liability, since, in that case, the seller’s expected utility is determined by the buyer’s action only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Thus, the second utility term can be approximated by using either the algorithm provided in Theorem 7 or that given in Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='12 The third term is related to the seller’s expected utility obtained by the transfers between the seller and the buyer, which include the charged price and the final payment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Such a utility term can be approximated by using a protocol that reveals all the information to the buyer while charging a carefully-chosen price for that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally, we prove the following main result: Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' There exists an algorithm that, given any α > 0 and ρ ∈ (0, 1/6] as input, computes a protocol without menus whose seller’s expected utility is greater than or equal to ρ OPT − 2−Ω(1/ρ) − α, where OPT is the seller’s expected utility in an optimal protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, the algorithm runs in time polynomial in Ilog m—where I is the size of the problem instance—when it is implemented with the algorithm in Theorem 7 as a subroutine, while it runs in time polynomial in Id when it is implemented with the algorithm in Theorem 8 as a subroutine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='2 Fixing the Number of Buyer’s Types Next, we study the problem of computing a seller-optimal protocol without menus when the number of buyer’s types is fixed, showing that it is possible to design a polynomial-time algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As a byproduct, the existence of such an algorithm shows that, for protocols without menus, the seller’s optimization problem always admits a maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As a preliminary result, we show that it is always possible to focus on protocols without menus (φ, p, π) that employ signals belonging to the set An, and define signaling schemes φ : Θ → ∆An and payment functions π : An × A → R+ such that, for every signal a ∈ An and k ∈ K, it holds ak ∈ Bk ξa,π, where ak ∈ A denotes the action corresponding to type k in a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Intuitively, in such protocols, a signal specifies an action recommendation for each buyer’s type, so that the buyer is always incentivized to follow such recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' With a slight abuse of notation, we say that protocols without menus (φ, p, π) as described above are generalized-direct and generalized-persuasive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally, we prove the following result: Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given a seller’s protocol without menus, there always exists another protocol without menus which is generalized-direct and generalized-persuasive, and achieves the same seller’s expected utility as the original protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 12Notice that we cannot employ Theorem 6 in place of Theorem 7, since the latter guarantees to achieve a seller’s expected utility that is arbitrarily close to that of the best protocol employing payment functions parametrized by β, and this is needed in order to derive the guarantees of our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Such a guarantee is not provided by Theorem 6, which only predicates on the quality of the returned protocol with respect to an optimal protocol without menus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 15 ARXIV PREPRINT - FEBRUARY 1, 2023 In order to prove the lemma, we observe that, given a protocol, if two signals induce the same best response for every buyer’s type, it is always possible to merge the two signals, retaining the same expected utility for both the seller and the buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, by iterating such a process, we recover a signaling scheme and a payment function employing An as set of signals .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As a second crucial step, we show that we can focus on protocols without menus (φ, p, π) whose price p is equal to the budget bk of one buyer’s type k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally: Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given a protocol without menus, there always exists another protocol (φ, p, π) such that p = bk for some k ∈ K, while achieving the same seller’s expected utility as the original protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, equipped with Lemma 9 and Lemma 10, we are ready to provide our polynomial-time algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Intuitively, since we can restrict the attention to protocols without menus (φ, p, π) that are generalized-direct and generalized- persuasive, and whose prices p belong to the set {bk}k∈K, we can solve the seller’s problem by iterating over all the possible price values p ∈ {bk}k∈K and, for each of them, over all the possible subsets R ⊆ K ∩ {k ∈ K : bk ≥ p} of buyer’s types that satisfy the IR constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This can be done in polynomial time since the number of buyer’s types n is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, for every price value p ∈ {bk}k∈K and set R ⊆ K ∩ {k ∈ K : bk ≥ p}, it is sufficient to solve the following optimization problem: sup φ≥0 π≥0 � k∈R λk � a∈An � θ∈Θ µθφθ(a) [us θ(ak) − π(a, ak)] + � k /∈R λk � θ∈Θ µθus θ(bk µ) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (6a) � θ∈Θ µθφθ(a) � uk θ(ak) + π(a, ak) � ≥ � θ∈Θ µθφθ(a) � uk θ(a′) + π(a, a′) � ∀k ∈ R, ∀a ∈ An, ∀a′ ̸= ak ∈ A (6b) � a∈An � θ∈Θ µθφθ(a) � uk θ(ak) + π(a, ak) � − bk ≥ � θ∈Θ µθuk θ(bk µ) ∀k ∈ R (6c) � a∈An � θ∈Θ µθφθ(a) � uk θ(ak) + π(a, ak) � − bk ≤ � θ∈Θ µθuk θ(bk µ) ∀k ̸∈ R (6d) � a∈An φθ(a) = 1 ∀θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (6e) By using techniques similar to those used in Section 3 for protocols with menus, we can show that Problem (6) is solvable in polynomial time by means of a suitable-defined LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This allows us to state our last results: Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Restricted to instances in which the number of buyer’s types n is fixed, the problem of computing a seller-optimal protocol without menus admits a polynomial-time algorithm.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Optimal Coordination in Generalized Principal-Agent Problems: A Revisit and Extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' arXiv preprint arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='01146 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Guru Guruganesh, Jon Schneider, and Joshua R Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Contracts under moral hazard and adverse selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In EC ’21: The 22nd ACM Conference on Economics and Computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 563–582.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Venkatesan Guruswami and Prasad Raghavendra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hardness of Learning Halfspaces with Noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 39, 2 (2009), 742–765.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='1137/070685798 Emir Kamenica and Matthew Gentzkow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Bayesian persuasion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' American Economic Review 101, 6 (2011), 2590–2615.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Shuze Liu, Weiran Shen, and Haifeng Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Optimal Pricing of Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Proceedings of the 22nd ACM Conference on Economics and Computation (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Yoav Shoham and Kevin Leyton-Brown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Multiagent systems: Algorithmic, game-theoretic, and logical founda- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Cambridge University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 17 ARXIV PREPRINT - FEBRUARY 1, 2023 A Proofs Omitted from Section 3 Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given any IC and IR seller’s protocol, it is always possible to recover an IC and IR seller’s protocol that is direct and persuasive, and it provides the seller with the same expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let { � φk, pk, πk � }k∈K be an IC and IR protocol such that there exist two signals s1, s2 ∈ S inducing the same best response ¯a ∈ A for a given receiver’s type ¯k ∈ K, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', b¯k ξs1 = b¯k ξs2 = ¯a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In the following, we show how to replace φ¯k and π¯k with a new signaling scheme ¯φ¯k and a new payment function ¯π¯k by merging s1, s2 into a single signal ¯s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally we define: ¯φ¯k θ(¯s) = φk θ(s1) + φk θ(s2) for each θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Similarly, we define: ¯π ¯k(¯s, ¯a) = � θ∈Θ µθφ¯k θ(s1)π¯k(s1, ¯a) + � θ∈Θ µθφ¯k θ(s2)π¯k(s2, ¯a) � θ∈Θ µθ(φ¯k θ(s1) + φ¯k θ(s2)) , Finally, we does not change all the other components of the protocol i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', we leave these components of { �¯φk, ¯pk, ¯πk � }k∈K equal to the one in { � φk, pk, πk � }k∈K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' To prove the lemma we show that the protocol { �¯φk, ¯pk, ¯πk � }k∈K achieves the same seller’s expected utility of { � φk, pk, πk � }k∈K, while satisfying the IC and IR constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As a first step, we observe that: � s∈S\\{s1,s2} � θ∈Θ µθ ¯φ ¯k θ(s) � u ¯k θ(b ¯k ξs,¯π) + ¯π¯k(s, b ¯k ξs,¯π) � + � θ∈Θ µθ ¯φ ¯k θ(¯s)[u ¯k θ(¯a) + ¯π¯k(¯s, ¯a)] − ¯p¯k = � s∈S\\{s1,s2} � θ∈Θ µθφ ¯k θ(s) � u ¯k θ(b ¯k ξs,π) + π¯k(s, b ¯k ξs,π) � + � θ∈Θ µθφ ¯k θ(s1)[u ¯k θ(¯a) + π¯k(s1, ¯a)]+ + � θ∈Θ µθφ ¯k θ(s2)[u ¯k θ(¯a) + π¯k(s2, ¯a)] − p¯k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The latter equality holds by linearity and proves that the protocol {(¯φk, ¯pk, ¯πk)}k∈K preserves the left-hand sides of the IR and IC constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, thanks to the convexity of the max operator, we can show that: max a∈A � θ∈Θ µθφ ¯k θ(s1)[u ¯k θ(a) + π¯k(s1, a)] + max a∈A � θ∈Θ µθφ ¯k θ(s2)[u ¯k θ(a) + π¯k(s2, a)] − p¯k ≥ max a∈A � θ∈Θ µθ ¯φ ¯k θ(¯s)[u ¯k θ(a) + ¯π¯k(¯s, a)] − p¯k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, by summing over the set (S ∪{¯s})\\{s1, s2}, we notice that the value of the right-hand side of the IC constraints achieved by the protocol {(¯φk, ¯pk, ¯πk)}k∈K is less or equal to the the value achieved by {(φk, pk, πk)}k∈K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Due to that, we can easily conclude that the new protocol preserves the IC and the IR constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, by observing that the following equality holds: � θ∈Θ µθ ¯φk θ(¯s)[u ¯k θ(¯a) + ¯π¯k(¯s, ¯a)] = � θ∈Θ µθφ ¯k θ(s1)[u ¯k θ(¯a) + π¯k(s1, ¯a)] + � θ∈Θ µθφ ¯k θ(s2)[u ¯k θ(¯a) + π¯k(s2, ¯a)], we can easily prove that the two protocols achieve the same seller’s expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, by iterating this procedure for each buyer’s type and couple of signals until there are no two signals inducing the same best response for that type, we get a protocol that employs direct and persuasive signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given an IC and IR protocol {(φk, pk, πk)}k∈K that is direct and persuasive, it is always possible to recover an IC and IR protocol {(φk, ˜pk, ˜πk)}k∈K such that: it is direct and persuasive, it provides the same seller’s expected utility as the original protocol, and, for every buyers’ type k ∈ K, it satisfies ˜pk = bk and ˜πk(a, a′) = 0 for all a ̸= a′ ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We first prove that by setting ˜πk(a, a′) = 0 for each a ̸= a′ ∈ A and k ∈ K, the seller’s expected utility does not change and the IC and IR constraints are preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' First, we show that by taking ˜πk(a, a′) = 0 for each a ̸= a′ ∈ A and k ∈ K, the signaling schemes φk remain persuasive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Indeed, we have that: � θ∈Θ µθφk θ(a)uk θ(a) + πk(a, a) ≥ � θ∈Θ µθφk θ(a′)uk θ(a′) + πk(a, a′) ≥ � θ∈Θ µθφk θ(a′)uk θ(a′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 18 ARXIV PREPRINT - FEBRUARY 1, 2023 Moreover, the left-hand side of all the equations defining the IR and IC constraints does not change since it depends only from the payments ˜πk(a, a) = πk(a, a) for each a′ ∈ A that remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Furthermore, by setting ˜πk(a, a′) = 0 for each a ̸= a′ ∈ A and k ∈ K, the right-hand sides of the IC constraints achieve smaller or equal values, since, intuitively, we are setting to zero non-negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, the IC constraints are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, by observing that the left-hand sides of the IR constraints and the seller’s expected utility do not embed terms ˜πk(a, a′) with a ̸= a′ ∈ A and k ∈ K, we conclude the first part of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In the second part of the proof we show that it is always possible to define a protocol in which the seller asks to each buyer’s type to deposit all their budget at the beginning of the interaction, achieving the same seller’s ex- pected utility and satisfying the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally, we show that given a protocol { � φk, pk, πk � }k∈K the protocol { � φk, ˜pk, ˜πk � }k∈K, with ˜pk = bk and ˜πk(a) = πk(a, a) + bk − pk for each k ∈ K and a ∈ A, achieves the same seller’s expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Indeed by linearity we have: � k∈K λk � � a∈A � θ∈Θ µθφk θ(a)us θ(a) − πk(a) + pk � = � k∈K λk � � a∈A � θ∈Θ µθφk θ(s)us θ(a) − πk(a) + bk − bk + pk � = � k∈K λk � � a∈A � θ∈Θ µθφk θ(a)us θ(a) − ˜πk(a) + bk � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' With similar arguments it is easy to check that the protocol { � φk, ˜pk, ˜πk � }k∈K satisfies the IC and IR constraints, concluding the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The optimal value of LP (3) is at least as large as the supremum in Problem (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We show that for each protocol { � φk, pk, πk � }k∈K that is feasible for Problem (2), we can derive a solution to LP (3) with at least the same value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This is sufficient to prove the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' By Lemma 1 and Lemma 2, we focus without loss of generality on protocols { � φk, pk, πk � }k∈K that are direct and persuasive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, we can build a solution (¯φ, ¯l, ¯y) to LP (3) letting ¯lk(a) = � θ∈Θ µθφθ(a)πk(a) for each k ∈ K and a ∈ A, and ¯yk,k′,a = max �� θ∈Θ µθφk′ θ (a) � uk θ(a) + πk′(a, a) � , max a′̸=a � θ∈Θ µθφk′ θ (a′)uk θ(a′) � for each k, k′ ∈ K and a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, we let ¯φ = φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' It is easy to verify that the solution (¯φ, ¯l, ¯y) results feasible for LP (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given a feasible solution to LP (3), it is possible to recover in polynomial time an IC and IR protocol whose seller’s expected utility is greater than or equal to the value of the solution to LP (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As a first step we show that from a feasible solution (φ, l, y) to LP (3) we can recover another solution with at least the same value in which if πk(a) > 0 then there exists a θ ∈ Θ such that φk θ(a) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Specifically, given a k ∈ K and an a ∈ A such that φk θ(a) = 0 for all θ ∈ Θ and lk(a) > 0, let ¯a ∈ A be (¯a, ¯θ) be any couple of an action and a state such that φk ¯θ(¯a) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We now define a new feasible solution (¯φ, ¯l, ¯y) as follows: ¯lk(a) = 0 ¯lk(¯a) = lk(a) + lk(¯a) ¯yk′,k,a = 0 ∀k′ ∈ K ¯yk′,k,¯a = yk′,k,¯a + lk(a) ∀k′ ∈ K, while we leave all the other terms variables equal to the ones in (φ, l, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' It is easy to check that the solution (¯φ, ¯l, ¯y)achieves the same seller’s expected utility while satisfying the constrains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Applying this procedure for each couple (k, a) such that φk θ(a) = 0 for all θ ∈ Θ and lk(a) > 0, we obtain a new solution (˜φ, ˜l, ˜y) such that if ˜lk(a) > 0 then there exists a θ ∈ Θ such that ˜φk θ(a) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 19 ARXIV PREPRINT - FEBRUARY 1, 2023 To recover a feasible protocol we just need to set payments as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' For each couple (k, a), if there exists a θ ∈ Θ such that ˜φk θ(a) > 0, we set ˜πk(a) = ˜lk(a)/(� θ∈Θ µθ ˜φk θ(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Otherwise, we set ˜πk(a) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Notice that the ratio ˜lk(a)/(� θ∈Θ µθ ˜φk θ(a)) is always well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, it is easy to see that {˜φk, ˜pk, ˜πk}k∈K, with ˜pk = bk for each k ∈ K is a feasible solution to Problem (2) with at least the same value as (φ, l, y) for LP (3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' There exists a polynomial-time algorithm that computes a protocol with menus that maximizes the seller’s expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The algorithm solves LP 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' By Lemma 3, this solution has value greater or equal to the supremum of Program 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, exploiting Lemma 4, we can recover in polynomial-time a protocol with at least the same utility, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', an optimal one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' B Proofs Omitted from Section 4 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given a posterior ξ ∈ ∆Θ, solving Problem (4) is equivalent to computing a contract maximizing the principal’s expected utility in an instance of observable-action principal-agent problem such that, for every agent’s type k ∈ K and action a ∈ A, the following holds: ck a = � θ∈Θ ξθ � uk θ(bk ξ) − uk θ(a) � and ra = � θ∈Θ ξθ us θ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, finding an optimal contract in any instance of observable-action principal-agent problem can be reduced in polynomial time to computing a seller-optimal protocol without menus in a problem instance in which the buyer has limited liability and there is only one state of nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We start proving the first part of the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given an instance of Problem (4), we build an instance of the observable-action principal-agent problem with ck a = � θ∈Θ ξθ � uk θ(bk ξ) − uk θ(a) � and ra = � θ∈Θ ξθ us θ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' To prove the equivalence between the two settings, we first show that the set of best-responses for the two problems coincides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Indeed, given a payment function π and a type k ∈ K, let a ∈ Bk ξ,π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, for each a′ ∈ A π(a) − ck a = π(a) − � θ∈Θ ξθ � uk θ(bk ξ) − uk θ(a) � = π(a) + � θ∈Θ ξθuk θ(a) − � θ∈Θ ξθuk θ(bk ξ) ≥ π(a′) + � θ∈Θ ξθuk θ(a′) − � θ∈Θ ξθuk θ(bk ξ) = π(a′) − ck a′, showing that a ∈ Bk π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Similarly, we can prove that if a ∈ Bk π, then a ∈ Bk ξ,π This implies that the set of best responses are equivalent for each payment function π, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', Bk π = Bk ξ,π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, argmax π � k∈K λk � rbkπ − π(bk π) � = argmax π � k∈K λk �� θ∈Θ ξθ us θ(bk π) − � θ∈Θ ξθ � uk θ(bk ξ) − uk θ(bk π) � � = argmax π � k∈K λk �� θ∈Θ ξθ us θ(bk π) + � θ∈Θ ξθuk θ(bk π) � = argmax π � k∈K λk �� θ∈Θ ξθ us θ(bk ξ,π) + � θ∈Θ ξθuk θ(bk ξ,π) � , showing the equivalence between the two problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This proves the first part of the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We now show that from an instance of the observable-action principal-agent problem we can always build an instance of the selling-information problem without menus and with only a single state of nature θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In particular, we set us θ(a) = ra for each a ∈ A, and uk θ(a) = 1 − ck a for each a ∈ A and k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Following a analysis similar to the first part of the proof, we can show that the two problems are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 20 ARXIV PREPRINT - FEBRUARY 1, 2023 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In observable-action principal-agent problems, the problem of computing a contract maximizing the principal’s expected utility is APX-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We reduce from vertex cover in cubic graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally, it is NP-hard to approximate the size of the minimum vertex cover in cubic graphs with an approximation (1 + ε), for a given constant ε > 0 Alimonti and Kann (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let η = ε/7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We show that an (1 − η)-approximation to the principal-agent problem with observable actions can be used to provide a (1 + ε) approximation to vertex cover, concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Consider an instance of vertex cover (V, E) with nodes V and edges E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let ρ = |V | and ℓ = |E|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given a vertex v ∈ V , we let E(v) be the set of edges e such that v is one of the extreme of the edge e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Similarly, given an edge e ∈ E, let V (e) be the set of vertexes v such that e is an edge with extreme v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We build an instance of the principal-agent problem with observable actions as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' For each vertex v ∈ V , there exists an agent’s type kv, while for each e ∈ E, there exists a type ke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' For each vertex v ∈ V , there exists an action av and an additional action a−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The cost of a type ke, e ∈ E, is cke a− = 0, cke av = 1 2 if e ∈ E(v) and 1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The cost of a type kv, v ∈ V , is ckv av = 0, and 1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, the principal’s utility is equal to 1 if the action is in {av}v∈V , while is equal to 0 otherwise, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', us av = 1 for each v ∈ V and us a− = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' All the types are equally probable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', λk = 1 ρ+ℓ for each k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' First, we show that if there exists a vertex cover V ⋆ of size ν, the value of the problem is at least (ρ−ν)+ ν 2 + ℓ 2 ρ+ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Consider the payment function such that π(av) = 1 2 if v ∈ V ⋆ and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' A type kv with v /∈ V ⋆ plays the action av and receives a payment of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' A type kv with v ∈ V ⋆ plays the action av and receives a payment of 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' A type ke plays an action av such that e ∈ E(v) (this action exists by construction) and receives a payment of 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' It is easy to see that the expected seller’s utility is (ρ−ν)+ ν 2 + ℓ 2 ρ+ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Suppose that there exists an algorithm that provides a 1 − η approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This implies that the algorithm returns a solution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', a payment function π, with value at least (1 − η) ρ− k 2 + ℓ 2 ρ+ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We show how to exploit the payment function π to build a vertex cover of size at most (1 + ε)ν in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In particular, given π we recover a vertex cover ¯V of the desired size as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' First, it is easy to see that we can set the payment π(a−) = 0 and payments π(av) ∈ {0, 1 2} for each v ∈ V without decreasing the utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Intuitively, payments are useful only to change the best response of a type ke, e ∈ E, from a− to av, v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' To do so, it is sufficient a payment of 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, let ¯E be the set of edges e ∈ E such that the best response of ke is a−, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', ¯E = {e ∈ E : bke π = a−}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Consider a edge e ∈ ¯E and a vertex v ∈ V (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Since e ∈ ¯E the payment π(av) = 0 and no type ke plays action av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, if we modify the payments by letting π(av) = 1 2 on the action av we have three effects: i) the type ke changes the best response to av, ii) some other types e′ ∈ E could change from action a− to av,13 iii) the payment of type kv increases by 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Overall the principal’s total utility increases by 1 2λke since ke changes from action a− with payment 0 to action av with payment 1 2, and it decreases by − 1 2λkv as the payment to type kv increases by 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, if other types ke′, e′ ̸= e change from a− with payment 0 to action av with payment 1 2 the principal’s utility increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This implies that the principal’s utility does not decrease with this procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, repeating this procedure we can build a payment function π with the same utility such that all the agent’s type plays actions av, v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, let ¯V be the set of vertexes with at least one agent of type ke, e ∈ E that plays this action, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', ¯V = {v ∈ V : bke π = av, e ∈ E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We show that ¯V is an vertex cover of size at most (1 + ǫ)ν, concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Notice that we can set payment π(av) = 0 for each v ∈ V \\ ¯V without decreasing the seller’s utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Removing the payment does not change any best response since the only type playing the action av is the type kv, the utility ukv(av) = 1, and the utility of playing any other action a ̸= av is ukv(a) + π(a) ≤ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, the principal’s utility is (ρ − | ¯V |) − 1 2| ¯V | + 1 2ℓ ρ + ℓ = ρ − 1 2| ¯V | + 1 2ℓ ρ + ℓ , since ρ − | ¯V | types kv, v ∈ V , play av and receive payment 0, | ¯V | types kv, v ∈ V , play av and receive payment 1 2, and all the types ke, e ∈ E, play an action av, v ∈ V and receive payment 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, since the principal’s utility is by assumption ρ− 1 2 | ¯V |+ 1 2 ℓ ρ+ℓ ≥ (1 − η) ρ− ν 2 + ℓ 2 ρ+ℓ , it holds | ¯V | ≤ η(2ρ + ℓ) + ν ≤ (1 + 7η)ν = (1 + 7η)ν = (1 + ε)ν, where the second inequality comes from ρ = 2 3ℓ and ℓ ≤ 3ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 13Notice that there could be other actions a′ v with π(av′) = 1 2 that provides the same utility of av for both the principal and the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 21 ARXIV PREPRINT - FEBRUARY 1, 2023 Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given a posterior ξ ∈ ∆Θ, for any ρ ∈ (0, 1/2], there exists a payment function π : ∆Θ × A → R+ such that π(ξ, a) = β � θ∈Θ ξθ us θ(a) for every a ∈ A, where β ∈ [0, 1] is an (action-independent) parameter, and, additionally, the following holds: � k∈K λk � θ∈Θ ξθ � us θ(bk ξ,π) − π(s, bk ξ,π) � ≥ ρ � k∈K λk max a∈A � θ∈Θ ξθ � us θ(a) + uk θ(a) − uk θ(bk ξ) � − 2Ω(1/ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, such a parameter β is equal to 1 − 2−i for some i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' , ⌊1/2ρ⌋}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As a first step, we notice that even if Castiglioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (2022a) show that linear contracts provide the desired approximation with respect to optimal contracts, their proof can be extended to show that linear contracts provide the same approximation with respect to the optimal social welfare, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', � k∈K λk maxa∈A[ra −ck a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' To prove this result, it is sufficient to follow all the steps of Theorem 3 of (Castiglioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', 2022a) except for the one in which Observation 1 is employed to upperboud the value of the optimal contract with the social welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, we can modify this result to hold in our setting exploiting Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' C Proofs Omitted from Section 5 Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given any ǫ, α > 0, a posterior ξ∗ ∈ ∆Θ, and a payment function π : ∆Θ × A → R+, there always exists a signaling scheme γ ∈ ∆Ξq with q = 2 log(2m/α)/ǫ2 such that: � ξ∈Ξq γξ �� θ∈Θ ξθ us θ(bk,ǫ ξ,π) − π(ξ, bk,ǫ ξ,π) � ≥ � θ∈Θ ξ∗ θ us θ(bk ξ∗,π) − π(ξ∗, bk ξ∗,π) − α, for every buyer’s type k ∈ K, where we let π : ∆Θ ×A → R+ be a payment function that is optimal in every posterior ξ ∈ Ξq when the buyer plays an ǫ-best response, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', π solves Problem (4) for every ξ ∈ Ξq with bk ξ,π replaced by bk,ǫ ξ,π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Furthermore, the signaling scheme γ satisfies: � ξ∈Ξq γξ ξθ = ξ∗ θ ∀θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let ˜ξ ∈ Ξq be the empirical mean of q i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' samples drawn according to ξ∗ ∈ ∆Θ, where each θ ∈ Θ has probability ξ∗ θ of being sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Therefore, ˜ξ ∈ Ξq is a random vector supported on q-uniform posteriors with expectation ξ∗ ∈ ∆Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, let γ ∈ ∆Ξq be a probability distribution such as, for each ξ ∈ Ξq, it holds γξ := Pr(˜ξ = ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We build a new payment function ˜π such that for each ξ ∈ ∆Θ and a ∈ A, we have ˜π(ξ, a) = π(ξ∗, a) Moreover, we let Ξq,ǫ be the set of posteriors such that ξ ∈ Ξq,ǫ if and only if for each a ∈ A it holds: ����� � θ∈Θ � ξθuk θ(a) − ξ∗ θuk θ(a) � ����� ≤ ǫ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (7) Then, for each ξ ∈ Ξq,ǫ, we have that Bk ξ∗,π ⊆ Bk,ǫ ξ,π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In particular, for any a∗ ∈ Bk ξ∗,π, ξ ∈ Ξq,ǫ and a ∈ A: � θ∈Θ ξθuk θ(a∗) + ˜π(ξ, a∗) ≥ � θ∈Θ ξ∗ θuk θ(a∗) + ˜π(ξ∗, a∗) − ǫ 2 (By Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (7) and the definition of Bk ξ∗,π) ≥ � θ∈Θ ξ∗ θuk θ(a) + ˜π(ξ∗, a) − ǫ 2 ≥ � θ∈Θ ξθuk θ(a) + ˜π(ξ, a) − ǫ (By Equation (7)) which is precisely the definition of Bk,ǫ ξ∗,π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' For each a ∈ A, let ˜tk a := � θ∈Θ ˜ξθuk θ(a)+˜π(˜ξ, a) and tk a := � θ∈Θ ξ∗ θuk θ(a)+˜π(ξ∗, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' By the Hoeffding’s inequality we have that, for each a ∈ A, Pr(|˜tk a − E[˜tk a]| ≥ ǫ 2) ≤ 2e−2q(ǫ/2)2 = 2e− log(2m/α) ≤ α m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (8) 22 ARXIV PREPRINT - FEBRUARY 1, 2023 Moreover, Equation (7) and the union bound yield the following: � ξ∈Ξq,ǫ γξ = Pr(˜ξ ∈ Ξq,ǫ) = Pr( � a∈A ��˜tk a − tk a �� ≤ ǫ 2) ≥ 1 − � a∈A Pr( ��˜tk a − tk a �� ≥ ǫ 2) ≥ 1 − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (By Equation (8)) Let ¯ξ be a d-dimensional vector defined as ¯αθ := � ξ∈Ξq\\Ξq,ǫ γξξθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' By definition and for the previous result we have: � θ∈Θ ¯ξθ ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' we can show: � ξ∈Ξq γξ � θ∈Θ ξθus θ(bk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='ǫ ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π′) − π′(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='ǫ ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π′) ≥ � ξ∈Ξq γξ � θ∈Θ ξθus θ(bk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='ǫ ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) − π(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='ǫ ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) ≥ � ξ∈Ξq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='ǫ γξ � θ∈Θ ξθus θ(bk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='ǫ ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) − π(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='ǫ ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) ≥ � ξ∈Ξq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='ǫ γξ � θ∈Θ ξθus θ(bk ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) − π(ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) (Bk ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π ⊆ Bk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='ǫ ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π for each ξ ∈ Ξq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='ǫ) = � θ∈Θ � us θ(bk ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) − π(ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) �� � ξ∈Ξq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='ǫ γξξθ � = � θ∈Θ � us θ(bk ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) − π(ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) �� � ξ∈Ξq γξξθ − ¯ξθ � (By definition of ¯α) = � θ∈Θ � ξθus θ(bk ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) − π(ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) �� � ξ∈Ξq γξξθ � − � θ∈Θ � us θ(bk ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) − π(ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) � ¯ξθ ≥ � θ∈Θ � us θ(bk ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) − π(ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) �� � ξ∈Ξq γξξθ � − � θ∈Θ ¯ξθ (Utilities in [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 1]) ≥ � θ∈Θ ξ∗ θus θ(bk ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) − π(ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, by definition of γ, we have that, for each θ ∈ Θ: � ξ∈Ξq γξξθ = ξ∗ θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Restricted to instances where the buyer has limited liability and the number of buyer’s actions m is fixed, there exists a polynomial-time algorithm that, given a posterior ξ ∈ ∆Θ as input, computes the payments π(ξ, a) for a ∈ A of a payment function π : ∆Θ × A → R+ optimal in ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given a posterior ξ ∈ ∆Θ and a tuple a ∈ A|K| we let Πa ⊆ Rm + be the set of payment functions π such that for each k ∈ K it holds ak ∈ Bk ξ,π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given an a ∈ A|K|, the problem of computing an optimal payment function restricted to payment functions in Πa can be formulated as follows: min π � k∈K λkπ(ξ, ak) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' � θ∈Θ ξθuk θ(ak) + π(ξ, ak) ≥ � θ∈Θ ξθuk θ(a′) + π(ξ, a′) ∀a′ ∈ A, k ∈ K π(ξ, a′) ≥ 0 ∀a′ ∈ A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 23 ARXIV PREPRINT - FEBRUARY 1, 2023 We observe that, for each tuple a ∈ A|K|, the vertexes of the regions Πa ⊆ Rm + are identified by m of the common O(nm2 + m) constraints: � θ∈Θ ξθuk θ(a′) + π(ξ, a′) ≥ � θ∈Θ ξθuk θ(a′′) + π(ξ, a′′)∀a′ ̸= a′′ ∈ A, ∀k ∈ K π(, ξ, a′) ≥ 0 ∀a′ ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, the total number of vertexes defining all the regions Πa, a ∈ A|K|, is at most �nm2+m m � = O((nm2 + m)m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, since the objective function is linear in Πa for each tuple a ∈ A|K|, given the optimal tuple of induced actions a∗ ∈ A|K| the optimum is attained in one of the vertexes of Πa∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, there are overall �� � a∈A|K| V (Πa) �� = O((km2 + m)m) vertexes, where V (·) denotes the set of vertexes of the polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, when m is fixed, it is possible to enumerate in polynomial time over all the vertexes in � a∈A|K| V (Πa) and compute the optimal payment function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Restricted to instances where the buyer has limited liability and the number of buyer’s actions m is fixed, the problem of computing a seller-optimal protocol without menus admits a PTAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given two arbitrary constants α, ǫ > 0 we let (γ, π) be an optimal protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We show that an (α+2√ǫ)-optimal protocol (γ∗, π∗) can be computed in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As a first step we define a signaling scheme γ∗ supported in Ξq as follows: γ∗ ˜ξ = � ξ∈supp(γ) γξγξ ˜ξ ∀˜ξ ∈ Ξq, where γξ ∈ ∆Ξq is the signaling scheme satisfying Lemma 6 with q = 2 log(2m/α) ǫ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' First we observe that γ∗ ∈ ∆Ξq satisfies the consistency constraints, indeed we have: � ˜ξ∈Ξq γ∗ ˜ξ ˜ξθ = � ξ∈supp(γ) γξ � ˜ξ∈Ξq γξ ˜ξ ˜ξθ = � ξ∈supp(γ) γξξθ = µθ ∀θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, let π∗ : ∆Θ × A → R+ be the optimal payment function in each ˜ξ ∈ Ξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We show that the protocol (γ∗, π∗, 0) is (α + 2√ǫ)-optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let π′′ : ∆Θ × A → R+ be the optimal payment function in each ξ ∈ Ξq when the buyer is playing an ǫ-best response, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', π′′(ξ, ·) ∈ arg max ˜π(ξ,·) � k∈K λk[ � θ ξθus θ(bk,ǫ ξ,˜π) − ˜π(ξ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, let π′ : ∆Θ ×A → R+ be the payment function such that π′(ξ, a) = (1−√ǫ)π′′(ξ, a)+√ǫ � θ∈θ ξθus θ(a) for each ξ and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' we have: � ˜ξ∈Ξq γ∗ ˜ξ � � θ∈Θ ˜ξθus θ(bk ˜ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π∗) − π∗(˜ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk ˜ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π∗) � ≥ � ˜ξ∈Ξq γ∗ ˜ξ � � θ∈Θ ˜ξθus θ(bk ˜ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π′) − π′(˜ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk ˜ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π′) � (Optimality of π∗) ≥ � ˜ξ∈Ξq γ∗ ˜ξ � � θ∈Θ ˜ξθus θ(bk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='ǫ ˜ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π′′) − π′′(˜ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='ǫ ˜ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π′′) � − 2√ǫ (By Proposition 3) ≥ � ξ∈supp(γ) γξ � � ˜ξ∈Ξq γξ ˜ξ � � θ∈Θ ˜ξθus θ(bk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='ǫ ˜ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π′′) − π′′(˜ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='ǫ ˜ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π′′) �� − 2√ǫ (By defintion of γ∗) ≥ � ξ∈supp(γ) γξ � � θ∈Θ ξθus θ(bk ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) − π(ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) � − α − 2√ǫ (By Lemma 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Notice that the optimal payment π∗ : ∆Θ × A → R+ in each ξ ∈ Ξq can be computed in polynomial time employing Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, to compute the optimal signaling scheme γ∗ ∈ ∆Ξq we can solve the following LP: � k∈K λk � ξ∈Ξq γξ � θ∈Θ ξθus θ(bk ξ,π∗) − π∗(ξ, bk ξ,π∗) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 24 ARXIV PREPRINT - FEBRUARY 1, 2023 � ξ∈supp(γ) γξξθ = µθ ∀θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Note that since |Ξq| = O(dq), all the payment function π∗ : Ξq × A → R+ can be precomputed in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, the LP has polynomially many variables and constraints and can be solved efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, the solution returned by the LP is α + 2√ǫ-optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Restricted to instances in which the buyer has limited liability, there exists an algorithm that, given any α, ǫ > 0 and ρ ∈ (0, 1/2] as input, returns a protocol without menus achieving a seller’s expected utility greater than or equal to ρ OPT − 2−Ω(1/ρ) − (α + 2√ǫ), where OPT is the seller’s expected utility in an optimal protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, the algorithm runs in time polynomial in Ilog m—where I is the size of the problem instance and m is the number of buyer’s actions—, and the seller’s expected utility in the returned protocol is greater than or equal to OPTLIN − (α + 2√ǫ), where OPTLIN is the best expected utility achieved by a protocol parametrized by β as in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The proof follows the same steps of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' However, it relies on Theorem 4 instead of Lemma 7 to compute an approximate payment function for all q-uniform posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given any α > 0, a posterior ξ∗ ∈ ∆Θ, and a payment function π : ∆Θ × A → R+ that is optimal in every posterior ξ ∈ Ξq with q = ⌈9d/α2⌉, there exists a signaling scheme γ ∈ ∆Ξq: � ξ∈Ξq γξ �� θ∈Θ ξθ us θ(bk ξ,π) − π(ξ, bk ξ,π) � ≥ � θ∈Θ ξ∗ θus θ(bk ξ∗,π) − π(ξ∗, bk ξ∗,π) − α, for every receiver’s type k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Furthermore, the signaling scheme γ satisfies: � ξ∈Ξq γξ ξθ = ξ∗ θ ∀θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We define a payment function π′ : ∆Θ ×A → R+ as follows: π′(ξ, a) = π(ξ∗, a) for each a ∈ A and ξ ∈ ∆Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Furthermore, we define: Iα(ξ∗) = � ξ ∈ ∆Θ : ∥ξ − ξ∗∥∞ ≤ α2 18d � , as the neighborhood of the given posterior ξ∗ ∈ ∆Θ and Ξ(ξ∗) = Iǫ(ξ∗) ∩ Ξq its intersection with the set Ξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Notice that if q ≥ 18d α2 , it holds ξ∗ ∈ co(Ξ(ξ∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 14 We show that for each ξ ∈ Iα(ξ∗), it holds bk ξ∗,π′ ∈ Bk,ǫ ξ,π′, where ǫ := α2/9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As a first step, by Hölder’s inequality we have that � θ∈Θ |(ξθ − ξ∗ θ)us θ(a)| ≤ d||ξ − ξ∗||∞ = ǫ/2 ∀a ∈ A, Moreover, by the definition of best response and the previous inequality, we have that: � θ∈Θ ξθuk θ(bk ξ∗,π′) + π′(ξ, bk ξ∗,π′) ≥ � θ∈Θ ξ∗ θuk θ(bk ξ∗,π′) + π′(ξ∗, bk ξ∗,π′) − ǫ/2 ≥ � θ∈Θ ξ∗ θuk θ(bk ξ,π′) + π′(ξ∗, bk ξ,π′) − ǫ/2 ≥ � θ∈Θ ξθuk θ(bk ξ,π′) + π′(ξ, bk ξ,π′) − ǫ ≥ � θ∈Θ ξθuk θ(a′) + π′(ξ, a′) − ǫ, for each a′ ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This shows that bk ξ∗,π′ ∈ Bk,ǫ ξ,π′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let π∗ : ∆Θ × A → R+ be the payment function prescribed by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, we have that: 14Given a finite set A we denote with co(A) the set containing all the convex combination of elements in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 25 ARXIV PREPRINT - FEBRUARY 1, 2023 � θ∈Θ ξθus θ(bk ξ,π) − π(ξ, bk ξ,π) ≥ � θ∈Θ ξθus θ(bk ξ,π∗) − π∗(ξ, bk ξ,π∗) (Optimality of π′) ≥ � θ∈Θ ξθus θ(bk,ǫ ξ,π′) − π′(ξ, bk,ǫ ξ,π′) − 2√ǫ (By Proposition 3 ) ≥ � θ∈Θ ξθus θ(bk ξ∗,π′) − π′(ξ, bk ξ∗,π′) − 2√ǫ ≥ � θ∈Θ ξ∗ θus θ(bk ξ∗,π′) − π′(ξ∗, bk ξ∗,π′) − 2√ǫ − ǫ = � θ∈Θ ξ∗ θus θ(bk ξ∗,π′) − π′(ξ∗, bk ξ∗,π′) − 3√ǫ = � θ∈Θ ξ∗ θus θ(bk ξ∗,π) − π(ξ∗, bk ξ∗,π) − 3√ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This shows that the expected seller’s utility decreases of at most 3√ǫ when we consider sufficiently close posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, by Caratheodory’s theorem we can decompose ξ∗ as follows: � ξ′∈Ξ(ξ) γξ∗ ξ′ ξ′ θ = ξ∗ θ ∀θ ∈ Θ with γξ∗ ∈ ∆Ξ(ξ∗), where we recall that ξ∗ ∈ co(Ξ(ξ∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We show now that such a decomposition decreases the expected seller’s utility only by the desired amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally, we have that: � ξ′∈Ξ(ξ) γξ∗ ξ′ � � θ∈Θ ξ′ θus θ(bk ξ′,π) + π(ξ′, bk ξ′,π) � ≥ � ξ′∈Ξ(ξ) γξ∗ ξ′ � � θ∈Θ ξ∗ θus θ(bk ξ∗,π) + π(ξ, bk ξ∗,π) − 3√ǫ � = � θ∈Θ ξ∗ θus θ(bk ξ∗,π) + π(ξ∗, bk ξ∗,π) − 3√ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Since 3√ǫ ≤ α, this concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Restricted to instances in which the buyer has limited liability and the number of states of nature d is fixed, there exists an algorithm that, given α > 0 and ρ ∈ (0, 1/2] as input, returns in polynomial time a protocol without menus achieving a seller’s expected utility greater than or equal to ρ OPT − 2−Ω(1/ρ) − α, where OPT is the seller’s expected utility in an optimal protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, the seller’s expected utility in the returned protocol is greater than or equal to ρ OPTLIN − 2−Ω(1/ρ) − α where OPTLIN is the best expected utility achieved by a protocol parametrized by β as in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given a constant α > 0 we let (γ, π) be an optimal protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As a first step, we show that there exists a protocol (γ∗, π∗) achieving a seller’s expected utility of at least APX ≥ ρOPT−2−Ω(1/ρ)−α, where OPT is the utility achieved with (γ, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, the payment function π∗ is a linear function with parameter β ∈ {1 − 2−i}i∈{i,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=',⌊ρ/2⌋}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We define a signaling scheme γ∗ supported in Ξq as follows: γ∗ ˜ξ = � ξ∈supp(γ) γξγξ ˜ξ ∀˜ξ ∈ Ξq, where γξ ∈ ∆Ξq is the signaling scheme satisfying Lemma 6 with q = 18d α2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' First we observe that γ∗ ∈ ∆Ξq satisfies the consistency constraints, indeed we have: � ˜ξ∈Ξq γ∗ ˜ξ ˜ξθ = � ξ∈supp(γ) γξ � ˜ξ∈Ξq γξ ˜ξ ˜ξθ = � ξ∈supp(γ) γξξθ = µθ ∀θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, we can define as π∗ : ∆Θ × A → R+ as the payment function computed (in polynomial time) with Corollary 2 in each q-uniform posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let π′ be the optimal payment function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We show that the protocol (γ∗, π∗) achieves the desired approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally: � ˜ξ∈Ξq γ∗ ˜ξ � � θ∈Θ ˜ξθus θ(bk ˜ξ,π∗) − π∗(˜ξ, bk ˜ξ,π∗) � 26 ARXIV PREPRINT - FEBRUARY 1, 2023 ≥ ρ \uf8eb \uf8ed � ˜ξ∈Ξq γ∗ ˜ξ � � θ∈Θ ˜ξθus θ(bk ˜ξ,π′) − π′(˜ξ, bk ˜ξ,π′) � \uf8f6 \uf8f8 − 2Ω(1/ρ) (Corollary 2) = � ξ∈supp(γ) γξ � � ˜ξ∈Ξq γξ ˜ξ � � θ∈Θ ˜ξθus θ(bk,ǫ ˜ξ,π) − π(˜ξ, bk,ǫ ˜ξ,π) �� (By defintion of γ∗) ≥ � ξ∈supp(γ) γξ � � θ∈Θ ξθus θ(bk ξ,π) − π(ξ, bk ξ,π) � − α (By Lemma 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This implies that since (γ∗, π∗) is feasible for the following LP, it has value at least ρOPT − 2Ω(1/ρ) − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' max γ≥0 � k∈K λk � ξ∈Ξq γξ � θ∈Θ ξθus θ(bk ξ,π∗) − π∗(ξ, bk ξ,π∗) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' � ξ∈supp(γ) γξξθ = µθ ∀θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, to find the desired approximation it is sufficient to compute π∗ in each q-uniform posterior and solve the LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Note that since |Ξq| = O(qd), the computation of the payment function π∗ : ∆Θ × A → R+ and the computation of the previous LP require polynomial time for each fixed α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, to prove the second part of the statement it is sufficient to notice that π∗ is optimal with respect to the desired set of linear payment functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' D Proofs Omitted from Section 6 Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The problem of computing a seller-optimal protocol without menus is APX-hard, even when the number of buyer’s actions m is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We introduce a reduction from LINEQ-MA(1 − ζ, δ) to the design of the optimal protocol, showing that for ζ and δ small enough, the following holds: Completeness: If an instance of LINEQ-MA(1 − ζ, δ) admits a 1 − ζ fraction of satisfiable equations when variables are restricted to lie in the hypercube {0, 1}nvar, then there exists a protocol that provides to the seller’s expected utility at least of η, where η will be defined in the following;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Soundness: If at most a δ fraction of the equations can be satisfied, then every protocol provides to the seller’s expected utility at most η − c, where c is a constant defined in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In the rest of the proof, given a vector of variables x ∈ Qnvar, for i ∈ [nvar], we denote with xi the component corresponding to the i-th variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Similarly, for j ∈ [neq], cj is the j-th component of the vector c, whereas, for i ∈ [nvar] and j ∈ [neq], the (j, i)-entry of A is denoted by Aji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Reduction As a preliminary step, we normalize the coefficients by letting ¯A := 1 τ A and ¯c := 1 τ 2 c, where we let τ := 2M max � maxi∈[nvar],j∈[neq] Aji, maxj∈[neq] cj, n2 var � and M will be defined in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' It is easy to see that the normalization preserves the number of satisfiable equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally, the number of satisfied equations of Ax = c is equal to the number of satisfied equations of ¯A¯x = ¯c, where ¯x = 1 τ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' For every variable i ∈ [nvar], we define a state of nature θi ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, we introduce three additional states θ0, θ1, θ2 ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The prior distribution µ ∈ int(∆Θ) is defined in such a way that µθi = 1 2n2var for every i ∈ [nvar], while µθ0 = nvar−1 2nvar , µθ1 = 1 4, and µθ2 = 1 4 (notice that � θ∈Θ µθ = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We define four buyer’s types k1 j , k2 j , k3 j , k4 j ∈ K for each equation j ∈ [neq], where the probability of observing each buyer’s type is 1 8neq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, we define an additional type k⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' All the types k ∈ K have budget bk = ν/2, where ν will be defined in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The buyer has 9 actions available, namely A := {a0, a1, a2, a3, a4, a5, a6, a7, a8}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, we define the utilities of the players, where the utility is 0 when not specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' For each k1 j , j ∈ [neq], the utilities are: u k1 j θi (a0) = 1 2 for each i ∈ [nvar], 27 ARXIV PREPRINT - FEBRUARY 1, 2023 u k1 j θi (a1) = 1 2 − ¯Aji + ¯cj for each i ∈ [nvar], u k1 j θi (a2) = 1 2 + ¯Aji − ¯cj for each i ∈ [nvar] u k1 j θ0 (a0) = 1 2, u k1 j θ0 (a1) = 1 2 + ¯cj, u k1 j θ0 (a2) = 1 2 − ¯cj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' u k1 j θ1 (a3) = 1 2 + 2ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' For each k2 j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' j ∈ [neq],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' the utilities are: u k2 j θi (a0) = 1 2 − ¯Aji + ¯cj for each i ∈ [nvar],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' u k2 j θi (a7) = 1 2 for each i ∈ [nvar] u k2 j θ0 (a0) = 1 2 + ¯cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' u k2 j θ1 (a3) = 1 2 + 2ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' For each type k3 j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' j ∈ [neq] the utilities are: u k3 j θi (a0) = 1 2 + ¯Aji − ¯cj for each i ∈ [nvar],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' u k3 j θi (a7) = 1 2 for each i ∈ [nvar] u k3 j θ0 (a1) = 1 2 − ¯cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' u k3 j θ1 (a3) = 1 2 + 2ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' For each type k4 j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' j ∈ [neq] the utilities are equivalent to the one of type k1 j but with the following differences: ukj θ (a5) = 1 2 for each θ ∈ Θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ukj θi (a7) = 1 2 for each i ∈ [nvar],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' the utilities of type k⋆ are: uk⋆ θ1 (a6) = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' uk⋆ θ (a1) = 1 for each θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, we let uk θ(a8) = 1 2 for every k ∈ K and θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, the utility of the seller is: us θ(a6) = 1 4 for each θ ∈ Θ, us θ(a5) = 4ν for each θ ∈ Θ, 28 ARXIV PREPRINT - FEBRUARY 1, 2023 us θ(a0) = ν for each θ ∈ Θ, us θ(a7) = 2ν for each θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We recall that the utility is 0 when not defined explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Suppose that there exists a vector ˆx ∈ {0, 1}nvar such that at least a fraction 1 − ζ of the equations in Aˆx = c are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let X1 ⊆ [nvar] be the set of variables i ∈ [nvar] with ˆxi = 1, while X0 := [nvar] \\ X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given the definition of ¯A and ¯c, there exists a vector ¯x ∈ {0, 1 τ }nvar such that at least a fraction 1 − ζ of the equations in ¯A¯x = ¯c are satisfied, and, additionally, ¯xi = 1 τ for all the variables in i ∈ X1, while ¯xi = 0 whenever i ∈ X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let us consider an (indirect) signaling scheme φ : Θ → ∆S where the set of signals is S := {s1, s2, s3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let q := nvar(nvar−1) τ−|X1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' For each i ∈ [nvar], let φθi(s1) = q and φθi(s2) = 1 − q if i ∈ X1, while φθi(s2) = 1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, let φθ0(s1) = 1, φθ1(s3) = 1 and φθ2(s2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, all the other probabilities φθ(s) are set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' It is easy to see that the signaling scheme is feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, we set the price p = ν/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, we set π(s3, a6) = 2ν and all the other payments π(s, a) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Now, we compute the expected seller’s utility due of each type of buyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The buyer of type k⋆ in the posterior ξs3 plays the action a6 and gets utility � θ∈θ ξs3 θ uk⋆ θ (a6) + π(s3, a6) = 1 + 2ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, in the other posteriors ξs1 and ξs2 the seller’s utility is at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, the protocol is IR for the buyer since the expected utility declining the protocol is 1 while accepting it is −π/2 + 1 · 3 4 + (1 + 2ν) 1 4 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, the expected principal utility when the buyer’s type is k⋆ is at least � s∈S � θ∈θ ξs θuk⋆ θ (bk⋆ ξs,π) = 1 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Consider a buyer k1 j , j ∈ [neq], such that the j-th equality is satisfied by the vector ˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Now, let us take the buyer’s posterior ξs1 ∈ ∆Θ induced by the signal s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let h := q n2var � i∈X1 q n2var + nvar−1 nvar .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, using the definition of ξs1, it is easy to check that ξ1 θi = h for every i ∈ Xs1, ξs1 θi = 0 for every i ∈ X0, while ξs1 θ0 = nvar−1 nvar � i∈X1 q n2var + nvar−1 nvar = 1 − h ��X1��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The buyer of type kj ∈ K experiences a utility of � θ∈Θ ξs1 θ ukj θ (a0) = 1 2 by playing action a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Instead,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' the utility she gets by playing a1 is defined as follows: � θ∈Θ ξs1 θ ukj θ (a1) = � i∈X1 h �1 2 − ¯Aji + ¯cj � + ξ1 θ0 �1 2 + ¯cj � = = h ��X1�� �1 2 + ¯cj � − h � i∈X1 ¯Aji + � 1 − h ��X1��� �1 2 + ¯cj � = = 1 2 + ¯cj − h � i∈X1 ¯Aji = 1 2 + ¯cj − 1 τ � i∈X1 ¯Aji = 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' where the second to last equality holds since h = 1 τ (by definition of h and q),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' while the last equality follows from the fact that the j-th equation is satisfied,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 1 τ � i∈X1 ¯Aji = ¯cj (recall that ¯xi = 1 τ for all i ∈ X1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Using similar arguments, we can write � θ∈Θ ξs1 θ ukj θ (a2) = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, all the other actions have utility 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, the buyer plays a0 in the posterior ξs1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In posterior ξs2 induced by signal s2, the utility of each action different from a8 is strictly smaller than 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, the buyer will play a8, while in posterior ξs3 induced by signal s3, the utility of action a3 is 1 2 + 2ν and the buyer will play a3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence the expected utility of the buyer is 3 4 1 2 + 1 4( 1 2 + 2ν) − p = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, the protocol is IR for the buyer since if she declines the protocol the utility is 1 2 while if she accepts the protocol the utility is 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, when the buyer’s type is k1 j the expected seller’s utility is νµθ0 + ν/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Consider a buyer k2 j or k3 j , j ∈ neq such that the j-th equality is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' A similar argument as before shows that in posterior ξs1 the buyer’s optimal action is a7, while in posterior ξs2, the optimal action is a8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In posterior ξs3, the optimal action is a3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, the expected buyer’s utility is 3 4 1 2 + 1 4( 1 2 + 2ν) − p = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, the protocol is IR for the buyer and provides expected seller’s utility at least 2νµθ0 + ν/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Consider a buyer k4 j , j ∈ [neq] such that the j-th equality is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The buyer has an utility similar to k1 j and plays the same best responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, it is indifferent in participating or not participating to the protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 29 ARXIV PREPRINT - FEBRUARY 1, 2023 We assume that they brake ties in favor of the seller and does not accept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' She plays action a5 and the expected seller’s utility is 2ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Since all the other buyer’s types provide positive utility —it never happens that the expected payment from the seller to the buyer exceeds the payment from the buyer to the seller—, the expected seller’s utility is at least η = 1 32 + (1 − ζ)1 8(µθ0ν + ν/2) + (1 − ζ)1 4(µθ02ν + ν/2) + (1 − ζ)1 82ν Soundness As a first step, we upperbound the expected seller’s utility from each type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' It is easy to see that the maximum expected utility that the seller can extract from the buyer’s type k⋆ is at most 1 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, the maximum expected utility that the seller can extract from a buyer of type k1 j , j ∈ [neq], is at most 1 8(ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The maximum expected utility that the seller can extract from a buyer of type k2 j or k3 j , j ∈ [neq] is at most 1 4 3 2ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, the maximum expected utility that the seller can extract from a buyer of type k4 j , j ∈ [neq], is 1 82ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Using the previous upperbounds, we can bound the component of the utility due to each set of types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' For each constant t < 1, there exist constants c = c(t), ζ = ζ(t) such that if the expected utility is greater than η − c then the expected utility from types k1 j , j ∈ [neq], is at least t 1 8(ν), the expected utility from types k2 j , j ∈ [neq], and k3 j , j ∈ [neq], is at least t 1 4 3 2ν, and the expected utility from types k4 j , j ∈ [neq], is at least t 1 82ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' To see that, consider for instance the types k1 j , j ∈ [neq].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' It must hold: 1 32 + ¯t1 8ν + 1 4 3 2ν + 1 82ν ≥ 1 32 + (1 − ζ)1 8(µθ0ν + ν/2) + (1 − ζ)1 4(µθ02ν + ν/2) + (1 − ζ)1 82ν Since for nvar large enough µθ0 is close to 1 2, for c(t), ζ(t) small enough constant the equation is satisfied for ¯t ≥ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' A similar result holds for every other set of types k2 j with j ∈ [neq], k3 j with j ∈ [neq], and k4 j with j ∈ [neq].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The next step is to show the existence of a posterior in which a t fraction of agent of types k1 j , j ∈ [neq], play a0 and the the same holds for each other set of types k2 j ,k3 j with action a7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Suppose by contradiction that there is no posterior in which a t fraction of k1 j , j ∈ [neq], plays a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' First, notice that the maximum payment is at most p = ν/2+ 1 M , otherwise all the buyer’s types k1 j are not IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, the seller’s utility minus payment is greater than 0 in a posterior only if the agent plays a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, it is easy to see that it is sufficient to consider signaling schemes that induce posteriors such that if ξθi > 0, then ξθ1 = 0 and ξθ2 = 0 since states ξθ1 and ξθ2 disincentivize the actions with high seller’s utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, the maximal utility from agents of types k1 j is at most 1 8 � ν/2 + 1 M + (t − 1/neq)1 2ν � < t1 8ν, for M large enough, reaching a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' A similar argument holds for the other types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This implies that there exists a set Q ⊆ [neq] and a posterior ξ such that for each j ∈ Q all the buyers k1 j , k2 j , and k3 j in the posterior play a0,a7, and a7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Notice that |Q| ≥ 1 − 3(1 − t) and for t large enough |Q| > δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Suppose that there exists a signal inducing a posterior ξ ∈ ∆Θ in which all the buyer’s types k1 j , j ∈ Q best respond by playing action a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We show that there exists at least one j ∈ Q such that it holds � θ∈Θ ξθu k1 j θ (a1) > � θ∈Θ ξθu k1 j θ (a0) or � θ∈Θ ξθu k1 j θ (a2) > � θ∈Θ ξθu k1 j θ (a0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' For every buyer’s type k1 j ∈ K, it holds � θ∈Θ ξθukj θ (a0) = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, it is the case that: � θ∈Θ ξθu k1 j θ (a1) = � i∈[nvar] ξθi �1 2 − ¯Aji + ¯cj � + ξθ0 �1 2 + ¯cj � = 1 2 + ¯cj − � i∈[nvar] ξθi ¯Aji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Similarly, it holds: � θ∈Θ ξθu k1 j θ (a2) = 1 2 − ¯cj + � i∈[nvar] ξθi ¯Aji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Suppose by contradiction that for every type k1 j , j ∈ Q, it is the case that � θ∈Θ ξθu k1 j θ (a0) ≥ � θ∈Θ ξθu k1 j θ (a1), which implies that ¯cj − � i∈[nvar] ξθi ¯Aji ≤ 0, whereas it holds � θ∈Θ ξθu k1 j θ (a0) ≥ � θ∈Θ ξθukj θ (a2), implying −¯cj + � i∈[nvar] ξθi ¯Aji ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Thus, � i∈[nvar] ξθi ¯Aji = ¯cj for every j ∈ Q and the vector ˆx ∈ Qnvar with ˆxi = ξθi for all i ∈ [nvar] satisfies at least a fraction δ of the equations, reaching a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Since we 30 ARXIV PREPRINT - FEBRUARY 1, 2023 have that t types k1 j play a0, this implies that π(ξ, a0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' However, at the same time we have that the buy- ers of type k2 j and k3 j plays action a7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Consider a j∗ ∈ Q such that � θ∈Θ ξθu k1 j∗ θ (a1) > � θ∈Θ ξθu k1 j∗ θ (a0) or � θ∈Θ ξθu k1 j∗ θ (a2) > � θ∈Θ ξθu k1 j∗ θ (a0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Recall that this buyer must play a7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' If the first inequality holds then it must hold � θ∈Θ ξθu k2 j∗ θ (a7) + π(ξ, a7) ≥ � θ∈Θ ξθu k2 j∗ θ (a0) + π(ξ, a0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, � θ∈Θ ξθu k2 j∗ θ (a7) = � θ∈Θ ξθu k1 j∗ θ (a0) < � θ∈Θ ξθu k1 j∗ θ (a1) = � θ∈Θ ξθu k2 j∗ θ (a0), implying π(ξ, a7) > π(ξ, a0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' A similar argument holds for the buyer k3 j∗ if the second inequality is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This implies that type k4 j∗ can play the same best responses of player k1 j in any posterior different from ξ and play action a7 in ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, the expected utility of buyer k4 j∗ is strictly greater than the one of k1 j∗ (that is IR), and hence it is strictly IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We conclude the proof showing that the utility of this buyer’s type is too small, reaching a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' First, notice that the seller must induce a posterior with ξθ1 ≥ 3 4 with probability at least 1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In all the other posteriors the seller’s utility from type k∗ is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' However, it must hold that the utility from type k⋆ is at least 1 64 for ν small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, playing posteriors with ξθ1 ≥ 3 4 with probability smaller than 1 8 the seller’s utility form type k∗ is at most 1 2 1 4 1 8 < 1 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Now consider the type k4 j∗ that is IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In a posterior ξ with ξθ1 ≥ 3 4, the seller’s utility when the type is k4 j∗ is at most 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, the total utility from this type is at most p + 7 84ν ≤ ν + 1/M, where the last inequality follows by the fact that the payment is at most ν 2 + 1/M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' For |Q| large enough, we have that a |Q|/neq − δ fraction of types k4 j provide seller’s utility at most ν + 1/M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, the total utility from type k4 j is at most 1 8[(|Q|/neq − δ)(ν + 1/M) + (1 − (|Q|/neq − δ))2ν] ≤ t 1 82ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Thus, we reach a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' There exists an algorithm that, given any α > 0 and ρ ∈ (0, 1/6] as input, computes a protocol without menus whose seller’s expected utility is greater than or equal to ρ OPT − 2−Ω(1/ρ) − α, where OPT is the seller’s expected utility in an optimal protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, the algorithm runs in time polynomial in Ilog m—where I is the size of the problem instance—when it is implemented with the algorithm in Theorem 7 as a subroutine, while it runs in time polynomial in Id when it is implemented with the algorithm in Theorem 8 as a subroutine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let (φ, p, π) be an optimal protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, the seller’s expected utility is given by: � k /∈Rφ,p,π λk � θ∈Θ µθus θ(bk µ) + � k∈Rφ,p,π λk �� s∈S � θ∈Θ µθφθ(s) � us θ(bk ξs,π) − π(s, bk ξs,π) � + p � , where we recall that Rφ,p,π is the set of buyer’s types for which the IR constraint is satisfied under protocol (φ, p, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given a signal s ∈ S and a type k ∈ K, let bk ξs ∈ arg maxa∈A � θ∈Θ µθφθ(s)uk θ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Intuitively, bk ξs is an opti- mal action for the buyer without considering the payment function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, the seller’s utility can be spitted in three components: (i) The utility from the buyer’s types that are not IR U1 := � k /∈Rφ,p,π λk � θ∈Θ µθus θ(bk µ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (ii) The maximum seller’s utility deriving from the buyer’s action U2 := � k∈Rφ,p,π λk �� s∈S � θ∈Θ µθφθ(s) � us θ(bk ξs) + uk θ(bk ξs,π) − uk θ(bk ξs) � � , where we use the fact that to incentivize action bk ξs,π over bk ξs the payment must be at least � s∈S � θ∈Θ µθφθ(s)(uk θ(bk ξs)−uk θ(bk ξs,π)) � θ∈Θ µθφθ(s) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (iii) The utility related to the overall payment that the seller’s can extract from the buyer given the price function π U3 := � k∈Rφ,p,π λk � p − � s � θ µθφθ(s) � π(s, bk s,π) + uk θ(bk s,π) − uk θ(bk ξs) � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 31 ARXIV PREPRINT - FEBRUARY 1, 2023 Notice that the term U2 + U3 is the utility deriving from buyer’s types for which the IR constraint is satisfied, where we add, respectively subtract, the term � k∈Rφ,p,π λk �� s∈S � θ∈Θ µθφθ(s) � uk θ(bk ξs,π) − uk θ(bk ξs) � � to U2, respectively U3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In the following, we design three protocols (φ1, p1, π1), (φ2, p2, π2), and (φ3, p3, π3), each with seller’s utility that approximates the corresponding utility terms U1, U2, and U3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We will show that this will implies that at least one protocol provides a good approximation of the overall seller’s utility, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', of U1 + U2 + U3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Approximate U1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The protocol (φ1, p1, π1) that provides no information, charges no price, and does not provides any payment has seller’s utility � k∈K � θ µθus θ(bk µ) ≥ � k /∈Rφ,p,π � θ µθus θ(bk µ) = U1 Approximate U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' By Corollary 2, we know that for each signal s ∈ S (inducing a posterior ξs) and ρ ∈ (0, 1/2], there exists a linear contract π′(s, ·) such that π′(s, a) = β � θ∈Θ ξs θus θ(a) with parameter β = 1 − 2−i, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' , ⌊ 1 2ρ⌋} that guarantees: � k∈K λk � � θ∈Θ ξs θ � us θ(bk ξs,π′) − π′(ξs, bk ξs,π′) � � (11a) ≥ ρ � k∈K λk �� θ ξs θ � us θ(bk ξs,π) + uk θ(bk ξs,π) − uk θ(bk ξs) � � − 2−Ω(1/ρ) (11b) ≥ ρ � k∈Rφ,p,π λk �� θ ξs θ � us θ(bk ξs,π) + uk θ(bk ξs,π) − uk θ(bk ξs) � � − 2−Ω(1/ρ) (11c) where the first inequality comes from Corollary 2, and the last one since we restrict the elements in the first summation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Now, we need a protocol (φ2, p2, π2) that approximate the utility obtained by the optimal protocol that uses only linear payment functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' When the number of states is fixed, we can approximate the optimal protocol that uses linear payment functions using Theorem 8 with an additive loss α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Otherwise, we can use Theorem 7 that is polynomial time when the number of actions is fixed, while it runs in quasi-polynomial time and provides a loss α when instantiated with sufficiently small parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, protocol (φ2, p2, π2) can be computed in time poly(min{Id, Ilog(m)}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Notice that both the algorithms returns a protocol such that p = 0 and hence p2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' we can show that the protocol (φ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' π2) has seller’s utility � k∈K λk �� s∈S � θ µθφθ(s) � us θ(bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π2) − π2(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π2) � � ≥ � k∈K λk �� s∈S � θ µθφθ(s) � us θ(bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π′) − π′(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π′ � � − α = � k∈K λk �� s∈S �� θ µθφθ(s) � � θ ξs θ � us θ(bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π′) − π′(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π′ � � − α = � s∈S �� θ µθφθ(s) � � k∈K λk �� θ ξs θ � us θ(bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π′) − π′(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π′ � � − α ≥ � s∈S �� θ µθφθ(s) � � ρ � k∈R λk � θ ξs θ � us θ(bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) + uk θ(bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) − uk θ(bk ξs) � − 2−Ω(1/ρ) � − α = � s∈S �� θ µθφθ(s) � \uf8ee \uf8f0ρ � k∈Rφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π λk � θ ξs θ � us θ(bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) + uk θ(bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) − uk θ(bk ξs) � \uf8f9 \uf8fb − 2−Ω(1/ρ) − α 32 ARXIV PREPRINT - FEBRUARY 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 2023 = ρ � k∈Rφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π λk � s∈S �� θ µθφθ(s) � �� θ ξs θ � us θ(bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) + uk θ(bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) − uk θ(bk ξs) � � − 2−Ω(1/ρ) − α = ρ � k∈Rφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π λk � s∈S � θ µθφθ(s) � us θ(bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) + uk θ(bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) − uk θ(bk ξs) � − 2−Ω(1/ρ) − α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' where the first inequality holds since π′ employs linear payments functions and (φ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' π2) has an additive loss α w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' any protocol that employs linear payments functions, while the second inequality comes from Equation (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Approximate U3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let δk := � θ µθuk θ(bk θ) − � θ µθuk θ(bk µ) for each k ∈ K, where bk θ is the best response of agent of type k ∈ K when the state of nature is θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' For each k ∈ Rφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' by the definition of IR it holds � s∈S � θ∈Θ µθφθ(s)[π(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) + uk θ(bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π)] − p ≥ � θ µθuk θ(bk µ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (12) Hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' p − � s∈S � θ∈Θ µθφθ(s) � π(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) + uk θ(bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) − uk θ(bk ξs) � = p − � s∈S � θ∈Θ µθφθ(s) � π(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) + uk θ(bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) � + � s∈S � θ∈Θ µθφθ(s)uk θ(bk ξs) ≤ p − � s∈S � θ∈Θ µθφθ(s) � π(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) + uk θ(bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) � + � s∈S � θ∈Θ µθφθ(s)uk θ(bk θ) = p − � s∈S � θ∈Θ µθφθ(s) � π(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) + uk θ(bk ξs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='π) � + � θ∈Θ µθuk θ(bk θ) ≤ − � θ∈Θ µθuk θ(bk θ) + � θ∈Θ µθuk θ(bk θ) ≤ δk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' where the first inequality follows by the optimality of action bk θ in state θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' and the second one by Equation (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Next, we show that for each ζ ∈ [0, 1] we can design a protocol with seller’s utility of at least ζ 2 � k∈K δk − 2−1/ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let Pζ := {2−i}i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=',⌊1/ζ⌋} ∪ {0}, and for each k ∈ K let pk be the greatest p ∈ Pζ such that p ≤ δk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, � k∈K λkpk ≥ � k∈K λk � δk/2 − 2−⌊1/ζ⌋� = � k∈K λkδk/2 − 2−⌊1/ζ⌋, where the inequality holds since either pk ≥ δk/2 or pk ≤ 2−⌊1/ζ⌋ Hence, � p∈Pζ p � k∈K:pk=p λk ≥ � k∈K λkδk/2 − 2−⌊1/ζ⌋, implying max p∈Pζ p � k∈K:pk=p λk ≥ 1 2|Pζ| � k∈K λkδk − 2−⌊1/ζ⌋ ≥ ζ 2 � k∈K λkδk − 2−⌊1/ζ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let p∗ = argmaxp∈Pζ p � k∈K:pk=p λk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Consider the protocol (φ3, p3, π3) that charges payment p3 = p∗, reveals all information with φ3 and set payment π3(s, a) = 0 for each s ∈ S and a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We show that this protocol satisfies the IR constraint for all the players such that pk = p∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Indeed, for all these types it holds � θ∈Θ µθuk θ(bk θ) − p∗ ≥ � θ∈Θ µθuk θ(bk θ) − δk = � θ∈Θ µθuk θ(bk θ) − �� θ µθuk θ(bk θ) − � θ µθuk θ(bk µ) � ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (14) Then, the utility of the protocol is at least the payment obtained by the buyers’ type in Rφ3,p3,π3 ⊇ {k ∈ K : pk = p∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In particular, it is at least p∗ � k∈K:pk=p∗ λk ≥ ζ 2 � k∈K λkδk − 2−⌊1/ζ⌋ 33 ARXIV PREPRINT - FEBRUARY 1, 2023 ≥ ζ 2 � k∈Rφ,p,π λkδk − 2−⌊1/ζ⌋ ≥ ζ 2 � k∈Rφ,p,π λk � p − � s∈S � θ µθφθ(s)[π(s, bk ξs,π) + uk θ(bk ξs,π) − uk θ(bk ξs)] � − 2−⌊1/ζ⌋ = ζ 2U3 − 2−⌊1/ζ⌋, where in the the first inequality we use Equation (14), and in the third inequality we use Equation (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Equivalently, setting ρ = ζ/2, we obtain that for each ρ ∈ [0, 1/2] there exists a protocol (φ3, p3, π3) that has seller’s utility at least ρU3 − 2−Ω(1/ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Wrapping up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let i = arg maxj∈{1,2,3} Uj and OPT be the seller’s utility with the optimal protocol (φ, p, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, since U1+U2+U3 = OPT, we have that Ui ≥ 1 3OPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, since for each ρ ∈ [0, 1/2] we can approximate each utility Ui, i ∈ {1, 2, 3} with a protocol with utility at least ρUi −2−Ω(1/ρ) −α, the seller’s utility of our approximation algorithm is at least ρUi − 2−Ω(1/ρ) − α ≥ ρOPT/3 − 2−Ω(1/ρ) − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, setting ρ′ = ρ/3, we obtain that for each ρ′ ∈ [0, 1/6] the utility of the designed protocol is at least OPT − 2−Ω(1/ρ) − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given a seller’s protocol without menus, there always exists another protocol without menus which is generalized-direct and generalized-persuasive, and achieves the same seller’s expected utility as the original protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let (φ, π, p) be a protocol and let be s1, s2 ∈ S be two signals such that bk ξs1 = bk ξs2 for each receiver’s type k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We show that it is always possible to define a new protocol (φ∗, π∗, p) that employs a single signal s∗ instead of s1 and s2 achieving the same seller’s expected utility while satisfying the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Formally, we define a new signaling scheme φ∗ as follows: �φ∗ θ(s∗) = φθ(s1) + φθ(s2) ∀θ ∈ Θ φ∗ θ(s) = φθ(s) ∀θ ∈ Θ, ∀s ∈ S \\ {s1, s2} and a new payment function π∗ as follows: �π∗(s∗, a) = zπ(s1, a) + (1 − z)π(s2, a) ∀a ∈ A π∗(s, a) = π(s, a) ∀a ∈ A, ∀s ∈ S \\ {s1, s2} with z = � θ∈Θ µθφθ(s1)/(� θ∈Θ µθ(φθ(s1) + φθ(s2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As a first step, we observe that for each k ∈ K it holds: � θ∈Θ µθ � φθ(s1) � us θ(bk ξs1,π) − π(s1, bk ξs1,π) � + φθ(s2) � us θ(bk ξs2,π) − π(s2, bk ξs2,π) � � = � θ∈Θ µθφ∗ θ(s∗) � us θ(bk ξs∗,π∗) − π∗(s∗, bk ξs∗,π∗) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, for each k ∈ K it holds: � θ∈Θ µθ � φθ(s1) � uk θ(bk ξs1 ,π) + π(s1, bk ξs1,π) � + φθ(s2) � uk θ(bk ξs2,π) + π(s2, bk ξs2,π) � � = � θ∈Θ µθφ∗ θ(s∗) � uk θ(bk ξs∗,π∗) + π∗(s∗, bk ξs∗,π∗) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, noticing that for each signal s ∈ S \\ {s1, s2} the seller’s utility and the buyer’s utility does not change from (φ, π, p) to (φ∗, π∗, p), the set R of buyer’s type for which the IR is satisfied does not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As a consequence, the two protocols achieve the same seller’s expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Applying this procedure to all the couples of signals that induces the same vector of best responses, we obtain a generalized-direct and generalized-persuasive protocol providing the same seller’s expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Given a protocol without menus, there always exists another protocol (φ, p, π) such that p = bk for some k ∈ K, while achieving the same seller’s expected utility as the original protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 34 ARXIV PREPRINT - FEBRUARY 1, 2023 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let (φ, π, p) be a protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We show that there exists a ˆk ∈ K and a payment function ˆπ such that the protocol (φ, ˆπ, bˆk) provides the same seller’s expected utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let ˆk ∈ arg min k∈Rφ,π,p:bk≥p{bk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We observe that all the buyer’s types k ∈ Rφ,π,p have enough budget to participate in the protocol,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', bk ≥ bˆk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Furthermore, we define ˆπ(s, a) = π(s, a) + bˆk − p for each s ∈ S and a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, we show that the set of types Rφ,π,p = Rφ,ˆπ,ˆp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Indeed, for each type k ∈ Rφ,ˆπ,ˆp it holds � θ∈Θ � s∈S µθφθ(s) � uk θ(bk ξs,ˆπ) + ˆπ(s, bk ξs,ˆπ) � − bˆk = � θ∈Θ � s∈S µθφθ(s) � uk θ(bk ξs,ˆπ) + π(s, bk ξs,ˆπ) + bˆk − p � − bˆk = � θ∈Θ � s∈S µθφθ(s) � uk θ(bk ξs,π) + π(s, bk ξs,π) � − p, and hence k ∈ Rφ,π,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Similarly, we can prove that each buyer’s type k /∈ Rφ,ˆπ,ˆp does not belong to Rφ,π,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' It follows that Rφ,π,p = Rφ,ˆπ,ˆp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, we can show that the seller’s utility results equal to the one in (φ, π, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Indeed, we have: � k∈Rφ,p,π λk � � θ∈Θ � s∈S µθφθ(s) � us θ(bk ξs,π) − π(s, bk ξs,π) � + p � + � k /∈Rφ,p,π λk � θ∈Θ µθus θ(bk µ) = � k∈Rφ,ˆ p,ˆπ λk � � θ∈Θ � s∈S µθφθ(s) � us θ(bk ξs,ˆπ) − ˆπ(s, bk ξs,ˆπ) � + bˆk � + � k /∈Rφ,ˆ p,ˆπ λk � � θ∈Θ µθus θ(bk µ) � This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Restricted to instances in which the number of buyer’s types n is fixed, the problem of computing a seller-optimal protocol without menus admits a polynomial-time algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' In the following, we present an algorithm to compute an optimal protocol that works in polynomial time when the number of buyer’s types is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As a first step, we observe that, thanks to Lemma 10, the initial payment required by the seller coincides with bk for some k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Furthermore, we can focus on direct protocols by Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, given a price p ∈ {bk}k∈K and a set of buyer’s types R ⊆ K ∩ {k ∈ K : bk ≥ p} for which the IR constraint is satisfied, the the problem of computing the optimal protocol can be formulated as Problem (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Similarly to Section 3, we can provide a linear relaxation of Problem (6) introducing a variable l(a, a′) that replaces � θ∈Θ µθφθ(a)π(a, a′) for each a ∈ An and a′ ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, we obtain the following LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' max φ≥0,l≥0 � k∈R λk � a∈An �� θ∈Θ µθφθ(a)us θ(ak) − l(a, ak) � + � k /∈R λk � θ∈Θ µθus θ(bk µ) (15a) � θ∈Θ µθφθ(a)uk θ(ak) + l(a, ak) ≥ � θ∈Θ µθφθ(a)uk θ(a′) + l(a, a′) ∀k ∈ R, ∀a ∈ An, ∀a′ ̸= ak ∈ A (15b) � a∈An �� θ∈Θ µθφθ(a)uk θ(ak) + l(a, ak) � − bk ≥ � θ∈Θ µθuk θ(bk µ) ∀k ∈ R (15c) � a∈An �� θ∈Θ µθφθ(a)uk θ(ak) + l(a, ak) � − bk ≤ � θ∈Θ µθuk θ(bk µ) ∀k ̸∈ R (15d) � a∈An φθ(a) = 1 ∀θ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' (15e) Hence, once we fix bk and R, LP (15) returns a solution that has the same value of the optimal protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 35 ARXIV PREPRINT - FEBRUARY 1, 2023 To compute the optimal protocol we can iterate over all the possible prices p ∈ {bk}k∈K and all the possible subsets R ⊆ K ∩ {k ∈ K : bk ≥ p} of receivers types for which the IR constraint is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Notice that, given a price p, the IR constraint can be satisfied only the buyer’s type k ∈ K with bk ≥ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Then, we solve LP (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, we return the solution with highest value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' As we show in the first part of the proof, this solution has the same value of the optimal protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, it is easy to check that the overall procedure requires to solve O(n2n) LPs, showing that the algorithm runs in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' To conclude the proof, we need to show how to modify the solution of LP 15 to obtain a protocol, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', a solution to Problem (6), with at least the same value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' To do so, we exploit a similar approach to the one presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let (φ, l) be the solution to LP (15) returned by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Suppose that there exists a couple (¯a, ¯k) such that l(¯a, a¯k) > 0 and � θ∈Θ µθφθ(¯a) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We show how to obtain a solution such that l(¯a, a) = 0 for each a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Notice that by Constraint (15b), it holds l(¯a, ¯ak) ≥ l(¯a, a) for each k ∈ K, a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This implies that l(¯a, ¯ak) = l(¯a, ¯ak′) for each k ̸= k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We denote this value with l(¯a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Let ˆa ∈ An be any signal such that � θ∈Θ µθφθ(ˆa) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Consider a assignment (φ, l′) to the variables such that l′(¯a, a) = 0 for each a ∈ A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' l′(ˆa, a) = l(ˆa, a) + l(¯a) for each a ∈ A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' l′(a) = l(a) for each a /∈ {¯a, ˆa}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We show that this solution is feasible to LP (15) and has the same objective value of (φ, l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Indeed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' it holds � k∈R λk � a∈An �� θ∈Θ µθφθ(a)us θ(ak) − l′(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ak) � + � k /∈R λk � θ∈Θ µθus θ(bk µ) = � k∈R λk � � a∈An\\{¯a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='ˆa} �� θ∈Θ µθφθ(a)us θ(ak) − l′(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ak) � + � θ∈Θ µθφθ(¯a)us θ(¯ak) + � θ∈Θ µθφθ(ˆa)us θ(ˆak) − (l(ˆa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ˆak) − l(¯a)) � + � k /∈R λk � θ∈Θ µθus θ(bk µ) = � k∈R λk � � a∈An\\{¯a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='ˆa} �� θ∈Θ µθφθ(a)us θ(ak) − l(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ak) � + � θ∈Θ µθφθ(¯a)us θ(¯ak) − l(¯a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ¯ak) + � θ∈Θ µθφθ(ˆa)us θ(ˆak) − l(ˆa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ˆak) � + � k /∈R λk � θ∈Θ µθus θ(bk µ) = � k∈R λk � a∈An �� θ∈Θ µθφθ(a)us θ(ak) − l(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ak) � + � k /∈R λk � θ∈Θ µθus θ(bk µ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' showing that the seller’s utility does not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, Constraints (15b) relative to ¯a are satisfied since have the form 0 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' The Constraints (15b) relative to ˆa continue to be satisfied since we add a term l(¯a) on both sides of the inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Finally, all the other Constraint (15b) are unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Consider Constraint (15c) relative to a buyer’s type k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' It holds � a∈An �� θ∈Θ µθφθ(a)uk θ(ak) + l′(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ak) � − bk = � a∈An\\{¯a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='ˆa} �� θ∈Θ µθφθ(a)uk θ(ak) + l(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ak) � + � θ∈Θ µθφθ(¯a)uk θ(¯ak) + � θ∈Θ µθφθ(ˆa)uk θ(ˆak) + l(ˆa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ˆak) + l(¯a) − bk = � a∈An\\{¯a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='ˆa} �� θ∈Θ µθφθ(a)uk θ(ak) + l(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ak) � + � θ∈Θ µθφθ(¯a)uk θ(¯ak) + l(¯a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ¯ak) + � θ∈Θ µθφθ(ˆa)uk θ(ˆak) + l(ˆa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ˆak) − bk 36 ARXIV PREPRINT - FEBRUARY 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 2023 = � a∈An �� θ∈Θ µθφθ(a)uk θ(ak) + l′(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' ak) � − bk ≥ � θ∈Θ µθuk θ(bk µ) Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' we can show that Constraints (15d) continue to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Hence, iteratively applying this procedure we ob- tain a solution with the same value of the optimal protocol and such that for each tuple (a, k) if l(a, ak) > 0 and � θ∈Θ µθφθ(a) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' We can convert this solution into an optimal protocol, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=', an optimal solution to Problem (6) setting π(a, ak) = l(a,ak) � θ∈Θ µθφθ(a) for each a ∈ An such that � θ∈Θ µθφθ(a) = 0 and k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' Moreover, we set all the other payments to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' It is easy to see that the obtained protocol is a feasible optimal solution to Problem (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} +page_content=' 37' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFST4oBgHgl3EQfZDim/content/2301.13790v1.pdf'} diff --git a/49E1T4oBgHgl3EQfmQS7/vector_store/index.pkl b/49E1T4oBgHgl3EQfmQS7/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..8a718c66b498118d943fed86542ca472016941d1 --- /dev/null +++ b/49E1T4oBgHgl3EQfmQS7/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:83c50802be53408269e9fe8c7a618fe0f5dc755690384244abf0e96970ed1bc2 +size 75432 diff --git a/59E3T4oBgHgl3EQfpgot/content/tmp_files/2301.04642v1.pdf.txt b/59E3T4oBgHgl3EQfpgot/content/tmp_files/2301.04642v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..78bd62d3bb74027bde6f3425934ef132e28c2feb --- /dev/null +++ b/59E3T4oBgHgl3EQfpgot/content/tmp_files/2301.04642v1.pdf.txt @@ -0,0 +1,855 @@ +Recursive Fermi-operator expansion strategies to accelerate subspace +diagonalization for large eigenvalue problems in density functional theory +Sameer Khadatkar1 and Phani Motamarri1 +Indian Institute of Science, Bengaluru, India. +(*Electronic mail: phanim@iisc.ac.in) +Quantum mechanical calculations for material modeling using density functional theory (DFT) involves solving a +large-scale nonlinear eigenvalue problem. These calculations are computationally demanding and have asymptotic +cubic scaling complexity with the number of electrons in the material system. The efficient computational strategies +used to solve these large nonlinear DFT eigenvalue problems rely on iterative orthogonal projection methods. The +Rayleigh-Ritz projection step and the subspace diagonalization incur the dominant computational cost in these projec- +tion methods. In this work, we explore scalable polynomial expansion based on recursive Fermi-operator expansion +approaches using mixed-precision arithmetic as an alternative to subspace diagonalization of the projected Hamiltonian +to reduce the computational cost. The performance and accuracy of these approaches have been thoroughly assessed by +comparing them with the explicit diagonalization approach using the state-of-the-art ELPA library on both multinode +CPUs and GPUs. +I. +INTRODUCTION +Eigenvalue problems are frequently encountered in many +scientific disciplines. For instance, the accurate and efficient +computation of eigenvectors and eigenvalues is critical in the +study of resonance, understanding the stability of fluid flows +subjected to small perturbations, obtaining insights into vibra- +tional modes of lattices, dimensionality reduction, and many +more. +Another well-known and challenging application of +eigenvalue problems is in the area of quantum modeling of +materials using Kohn-Sham density functional theory (DFT)1, +which has been immensely successful in providing critical in- +sights into various ground-state material properties. To com- +pute the ground-state electronic structure in DFT, one is con- +fronted with solving a large-scale nonlinear eigenvalue prob- +lem using a self-consistent field iteration procedure (SCF) for +N smallest eigenvalue/eigenvector pairs, with N being pro- +portional to the number of electrons in the material system. +This results in asymptotic cubic complexity O(N3) with the +number of electrons for DFT, making these calculations com- +putationally demanding and often restrictive in terms of sys- +tem sizes that can be handled using widely used DFT codes. +Many of these codes employ plane-wave basis sets, which re- +strict simulation domains to periodic or atomic-orbital type +basis sets, which are not systematically convergent, and these +basis sets are not amenable for massive parallelization on par- +allel computing architectures. To extend the range of system +sizes to be studied, numerous past efforts have focused on de- +veloping systematically convergent real-space computational +methodologies 2–6 that have focused on reducing the prefac- +tor associated with the cubic computational complexity along- +side improving the parallel scalability, thereby enabling large- +scale DFT calculations up to 100,000 electrons. These real- +space DFT discretization approaches result in large sparse +Hermitian eigenvalue problems of the form Hψψψi = εh +i ψψψi to +be solved for N smallest eigenvalue/eigenvector pairs, with N +being proportional to M, the dimension of the sparse Hamil- +tonian matrix H (M ≈ 105 −107). We note that N depends on +the number of electrons in the material system and is usually +0.1 − 0.5% of M, the degrees of freedom (DoFs) used in the +simulation. +In the electronic structure community, the most popular +eigensolver strategies employed to solve these large DFT +eigenvalue problems include the Davidson approach, Lo- +cally Optimal Block Pre-conditioned Conjugate Gradient +(LOBPCG) method, or the Chebyshev filtered subspace it- +eration (ChFSI) approach. These eigensolvers belong to the +category of iterative orthogonal projection methods (IOP) +wherein the matrix H is orthogonally projected onto a care- +fully constructed subspace rich in the wanted eigenvectors +(Rayleigh-Ritz step), and subsequently, the resulting smaller +dense projected Hamiltonian Hp is explicitly diagonalized +(subspace diagonalization) to approximate the desired eigen- +value/eigenvector pairs of the H matrix. +The cubic scal- +ing computational cost of this subspace diagonalization step +dominates for medium to large-scale material systems (N > +20,000) in comparison to the costs associated with subspace +construction and Rayleigh-Ritz steps in IOP methods. For in- +stance, the authors6 employing the ChFSI approach have re- +ported that the subspace diagonalization constitutes roughly +30% of the total ChFSI cost for N ≈ 30,000, whereas it ac- +counts for around 56% of the total cost for N ≈ 50,000. To +this end, the current work explores recursive polynomial ex- +pansion approaches based on Fermi-operator expansion as an +alternative to the subspace diagonalization procedure to im- +prove the computational efficiency, thereby reducing the com- +putational prefactor associated with the cubic complexity of +the subspace diagonalization approach. Furthermore, the en- +ergy efficiency and parallel scaling efficiency of these ap- +proaches is examined on both multinode CPUs and multinode +GPUs. +Recursive polynomial expansion approaches (RPE) rely +on the key idea that constructing a density matrix (projec- +tor matrix corresponding to N smallest eigenvectors) suffices +to compute ground-state electronic structure in DFT at zero- +temperature without the necessity of knowing explicit eigen- +values and eigenvectors. These RPE approaches 7–12 have +been explored in the past for conducting ground-state DFT +calculations using atomic-orbital basis. However, the compu- +tational efficiency, scaling and energy efficiency of these ap- +proaches have not been explored in comparison to subspace +arXiv:2301.04642v1 [physics.comp-ph] 11 Jan 2023 + +2 +diagonalization procedures for their use in iteration orthogo- +nal projection methods on multinode CPU and GPU architec- +tures. The evolving computing architectures in today’s ex- +ascale era place a heavy emphasis on scalable methodolo- +gies with a focus on reduced data movement and increased +arithmetic intensity, with an equal emphasis on using energy- +efficient algorithms. The current work assumes significance +in this regard and is useful for solving large-scale eigenvalue +problems arising from the discretization of DFT using sys- +tematically convergent basis sets employing IOP methods. To +this end, the key contributions of our current work, as de- +scribed in the subsequent sections, include – (a) efficient im- +plementation strategies of various recursive polynomial ex- +pansion (RPE) techniques based on Fermi-operator expansion +on both multinode CPU and GPU architectures for both zero- +temperature case and the finite-temperature case of Fermi- +dirac smearing of the occupancy function (b) design of mixed +precision strategies in conjunction with RPE to reduce com- +pute and data access costs (c) assessing accuracy, scaling effi- +ciency and energy efficiency of the proposed implementation +procedures by comparing it with explicit diagonalization al- +gorithms provided by state-of-the-art ELPA library13. +II. +RELATED WORK AND BACKGROUND +This section discusses key ideas central to recursive poly- +nomial expansion approaches which are used to approximate +the density matrix. +A. +Density matrix +At zero electronic temperature, the density matrix (D) can +be defined as a projector matrix corresponding to the lowest +occupied (Nocc ≤ N) eigenvectors of the Kohn-Sham Hamil- +tonian H matrix. +Mathematically, it takes the form of a +shifted Heaviside step function, θ(.), given by D = θ[µI−H]. +The density matrix (D) in the case of finite-temperature is a +smeared version of zero-temperature density matrix and math- +ematically represented by a Fermi-operator matrix function +given by D = [eβ(H−µI) + I]−1, where, I denotes the identity +matrix, β = 1/(kBTe) is the inverse electronic temperature, µ +is the Fermi-energy, and H is the Hamiltonian matrix. Note +that the eigenvalues fi of D are referred to as occupancies. fi +is either 0 or 1 for a zero-temperature case whereas for the +case of a finite-temperature case, fi ∈ [0,1]. +B. +Recursive polynomial expansion techniques to +approximate the density matrix +Two types of polynomial expansion schemes can be used +to approximate the density matrix – (a) Serial Fermi-operator +expansion schemes (Chebyshev Fermi-operator expansion +scheme14, Green’s function expansion scheme15, etc), (b) +Recursive Fermi-operator expansion schemes 8–12. +In this +work, we employ the latter approach i.e., the recursive Fermi- +operator expansion schemes as they are shown to be more ef- +ficient and can be used to approximate the density matrix for +both zero-temperature, and finite-temperature cases as well. +1. +Recursive Fermi-operator expansion for zero-temperature +density matrix (Z-RFOE) +The recursive Fermi-operator expansion8 involves succes- +sive projections of a matrix Xn, where X0 = H and Xn+1 = +F(Xn). The functions F(Xn) are chosen to project the eigen- +value spectrum of Xn to eigenvalues closer either to 1 or to 0. +Mathematically this can be represented as +D = θ(µI−H) ≈ Fm(Fm−1(...F0(H)...)) +(1) +One of the most efficient techniques in Z-RFOE is to use the +second-order projection polynomials (SP2) 9 given by Xn+1 = +Fn(Xn) = Xn ± (Xn − X2 +n). The SP2 here are continuously +increasing and decreasing functions in the interval [0, 1]. The +± sign is chosen to adjust the trace of Xn+1 in each projection +such that it converges to Nocc. +2. +Accelerated recursive polynomial expansion for +zero-temperature density matrix (A-Z-RFOE) +This technique works on the concept of shifting and scaling. +In Z-RFOE, we used SP2 polynomials, which either took the +form F(X) = X2 or F(X) = 2X−X2. In A-Z-RFOE, instead +of restricting ourselves to these fixed expansion functions, we +give it some freedom to choose the expansion functions such +that it moves the eigenvalues closer to either 1 or 0 faster. To +optimize the convergence, we chose the polynomial such that +each iteration gives the highest slope of projection around the +eigenvalues, which are rescaled values of the HOMO (Highest +Occupied Molecular Orbital) and LUMO (Lowest Unoccu- +pied Molecular Orbital) eigenvalues and done such that there +is no risk of eigenvalues switching the places between the oc- +cupied and the unoccupied states10. +3. +Recursive Fermi-operator expansion scheme for +finite-temperature cases (T-RFOE) +Finite-temperature density matrix has occupancies fi ∈ +[0,1] and the SP2 recursion scheme discussed above is not +well suited for approximating density matrix with fractional +occupancies. +To this end, an intermediate function gener- +ated in Z-RFOE that is obtained before the convergence of +the algorithm to the projector matrix is used. This serves as a +smeared function to zero-temperature density matrix (Heavi- +side step function). To this end, the truncated expansion for +computing, the density matrix D can be given by the expres- +sion in (2). +Gm(H) = Fm(Fm−1(...F0(H)...)) +(2) +Lower the electronic temperature Te higher will be the β value +(refer to Sec. IIA), and more recursion steps m will be re- +quired to approximate the density matrix11,12. +C. +Accuracy of the polynomial expansion procedures +The accuracy of the aforementioned polynomial expansion +procedures is given in terms of the degree npl of the polyno- +mial needed to approximate the density matrix and is given +by npl ∝ (εN −ε1) with ε1,εN being spectral bound estimates + +3 +of H16. It is well known that DFT discretized Hamiltonian H +using real-space approaches has large spectral width εN − ε1 +resulting in higher npl for approximating the density matrix. +Often this leads to an inefficient computational procedure to +approximate the density matrix since the dimension of H can +be of O(105 −107). +III. +COMPUTATIONAL METHODOLOGY AND +IMPLEMENTATION +A. +Proposed methodology +Due to the aforementioned limitations of employing the re- +cursive polynomial expansion procedures on the real-space +discretized DFT Hamiltonian (H), we resort to iterative or- +thogonal projection (IOP) methods of solving a large sparse +eigenvalue problem and choose to work with the smaller dense +projected Hamiltonian Hp in the subspace rich with eigenvec- +tors of H. To this end, we employ the recursive polynomial +expansion procedures on Hp to approximate the density ma- +trix in the subspace as an alternative to explicit subspace diag- +onalization. Since the spectral width of Hp is commensurate +with spectral width corresponding to occupied eigenstates, it +is small and the proposed approach is computationally effi- +cient as demonstrated subsequently. +B. +Algorithmic details +Using Hp, Z-RFOE and A-Z-RFOE schemes employing +SP2 polynomials for approximating zero-temperature density +matrix and the T-RFOE scheme for the finite-temperature den- +sity matrix have been implemented in a distributed setting. +Figure 1 shows the schematic of the RFOE algorithm imple- +mented in the current work. +Furthermore, we also explored mixed-precision strategies +in conjunction with the RFOE schemes implemented in this +work. To this end, we rely on the fact that far away from +RFOE convergence to the appropriate density matrix, the +floating point operations involved in the initial RFOE itera- +tions can be performed in single precision (FP32) and switch- +ing to FP64 operations thereafter. The criteria to decide the +number of initial FP32 iterations is linked to relative trace +change of Xn (εtr) of two successive RFOE iterations. An es- +timate of εtr could be obtained by examining the dependence +of εtr on the relative change in occupied eigensubspace be- +tween the starting matrix X0 = Hp and the intermediate ma- +trices Xn generated during the course of RFOE. Our numerical +studies on smaller size representative Hp arising in DFT show +that εtr ≈ O(10−4) gives an acceptable error of O(10−7) with +respect to fully double precision (FP64) computation of the +density matrix. +C. +Implementation details +The multinode parallel implementation of RFOE codes was +done in C++ employing Message Passing Interface (MPI) +library. +Software for Linear Algebra Targeting Exascale +(SLATE) library17 was used for storing the parallel matrices +encountered during the course of RFOE. SLATE stores the +matrix in a 2-D block-cyclic manner on both CPUs and GPUs +within a node. The tile size is the most basic parameter that +can affect the SLATE routines’ performance. Numerical ex- +periments were conducted by varying the tile size to decide +the optimal tile size. +Some of the key aspects of the implementation are high- +lighted below: +1. +Trace Calculations +Traces of matrix squares are required during the course of +RFOE iterations and are computed by evaluating the square of +the Frobenius norm of the given symmetric matrix (Tr(A2) = +||A||2 +F). To this end, Frobenius norm function available in the +SLATE library was used. Further, the computations of matrix +traces which was required only in the beginning and end of +RFOE involved a traversal through the diagonal elements of +the global matrix and the use of an MPI collective function. +2. +Matrix-matrix multiplication +The computationally dominant step in all the RFOE algo- +rithms implemented is the matrix-matrix multiplication step. +We used the SLATE library functions to perform this step +in parallel across multinode CPUs and GPUs. The perfor- +mance of the Communication-optimal Matrix Multiplication +(COSMA)18 and cuBLASMg (made available from CUDA +Math Library Early Access Program19) library was also ex- +plored to compute parallel matrix-matrix multiplications on +CPUs and GPUs. +Our studies indicates that COSMA was +slower in terms of computational times compared to the +SLATE library. And the cuBLASMg library is restricted to +multi-GPUs within a single node. +D. +Metrics for accuracy benchmarking of RFOE +For accuracy benchmarking of the RFOE methods imple- +mented, we computed two errors: (a) Relative error between +the exact and approximated density matrix (f(H)) using the +Frobenius norm, i.e ε1 = (||D − Dref ||F)/||Dref ||F, (b) Rela- +tive error between the trace of actual and the approximated +f(H)H, ε2 = (tr(DH) − tr(Dref H))/tr(Dref H). +Dref was +computed by explicit diagonalization using ELPA library13. +IV. +RESULTS AND DISCUSSION +In order to assess the accuracy and performance of the pro- +posed methods, we employ synthetic matrices representative +of the subspace projected Hamiltonians (Hp) arising in DFT +calculations. +To this end, the matrix Hp is constructed in +such a way that the spectral width is smaller and remains con- +stant with increase in matrix size. We choose H p +ij = H p +ji = +e−d∗|i−j| ∗ sin(i + 1), and the matrix sizes used were 8192 × +8192, 16384 × 16384, 32768 × 32768, and 65536 × 65536. +The multinode CPU study was done on PARAM Pravega hav- +ing Intel Xeon Cascade Lake 8268 CPU (2.9 GHz) with 48 +cores (96 threads) on each node, while multinode GPU study +was done on a local lab cluster having 16x (8 on each node) +NVIDIA Tesla V100 GPUs with 32 GB of memory. + +4 +FIG. 1: General implementation details flowchart for all the +RFOE codes +The performance metrics used for comparisons are: +• Node-hrs ⇒ Execution time (in hours) × the number +of nodes taken in the best scaling regime. It gives a +measure of computational efficiency on CPUs. +• GPU-hrs ⇒ Execution time (in hours) × the number +of GPUs taken in the best scaling regime. It gives a +measure of computational efficiency on GPUs. +• Minimum walltime ⇒ Least possible time for the job +execution using as many resources as possible. It is a +measure of scaling efficiency of the implementation. +• Energy consumption ⇒ Upper bound of the energy re- +quired by the job in kWh to run it on the supercomputer. +Indicative of the rupee cost required for the calculations +on the supercomputer. For the energy consumption cal- +culation we used the Thermal Design Power (TDP) rat- +ings for both CPUs and GPUs. +1. +Multinode CPU comparisons +Figure 2 shows that, all our RFOE implementations for +zero-temperature case are better than ELPA in terms of node- +hrs, which indicates that all of our implementations are com- +putationally efficient compared to ELPA. For instance, the A- +Z-RFOE results in a speedup of around 2x in comparison to +ELPA for the 65536 size matrix. In the minimum walltime +regime, we find that ELPA is slightly faster than the RFOE im- +plementation for the matrix sizes considered. Figure 3 shows +(a) +(b) +FIG. 2: (a) Node-hrs vs. matrix size plot, and (b) Min. +walltime vs. matrix size plot (Note: c stands for CPUs, n +stands for nodes) for different implementations of RFOEs for +zero-temperature case on multinode CPUs. +that, both our T-RFOE and mixed-precision T-RFOE imple- +mentations for finite-temperature case are better than ELPA in +terms of node-hrs (4.2x speedup of mixed-precision T-RFOE +implementation over ELPA for the 65536 size matrix), which +indicates that both of our implementations are computation- +ally efficient compared to ELPA. And, even in the minimum +walltime, mixed precision T-RFOE was found to be slightly +better than ELPA. The number of CPUs on which we got min- +imum walltime for different matrix sizes is also shown on the +minimum walltime plots. +2. +Multinode GPU comparisons +Figure 4 shows that, all our implementations for zero- +temperature case are better than ELPA up to 16384 size ma- +trix, and beyond this size, the mixed-precision Z-RFOE and +A-Z-RFOE are better than ELPA for GPU-hrs timings, which +indicates that both of our implementations are computation- +ally efficient compared to ELPA. Our A-Z-RFOE implemen- +tation gave 1.5x speedup over ELPA for the 32768 size ma- +trix. Due to the memory issue of ELPA, it does not work +on multinode GPUs up to 16 GPUs for the 65536 size ma- +trix, which indicates that the RFOE implementations use the +memory efficiently compared to ELPA. And for minimum +walltime, ELPA shows a better behaviour, suggesting that +the RFOE implementations are not scaling well on multinode +GPUs. Figure 5 shows that, our T-RFOE and mixed-precision +T-RFOE implementations for finite-temperature case are bet- + +Initializations +Nocc : Occupancy number +So= Hp : Hamiltonian +(Defined on Distributed System) +Xo = WoSo+bol (on Parallel architectures) +Initial scaling using spectral bound estimates of H +Ns = Tr[Xo] (Using MPl_Allreduce) +While +(TrEr < Tolerance) +Final D Matrix +Xn = WXn-12 + bXn-1 +Nx = IIXn-1ll-=2 +FP32/FP64 GEMM for Xn-12 +( IIXn-1]l==[Tr[Xn-13]1/2) +Xn and Xn-1 stored in 2-D +block-cyclic manner on +CPUs/GPUs +Compute w = f(Ns, Nx, Nocc) +and b = g(Ns, Nx, Nocc) +where f() and g(.) depends on +Update Ns using Nx and Ns +the expansion scheme (Z-RFOE +TrEr = abs(Ns - Nocc) +A-Z-RFOE, T-RFOE)5 +ELPA +Double-PrecisionZ-RFOE +4 +Mixed-Precision Z-RFOE +node-hrs +Double-Precision A-Z-RFOE +m +1 +0 +8192 +16384 +32768 +65536 +Matrix Size350 +ELPA +(secs) +300 +Double-Precision Z-RFOE +250 +Mixed-PrecisionZ-RFOE +(6144c128n) +Double-Precision A-Z-RFOE +walltime +200 +150 +100 +Min. +(6144c 128n) +(6144c128n) +（12288c256n) +50 +(3072c64n) +←(12288c256n) +0 +（3072c64n） +(12288c256n) +8192 +16384 +32768 +65536 +Matrix Size5 +(a) +(b) +FIG. 3: (a) Node-hrs vs. matrix size plot, and (b) Min. +walltime vs. matrix size plot (Note: c stands for CPUs, n +stands for nodes) for different implementations of RFOEs for +finite-temperature case on multinode CPUs. +ter than ELPA for GPU-hrs timings (2.5x speedup of mixed- +precision T-RFOE implementation over ELPA for the 65536 +size matrix), indicating that both implementations are com- +putationally efficient compared to ELPA. And, for minimum +walltime, our mixed-precision T-RFOE implementation is al- +most similar to or better than ELPA. The number of GPUs on +which we got minimum walltime for different matrix sizes is +shown on the minimum walltime plot. +3. +Energy consumption comparisons +Figure 6 shows that, in both the regimes of node-hrs/GPU- +hrs and minimum walltime, we are better than ELPA in terms +of energy consumption for zero-temperature case. Figure 7 +shows that, in both the regimes of node-hrs/GPU-hrs and min- +imum walltime case, we are better than ELPA in terms of en- +ergy consumption for finite-temperature case. This indicates +that the rupee cost required for our calculations on the super- +computer will be less than ELPA for both zero-temperature +case and finite-temperature case of approximating the density +matrix. +4. +Accuracy benchmarking +The errors ε1 and ε2 (defined earlier) were of the O(10−10) +and O(10−09) for double-precision implementation of Z- +RFOE and A-Z-RFOE. And, were of the O(10−07) and +O(10−09) for mixed-precision implementation of Z-RFOE. +(a) +(b) +FIG. 4: (a) GPU-hrs vs. matrix size plot, and (b) Min. +walltime vs. matrix size plot for different implementations of +RFOEs for zero-temperature case on multinode GPUs. +(a) +(b) +FIG. 5: (a) GPU-hrs vs. matrix size plot, and (b) Min. +walltime vs. matrix size plot for different implementations of +RFOEs for finite-temperature case on multinode GPUs. + +5 +ELPA +Double-PrecisionT-RFOE +4 +Mixed-Precision T-RFOE +node-hrs +m +N +1 +0 +8192 +16384 +32768 +65536 +Matrix Size160 +ELPA +(12288c256n) +(secs) +140 +Double-Precision T-RFOE +Mixed-PrecisionT-RFOE +120 +(6144c128n) +Min. walltime +100 +80 +60 +(6144c128n) +40 +(6144c128n) +20 +(3072c 64n) +(12288c256n) +0 +(3072c64n) +(12288c256n） +8192 +16384 +32768 +65536 +Matrix Size1.75 +ELPA +1.50 +Double-Precision Z-RFOE +1.25 +Mixed-PrecisionZ-RFOE +GPU-hrs +Double-Precision A-Z-RFOE +1.00 +0.75 +0.50 +0.25 +0.00 +8192 +16384 +32768 +65536 +Matrix size700 +ELPA +(SDos) +Double-Precision Z-RFOE +600 +Mixed-Precision Z-RFOE +(8 GPUs) +Min. walltime +500 +Double-Precision A-Z-RFOE +400 +300 +200 +(8 GPUs) +100 +(4 GPUs) +(6 GPUs) +(16 GPUS) +0 +（6GPUS) +(16GPUs) +8192 +16384 +32768 +65536 +Matrix size0.8 +ELPA +0.7 +Double-Precision T-RFOE +0.6 +Mixed-PrecisionT-RFOE +GPU-hrs +0.5 +0.4 +0.3 +0.2 +0.1 +0.0 +8192 +16384 +32768 +65536 +Matrix Size350 +ELPA +(secs) +300 +Double-PrecisionT-RFOE +Mixed-Precision T-RFOE +250 +(8 GPUs) +Min. walltime +200 +150 +(8 GPUs) +100 +(6 GPUs) +50 +(4 GPUS) +(16GPUS) +0 +(6GPUs) +(16-GPUs) +8192 +16384 +32768 +65536 +Matrix size6 +(a) +(b) +FIG. 6: Energy consumption (kWh) vs. matrix size plot in +terms of (a) Node-hrs/GPU-hrs, and (b) Min. walltime for the +best implementation of RFOE for zero-temperature case. +(a) +(b) +FIG. 7: Energy consumption (kWh) vs. matrix size plot in +terms of (a) Node-hrs/GPU-hrs, and (b) Min. walltime for the +best implementation of RFOE for finite-temperature case. +For T-RFOE, the error ε1 was of the O(10−03) and ε2 was +of O(10−06). The density matrix approximated by T-RFOE +has an higher error compared to the Fermi-Dirac based den- +sity matrix. However, in DFT, the computation of material +properties relies on differences in energies and hence, the T- +RFOE approach can be viewed as an alternative approach to +smearing the zero-temperature density matrix, which can be +practically helpful in approximating finite-temperature den- +sity matrix. +V. +CONCLUSIONS +RFOE schemes, as expected, had a lesser computational +prefactor which made them computationally efficient com- +pared to ELPA in the node-hrs/GPU-hrs regime. In the case of +minimum walltimings, ELPA timings were better as it scaled +better than our RFOE implementations. Energy efficiency- +wise, the RFOE implementations were better on both multin- +ode CPUs and GPUs, which is directly proportional to the cost +required for the computations. In terms of memory utiliza- +tion, multinode GPU implementations of RFOE were better +than ELPA. 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Gisin, “High +resolution fiber distributed measurements with coherent OFDR,” in Proc. +ECOC’00 (2000) p. 109. +40cuBLAS Library, . +41cuSOLVER Library, . + diff --git a/59E3T4oBgHgl3EQfpgot/content/tmp_files/load_file.txt b/59E3T4oBgHgl3EQfpgot/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e39105de90ab999b5ea6d9ebb5cdc6c12af27027 --- /dev/null +++ b/59E3T4oBgHgl3EQfpgot/content/tmp_files/load_file.txt @@ -0,0 +1,515 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf,len=514 +page_content='Recursive Fermi-operator expansion strategies to accelerate subspace diagonalization for large eigenvalue problems in density functional theory Sameer Khadatkar1 and Phani Motamarri1 Indian Institute of Science, Bengaluru, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' (*Electronic mail: phanim@iisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='in) Quantum mechanical calculations for material modeling using density functional theory (DFT) involves solving a large-scale nonlinear eigenvalue problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' These calculations are computationally demanding and have asymptotic cubic scaling complexity with the number of electrons in the material system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' The efficient computational strategies used to solve these large nonlinear DFT eigenvalue problems rely on iterative orthogonal projection methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' The Rayleigh-Ritz projection step and the subspace diagonalization incur the dominant computational cost in these projec- tion methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' In this work, we explore scalable polynomial expansion based on recursive Fermi-operator expansion approaches using mixed-precision arithmetic as an alternative to subspace diagonalization of the projected Hamiltonian to reduce the computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' The performance and accuracy of these approaches have been thoroughly assessed by comparing them with the explicit diagonalization approach using the state-of-the-art ELPA library on both multinode CPUs and GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' INTRODUCTION Eigenvalue problems are frequently encountered in many scientific disciplines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' For instance, the accurate and efficient computation of eigenvectors and eigenvalues is critical in the study of resonance, understanding the stability of fluid flows subjected to small perturbations, obtaining insights into vibra- tional modes of lattices, dimensionality reduction, and many more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Another well-known and challenging application of eigenvalue problems is in the area of quantum modeling of materials using Kohn-Sham density functional theory (DFT)1, which has been immensely successful in providing critical in- sights into various ground-state material properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' To com- pute the ground-state electronic structure in DFT, one is con- fronted with solving a large-scale nonlinear eigenvalue prob- lem using a self-consistent field iteration procedure (SCF) for N smallest eigenvalue/eigenvector pairs, with N being pro- portional to the number of electrons in the material system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' This results in asymptotic cubic complexity O(N3) with the number of electrons for DFT, making these calculations com- putationally demanding and often restrictive in terms of sys- tem sizes that can be handled using widely used DFT codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Many of these codes employ plane-wave basis sets, which re- strict simulation domains to periodic or atomic-orbital type basis sets, which are not systematically convergent, and these basis sets are not amenable for massive parallelization on par- allel computing architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' To extend the range of system sizes to be studied, numerous past efforts have focused on de- veloping systematically convergent real-space computational methodologies 2–6 that have focused on reducing the prefac- tor associated with the cubic computational complexity along- side improving the parallel scalability, thereby enabling large- scale DFT calculations up to 100,000 electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' These real- space DFT discretization approaches result in large sparse Hermitian eigenvalue problems of the form Hψψψi = εh i ψψψi to be solved for N smallest eigenvalue/eigenvector pairs, with N being proportional to M, the dimension of the sparse Hamil- tonian matrix H (M ≈ 105 −107).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' We note that N depends on the number of electrons in the material system and is usually 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='5% of M, the degrees of freedom (DoFs) used in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' In the electronic structure community, the most popular eigensolver strategies employed to solve these large DFT eigenvalue problems include the Davidson approach, Lo- cally Optimal Block Pre-conditioned Conjugate Gradient (LOBPCG) method, or the Chebyshev filtered subspace it- eration (ChFSI) approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' These eigensolvers belong to the category of iterative orthogonal projection methods (IOP) wherein the matrix H is orthogonally projected onto a care- fully constructed subspace rich in the wanted eigenvectors (Rayleigh-Ritz step), and subsequently, the resulting smaller dense projected Hamiltonian Hp is explicitly diagonalized (subspace diagonalization) to approximate the desired eigen- value/eigenvector pairs of the H matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' The cubic scal- ing computational cost of this subspace diagonalization step dominates for medium to large-scale material systems (N > 20,000) in comparison to the costs associated with subspace construction and Rayleigh-Ritz steps in IOP methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' For in- stance, the authors6 employing the ChFSI approach have re- ported that the subspace diagonalization constitutes roughly 30% of the total ChFSI cost for N ≈ 30,000, whereas it ac- counts for around 56% of the total cost for N ≈ 50,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' To this end, the current work explores recursive polynomial ex- pansion approaches based on Fermi-operator expansion as an alternative to the subspace diagonalization procedure to im- prove the computational efficiency, thereby reducing the com- putational prefactor associated with the cubic complexity of the subspace diagonalization approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Furthermore, the en- ergy efficiency and parallel scaling efficiency of these ap- proaches is examined on both multinode CPUs and multinode GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Recursive polynomial expansion approaches (RPE) rely on the key idea that constructing a density matrix (projec- tor matrix corresponding to N smallest eigenvectors) suffices to compute ground-state electronic structure in DFT at zero- temperature without the necessity of knowing explicit eigen- values and eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' These RPE approaches 7–12 have been explored in the past for conducting ground-state DFT calculations using atomic-orbital basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' However, the compu- tational efficiency, scaling and energy efficiency of these ap- proaches have not been explored in comparison to subspace arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='04642v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='comp-ph] 11 Jan 2023 2 diagonalization procedures for their use in iteration orthogo- nal projection methods on multinode CPU and GPU architec- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' The evolving computing architectures in today’s ex- ascale era place a heavy emphasis on scalable methodolo- gies with a focus on reduced data movement and increased arithmetic intensity, with an equal emphasis on using energy- efficient algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' The current work assumes significance in this regard and is useful for solving large-scale eigenvalue problems arising from the discretization of DFT using sys- tematically convergent basis sets employing IOP methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' To this end,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' the key contributions of our current work,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' as de- scribed in the subsequent sections,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' include – (a) efficient im- plementation strategies of various recursive polynomial ex- pansion (RPE) techniques based on Fermi-operator expansion on both multinode CPU and GPU architectures for both zero- temperature case and the finite-temperature case of Fermi- dirac smearing of the occupancy function (b) design of mixed precision strategies in conjunction with RPE to reduce com- pute and data access costs (c) assessing accuracy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' scaling effi- ciency and energy efficiency of the proposed implementation procedures by comparing it with explicit diagonalization al- gorithms provided by state-of-the-art ELPA library13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' RELATED WORK AND BACKGROUND This section discusses key ideas central to recursive poly- nomial expansion approaches which are used to approximate the density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Density matrix At zero electronic temperature, the density matrix (D) can be defined as a projector matrix corresponding to the lowest occupied (Nocc ≤ N) eigenvectors of the Kohn-Sham Hamil- tonian H matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Mathematically, it takes the form of a shifted Heaviside step function, θ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' ), given by D = θ[µI−H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' The density matrix (D) in the case of finite-temperature is a smeared version of zero-temperature density matrix and math- ematically represented by a Fermi-operator matrix function given by D = [eβ(H−µI) + I]−1, where, I denotes the identity matrix, β = 1/(kBTe) is the inverse electronic temperature, µ is the Fermi-energy, and H is the Hamiltonian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Note that the eigenvalues fi of D are referred to as occupancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' fi is either 0 or 1 for a zero-temperature case whereas for the case of a finite-temperature case, fi ∈ [0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Recursive polynomial expansion techniques to approximate the density matrix Two types of polynomial expansion schemes can be used to approximate the density matrix – (a) Serial Fermi-operator expansion schemes (Chebyshev Fermi-operator expansion scheme14, Green’s function expansion scheme15, etc), (b) Recursive Fermi-operator expansion schemes 8–12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' In this work, we employ the latter approach i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=', the recursive Fermi- operator expansion schemes as they are shown to be more ef- ficient and can be used to approximate the density matrix for both zero-temperature, and finite-temperature cases as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Recursive Fermi-operator expansion for zero-temperature density matrix (Z-RFOE) The recursive Fermi-operator expansion8 involves succes- sive projections of a matrix Xn, where X0 = H and Xn+1 = F(Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' The functions F(Xn) are chosen to project the eigen- value spectrum of Xn to eigenvalues closer either to 1 or to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Mathematically this can be represented as D = θ(µI−H) ≈ Fm(Fm−1(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='F0(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=')) (1) One of the most efficient techniques in Z-RFOE is to use the second-order projection polynomials (SP2) 9 given by Xn+1 = Fn(Xn) = Xn ± (Xn − X2 n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' The SP2 here are continuously increasing and decreasing functions in the interval [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' The ± sign is chosen to adjust the trace of Xn+1 in each projection such that it converges to Nocc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Accelerated recursive polynomial expansion for zero-temperature density matrix (A-Z-RFOE) This technique works on the concept of shifting and scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' In Z-RFOE, we used SP2 polynomials, which either took the form F(X) = X2 or F(X) = 2X−X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' In A-Z-RFOE, instead of restricting ourselves to these fixed expansion functions, we give it some freedom to choose the expansion functions such that it moves the eigenvalues closer to either 1 or 0 faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' To optimize the convergence, we chose the polynomial such that each iteration gives the highest slope of projection around the eigenvalues, which are rescaled values of the HOMO (Highest Occupied Molecular Orbital) and LUMO (Lowest Unoccu- pied Molecular Orbital) eigenvalues and done such that there is no risk of eigenvalues switching the places between the oc- cupied and the unoccupied states10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Recursive Fermi-operator expansion scheme for finite-temperature cases (T-RFOE) Finite-temperature density matrix has occupancies fi ∈ [0,1] and the SP2 recursion scheme discussed above is not well suited for approximating density matrix with fractional occupancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' To this end, an intermediate function gener- ated in Z-RFOE that is obtained before the convergence of the algorithm to the projector matrix is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' This serves as a smeared function to zero-temperature density matrix (Heavi- side step function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' To this end, the truncated expansion for computing, the density matrix D can be given by the expres- sion in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Gm(H) = Fm(Fm−1(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='F0(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=')) (2) Lower the electronic temperature Te higher will be the β value (refer to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' IIA), and more recursion steps m will be re- quired to approximate the density matrix11,12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Accuracy of the polynomial expansion procedures The accuracy of the aforementioned polynomial expansion procedures is given in terms of the degree npl of the polyno- mial needed to approximate the density matrix and is given by npl ∝ (εN −ε1) with ε1,εN being spectral bound estimates 3 of H16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' It is well known that DFT discretized Hamiltonian H using real-space approaches has large spectral width εN − ε1 resulting in higher npl for approximating the density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Often this leads to an inefficient computational procedure to approximate the density matrix since the dimension of H can be of O(105 −107).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' COMPUTATIONAL METHODOLOGY AND IMPLEMENTATION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Proposed methodology Due to the aforementioned limitations of employing the re- cursive polynomial expansion procedures on the real-space discretized DFT Hamiltonian (H), we resort to iterative or- thogonal projection (IOP) methods of solving a large sparse eigenvalue problem and choose to work with the smaller dense projected Hamiltonian Hp in the subspace rich with eigenvec- tors of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' To this end, we employ the recursive polynomial expansion procedures on Hp to approximate the density ma- trix in the subspace as an alternative to explicit subspace diag- onalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Since the spectral width of Hp is commensurate with spectral width corresponding to occupied eigenstates, it is small and the proposed approach is computationally effi- cient as demonstrated subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Algorithmic details Using Hp, Z-RFOE and A-Z-RFOE schemes employing SP2 polynomials for approximating zero-temperature density matrix and the T-RFOE scheme for the finite-temperature den- sity matrix have been implemented in a distributed setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Figure 1 shows the schematic of the RFOE algorithm imple- mented in the current work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Furthermore, we also explored mixed-precision strategies in conjunction with the RFOE schemes implemented in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' To this end, we rely on the fact that far away from RFOE convergence to the appropriate density matrix, the floating point operations involved in the initial RFOE itera- tions can be performed in single precision (FP32) and switch- ing to FP64 operations thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' The criteria to decide the number of initial FP32 iterations is linked to relative trace change of Xn (εtr) of two successive RFOE iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' An es- timate of εtr could be obtained by examining the dependence of εtr on the relative change in occupied eigensubspace be- tween the starting matrix X0 = Hp and the intermediate ma- trices Xn generated during the course of RFOE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Our numerical studies on smaller size representative Hp arising in DFT show that εtr ≈ O(10−4) gives an acceptable error of O(10−7) with respect to fully double precision (FP64) computation of the density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Implementation details The multinode parallel implementation of RFOE codes was done in C++ employing Message Passing Interface (MPI) library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Software for Linear Algebra Targeting Exascale (SLATE) library17 was used for storing the parallel matrices encountered during the course of RFOE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' SLATE stores the matrix in a 2-D block-cyclic manner on both CPUs and GPUs within a node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' The tile size is the most basic parameter that can affect the SLATE routines’ performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Numerical ex- periments were conducted by varying the tile size to decide the optimal tile size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Some of the key aspects of the implementation are high- lighted below: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Trace Calculations Traces of matrix squares are required during the course of RFOE iterations and are computed by evaluating the square of the Frobenius norm of the given symmetric matrix (Tr(A2) = ||A||2 F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' To this end, Frobenius norm function available in the SLATE library was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Further, the computations of matrix traces which was required only in the beginning and end of RFOE involved a traversal through the diagonal elements of the global matrix and the use of an MPI collective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Matrix-matrix multiplication The computationally dominant step in all the RFOE algo- rithms implemented is the matrix-matrix multiplication step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' We used the SLATE library functions to perform this step in parallel across multinode CPUs and GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' The perfor- mance of the Communication-optimal Matrix Multiplication (COSMA)18 and cuBLASMg (made available from CUDA Math Library Early Access Program19) library was also ex- plored to compute parallel matrix-matrix multiplications on CPUs and GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Our studies indicates that COSMA was slower in terms of computational times compared to the SLATE library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' And the cuBLASMg library is restricted to multi-GPUs within a single node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Metrics for accuracy benchmarking of RFOE For accuracy benchmarking of the RFOE methods imple- mented, we computed two errors: (a) Relative error between the exact and approximated density matrix (f(H)) using the Frobenius norm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='e ε1 = (||D − Dref ||F)/||Dref ||F, (b) Rela- tive error between the trace of actual and the approximated f(H)H, ε2 = (tr(DH) − tr(Dref H))/tr(Dref H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Dref was computed by explicit diagonalization using ELPA library13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' RESULTS AND DISCUSSION In order to assess the accuracy and performance of the pro- posed methods, we employ synthetic matrices representative of the subspace projected Hamiltonians (Hp) arising in DFT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' To this end, the matrix Hp is constructed in such a way that the spectral width is smaller and remains con- stant with increase in matrix size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' We choose H p ij = H p ji = e−d∗|i−j| ∗ sin(i + 1), and the matrix sizes used were 8192 × 8192, 16384 × 16384, 32768 × 32768, and 65536 × 65536.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' The multinode CPU study was done on PARAM Pravega hav- ing Intel Xeon Cascade Lake 8268 CPU (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='9 GHz) with 48 cores (96 threads) on each node, while multinode GPU study was done on a local lab cluster having 16x (8 on each node) NVIDIA Tesla V100 GPUs with 32 GB of memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' 1: General implementation details flowchart for all the RFOE codes The performance metrics used for comparisons are: Node-hrs ⇒ Execution time (in hours) × the number of nodes taken in the best scaling regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' It gives a measure of computational efficiency on CPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' GPU-hrs ⇒ Execution time (in hours) × the number of GPUs taken in the best scaling regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' It gives a measure of computational efficiency on GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Minimum walltime ⇒ Least possible time for the job execution using as many resources as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' It is a measure of scaling efficiency of the implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Energy consumption ⇒ Upper bound of the energy re- quired by the job in kWh to run it on the supercomputer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Indicative of the rupee cost required for the calculations on the supercomputer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' For the energy consumption cal- culation we used the Thermal Design Power (TDP) rat- ings for both CPUs and GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Multinode CPU comparisons Figure 2 shows that, all our RFOE implementations for zero-temperature case are better than ELPA in terms of node- hrs, which indicates that all of our implementations are com- putationally efficient compared to ELPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' For instance, the A- Z-RFOE results in a speedup of around 2x in comparison to ELPA for the 65536 size matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' In the minimum walltime regime, we find that ELPA is slightly faster than the RFOE im- plementation for the matrix sizes considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Figure 3 shows (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' 2: (a) Node-hrs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' matrix size plot, and (b) Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' walltime vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' matrix size plot (Note: c stands for CPUs, n stands for nodes) for different implementations of RFOEs for zero-temperature case on multinode CPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' that, both our T-RFOE and mixed-precision T-RFOE imple- mentations for finite-temperature case are better than ELPA in terms of node-hrs (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='2x speedup of mixed-precision T-RFOE implementation over ELPA for the 65536 size matrix), which indicates that both of our implementations are computation- ally efficient compared to ELPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' And, even in the minimum walltime, mixed precision T-RFOE was found to be slightly better than ELPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' The number of CPUs on which we got min- imum walltime for different matrix sizes is also shown on the minimum walltime plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Multinode GPU comparisons Figure 4 shows that, all our implementations for zero- temperature case are better than ELPA up to 16384 size ma- trix, and beyond this size, the mixed-precision Z-RFOE and A-Z-RFOE are better than ELPA for GPU-hrs timings, which indicates that both of our implementations are computation- ally efficient compared to ELPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Our A-Z-RFOE implemen- tation gave 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='5x speedup over ELPA for the 32768 size ma- trix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Due to the memory issue of ELPA, it does not work on multinode GPUs up to 16 GPUs for the 65536 size ma- trix, which indicates that the RFOE implementations use the memory efficiently compared to ELPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' And for minimum walltime, ELPA shows a better behaviour, suggesting that the RFOE implementations are not scaling well on multinode GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Figure 5 shows that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' our T-RFOE and mixed-precision ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='T-RFOE implementations for finite-temperature case are bet- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='Initializations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='Nocc : Occupancy number ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='So= Hp : Hamiltonian ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='(Defined on Distributed System) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='Xo = WoSo+bol (on Parallel architectures) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='Initial scaling using spectral bound estimates of H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='Ns = Tr[Xo] (Using MPl_Allreduce) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='While ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='(TrEr < Tolerance) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='Final D Matrix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='Xn = WXn-12 + bXn-1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='Nx = IIXn-1ll-=2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='FP32/FP64 GEMM for Xn-12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='( IIXn-1]l==[Tr[Xn-13]1/2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='Xn and Xn-1 stored in 2-D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='block-cyclic manner on ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='CPUs/GPUs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='Compute w = f(Ns,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Nx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Nocc) and b = g(Ns,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Nx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Nocc) where f() and g(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=') depends on Update Ns using Nx and Ns the expansion scheme (Z-RFOE TrEr = abs(Ns - Nocc) A-Z-RFOE, T-RFOE)5 ELPA Double-PrecisionZ-RFOE 4 Mixed-Precision Z-RFOE node-hrs Double-Precision A-Z-RFOE m 1 0 8192 16384 32768 65536 Matrix Size350 ELPA (secs) 300 Double-Precision Z-RFOE 250 Mixed-PrecisionZ-RFOE (6144c128n) Double-Precision A-Z-RFOE walltime 200 150 100 Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' (6144c 128n) (6144c128n) （12288c256n) 50 (3072c64n) ←(12288c256n) 0 （3072c64n） (12288c256n) 8192 16384 32768 65536 Matrix Size5 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' 3: (a) Node-hrs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' matrix size plot, and (b) Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' walltime vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' matrix size plot (Note: c stands for CPUs, n stands for nodes) for different implementations of RFOEs for finite-temperature case on multinode CPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' ter than ELPA for GPU-hrs timings (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='5x speedup of mixed- precision T-RFOE implementation over ELPA for the 65536 size matrix), indicating that both implementations are com- putationally efficient compared to ELPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' And, for minimum walltime, our mixed-precision T-RFOE implementation is al- most similar to or better than ELPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' The number of GPUs on which we got minimum walltime for different matrix sizes is shown on the minimum walltime plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Energy consumption comparisons Figure 6 shows that, in both the regimes of node-hrs/GPU- hrs and minimum walltime, we are better than ELPA in terms of energy consumption for zero-temperature case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Figure 7 shows that, in both the regimes of node-hrs/GPU-hrs and min- imum walltime case, we are better than ELPA in terms of en- ergy consumption for finite-temperature case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' This indicates that the rupee cost required for our calculations on the super- computer will be less than ELPA for both zero-temperature case and finite-temperature case of approximating the density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Accuracy benchmarking The errors ε1 and ε2 (defined earlier) were of the O(10−10) and O(10−09) for double-precision implementation of Z- RFOE and A-Z-RFOE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' And, were of the O(10−07) and O(10−09) for mixed-precision implementation of Z-RFOE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' 4: (a) GPU-hrs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' matrix size plot, and (b) Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' walltime vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' matrix size plot for different implementations of RFOEs for zero-temperature case on multinode GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' 5: (a) GPU-hrs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' matrix size plot, and (b) Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' walltime vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' matrix size plot for different implementations of RFOEs for finite-temperature case on multinode GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' 5 ELPA Double-PrecisionT-RFOE 4 Mixed-Precision T-RFOE node-hrs m N 1 0 8192 16384 32768 65536 Matrix Size160 ELPA (12288c256n) (secs) 140 Double-Precision T-RFOE Mixed-PrecisionT-RFOE 120 (6144c128n) Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' walltime 100 80 60 (6144c128n) 40 (6144c128n) 20 (3072c 64n) (12288c256n) 0 (3072c64n) (12288c256n） 8192 16384 32768 65536 Matrix Size1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='75 ELPA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='50 Double-Precision Z-RFOE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='25 Mixed-PrecisionZ-RFOE GPU-hrs Double-Precision A-Z-RFOE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='00 8192 16384 32768 65536 Matrix size700 ELPA (SDos) Double-Precision Z-RFOE 600 Mixed-Precision Z-RFOE (8 GPUs) Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' walltime 500 Double-Precision A-Z-RFOE 400 300 200 (8 GPUs) 100 (4 GPUs) (6 GPUs) (16 GPUS) 0 （6GPUS) (16GPUs) 8192 16384 32768 65536 Matrix size0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='8 ELPA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='7 Double-Precision T-RFOE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='6 Mixed-PrecisionT-RFOE GPU-hrs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content='0 8192 16384 32768 65536 Matrix Size350 ELPA (secs) 300 Double-PrecisionT-RFOE Mixed-Precision T-RFOE 250 (8 GPUs) Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' walltime 200 150 (8 GPUs) 100 (6 GPUs) 50 (4 GPUS) (16GPUS) 0 (6GPUs) (16-GPUs) 8192 16384 32768 65536 Matrix size6 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' 6: Energy consumption (kWh) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' matrix size plot in terms of (a) Node-hrs/GPU-hrs, and (b) Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' walltime for the best implementation of RFOE for zero-temperature case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' 7: Energy consumption (kWh) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' matrix size plot in terms of (a) Node-hrs/GPU-hrs, and (b) Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' walltime for the best implementation of RFOE for finite-temperature case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' For T-RFOE, the error ε1 was of the O(10−03) and ε2 was of O(10−06).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' The density matrix approximated by T-RFOE has an higher error compared to the Fermi-Dirac based den- sity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' However, in DFT, the computation of material properties relies on differences in energies and hence, the T- RFOE approach can be viewed as an alternative approach to smearing the zero-temperature density matrix, which can be practically helpful in approximating finite-temperature den- sity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' CONCLUSIONS RFOE schemes, as expected, had a lesser computational prefactor which made them computationally efficient com- pared to ELPA in the node-hrs/GPU-hrs regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' In the case of minimum walltimings, ELPA timings were better as it scaled better than our RFOE implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Energy efficiency- wise, the RFOE implementations were better on both multin- ode CPUs and GPUs, which is directly proportional to the cost required for the computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' In terms of memory utiliza- tion, multinode GPU implementations of RFOE were better than ELPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' From all the observations we can conclude that, these techniques can be used whenever we have fewer com- putational resources and have cost constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' 1W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Kohn and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' Sham, “Self-consistent equations including exchange and correlation 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' 40cuBLAS Library, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} +page_content=' 41cuSOLVER Library, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E3T4oBgHgl3EQfpgot/content/2301.04642v1.pdf'} diff --git a/5NE1T4oBgHgl3EQfBAIU/content/tmp_files/2301.02845v1.pdf.txt b/5NE1T4oBgHgl3EQfBAIU/content/tmp_files/2301.02845v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..cb86b17e6f97e8186251221e562d1d6e0c3d5af8 --- /dev/null +++ b/5NE1T4oBgHgl3EQfBAIU/content/tmp_files/2301.02845v1.pdf.txt @@ -0,0 +1,1784 @@ +A one-dimensional model for axisymmetric deformations of an +inflated hyperelastic tube of finite wall thickness +Xiang Yua,˚, Yibin Fub +aSchool of Computer Science and Technology, Dongguan University of Technology, Dongguan, China +bSchool of Computing and Mathematics, Keele University, Staffordshire ST5 5BG, UK +Abstract +We derive a one-dimensional (1d) model for the analysis of bulging or necking in an inflated hyper- +elastic tube of finite wall thickness from the three-dimensional finite elasticity theory by applying +the dimension reduction methodology proposed by Audoly and Hutchinson (J. Mech. Phys. Solids, +97, 2016). The 1d model makes it much easier to characterize fully nonlinear axisymmetric defor- +mations of a thick-walled tube using simple numerical schemes such as the finite difference method. +The new model recovers the diffuse interface model for analyzing bulging in a membrane tube and +the 1d model for investigating necking in a stretched solid cylinder as two limiting cases. It is con- +sistent with, but significantly refines, the exact linear and weakly nonlinear bifurcation analyses. +Comparisons with finite element simulations show that for the bulging problem, the 1d model is +capable of describing the entire bulging process accurately, from initiation, growth, to propagation. +The 1d model provides a stepping stone from which similar 1d models can be derived and used to +study other effects such as anisotropy and electric loading, and other phenomena such as rupture. +Keywords: +localized bulging; necking; reduced models; tubes; stability; nonlinear elasticity +1. Introduction +Hyperelastic tubes are commonly found in various applications ranging from soft robotics (Ma +et al., 2015; Lu et al., 2015, 2020) to energy harvesting (Lu & Suo, 2012; Bucchi & Hearn, 2013; +Smith, 2016). They are also used to model human arteries in order to understand pathological +conditions such as aneurysms (Fu et al., 2012; Alhayani et al., 2014; Demirkoparan & Merodio, +2017; Varatharajan & DasGupta, 2017; Hejazi et al., 2021). Inflation of a hyperelastic tube is one +of the few boundary value problems in nonlinear elasticity that have closed-form solutions, and it +provides the simplest setup to explain bifurcation, localization, loss of convexity, and “two-phase” +deformations. Thus, understanding this problem is not only important for applications, but may +also shed light on other more complicated stability and bifurcation problems. +˚Corresponding author +Email address: yuxiang@dgut.edu.cn (Xiang Yu) +Preprint submitted to Journal of the Mechanics and Physics of Solids +January 10, 2023 +arXiv:2301.02845v1 [cond-mat.soft] 7 Jan 2023 + +Simple inflation experiments with commercially available latex rubber tubes show that localized +bulging is the dominant deformation form. For almost all realistic constitutive models for rubber, +the pressure versus volume curve has an N shape under the condition of fixed resultant axial force +(Green & Adkins, 1960). This led Yin (1977) and Chater & Hutchinson (1984) to analyze the final +observable configuration as that corresponding to a “two-phase” deformation. +The subsequent +experimental studies carried out by Kyriakides & Chang (1990, 1991), Pamplona et al. (2006) and +Goncalves et al. (2008) have provided a clear picture on how a localized bulge initiates, grows and +then propagates under fixed axial force or fixed-ends conditions. +When the membrane assumption is made, the governing equations for tube inflation can be +viewed as a finite-dimensional spatial dynamical system that has two conservation laws/integrals +(Pipkin, 1968). This realization enabled Fu et al. (2008) to demonstrate explicitly how a localized +solution initiates as a zero-wave-number mode from the uniform deformation and how it evolves +into a “two-phase” state. The stability of bulging solutions and their sensitivity to imperfections +have been studied under the same framework (Pearce & Fu, 2010; Fu & Il’ichev, 2015; Fu & Xie, +2010). Fresh analytical insight into the case of fixed ends has also been obtained. It is shown +that the bifurcation condition for this case corresponds to the axial force reaching a maximum at +a fixed pressure (Fu & Il’ichev, 2015); in other words, as pressure is increased, the critical pressure +is the value of pressure at which the axial force reaches a maximum when viewed as a function of +the axial stretch. Also, in contrast with the case of fixed axial force where the measured pressure +approaches a constant value (the propagation pressure), the measured pressure in the case of fixed +ends has an N shape where the right branch approaches a master curve that is independent of the +pre-axial-stretch or the tube length (Guo et al., 2022). +In some practical applications, however, the tube wall may be of moderate or even large thickness +and the membrane model no longer applies. For example, in the context of aneurysm formation, +a human artery can be as thick as a quarter of its outer radius (M¨uller et al., 2008), and fiber- +reinforcement also seems to reduce the range of validity of the membrane assumption (Wang & +Fu, 2018). Thus, recent studies have begun to consider hyperelastic tubes of finite wall thickness. +Fu et al. (2016) showed that the associated bifurcation condition for localized bulging corresponds +to the vanishing of the Jacobian determinant of the internal pressure and resultant axial force as +functions of the azimuthal stretch and the axial stretch; see also Yu & Fu (2022) for an alternative +derivation. +This provides a framework under which additional effects such as rotation (Wang +et al., 2017), double-fiber-reinforcement (Wang & Fu, 2018), bi-laying (Liu et al., 2019; Ye et al., +2019), torsion (Althobaiti, 2022), and surface tension (Emery & Fu, 2021a,b,c; Emery, 2023) can +be assessed in a systematic manner. Ye et al. (2020) conducted a weakly non-linear analysis and +derived the bulging solution explicitly. The analytic predictions were corroborated by numerical +simulations and experiments (Wang et al., 2019). +For tubes of finite wall thickness, the equations that govern their axisymmetric deformations +are coupled nonlinear partial differential equations. Although analytic solutions can be obtained in +2 + +the near-critical regime using asymptotic methods (Ye et al., 2020), the complexity of the governing +equations forbids any further analytic attempts to understand the bulging evolution further away +from the bifurcation point. The post-bifurcation behavior in the fully nonlinear regime has so far +only been investigated by resorting to Abaqus simulations (Wang et al., 2019; Lin et al., 2020). +This is not satisfactory since the insight provided by full-scale simulations tends to be limited and +there are situations where repeated calculations of the bulging profile are required (e.g. in the +assessment of the rupture potential (Hejazi et al., 2021)). +A recent series of studies by Audoly and coworkers has opened the possibility that a 1d reduced +model can be derived to describe the fully nonlinear evolution of bulging or necking. In the first +of this series, Audoly & Hutchinson (2016), the authors derived a 1d model for tensile necking +localization in a 3d prismatic solid of arbitrary cross-section. The key idea of their derivation is a +dimension reduction assuming slow variation in the axial direction that respects self-consistency. +In terms of the language of perturbation analysis, the leading-order solution is almost correct and +higher-order terms are only added to restore self-consistency. The method was later applied by +Lestrigant and Audoly to obtain a diffuse interface model for the characterization of propagating +bulges in membrane tubes (Lestringant & Audoly, 2018) and a 1d model for predicting surface +tension-driven necking in soft elastic cylinders (Lestringant & Audoly, 2020b). It has also been +used recently to derive a 1d model for elastic ribbons (Audoly & Neukirch, 2021) and for tape springs +(Kumar et al., 2022). The systematic reduction method for deriving 1d strain-gradient models for +nonlinear slender structures was further generalized by Lestringant & Audoly (2020a). It is worth +pointing out that although the 1d models are built on the assumption that localized solutions +vary slowly in the longitudinal direction, it is surprisingly accurate, even in the region where the +localization is well developed. This is illustrated by the numeric examples in the aforementioned +work and in the comparative studies by Wang & Fu (2021) and Fu et al. (2021). +This work aims to extend the diffuse interface model of Lestringant & Audoly (2018) for mem- +brane tubes to tubes of finite wall thickness, in a similar spirit as the previous work Fu et al. (2016) +and Ye et al. (2020) that extend the bifurcation condition and the weakly nonlinear analysis from +membrane tubes to thick-walled tubes. In contrast with the case under the membrane assumption +where the original governing equations are already one-dimensional, the governing equations for the +current case are two-dimensional, and the uniformly inflated deformation is no longer homogeneous +since the solution depends on the radial variable. It will be shown that a 1d reduced model can +still be derived and simplified to the form +E1dras “ +ż L +´L +´ +Gpa, λpaqq ` 1 +2Dpaqa1pZq2¯ +dZ ` Cpaqa1pZq|L +´L, +(1.1) +where L is the initial half length, Z is the axial coordinate, apZq is the azimuthal stretch on the inner +surface (a constant multiple of the deformed inner radius ) and the expressions for Gpa, λpaqq, Dpaq +and Cpaq are given in (3.10), (4.21) and (4.22), respectively. The first term G in (1.1) corresponds +to the energy of the uniform deformation, which determines the amplitudes of the two phases in +3 + +the bulge propagation stage; the second term accounts for the contribution of the strain gradient +to the total energy, which describes how these two phases are connected. +The Euler-Lagrange +equation associated with the energy functional (1.1) is a second-order nonlinear ode for apZq, +which is a drastic simplification from the original nonlinear partial differential equations. This 1d +model is validated by comparison with finite element simulations, showing excellent agreement with +numerical results even for the propagation stage. +The outline of this paper is as follows. In Sect. 2, we formulate the 3d axisymmetric finite- +strain model for a tube of finite wall thickness under inflation and axial stretching. In Sect. 3, we +summarize solutions corresponding to uniform inflation of the tube, making preparation for the +subsequent dimension reduction. In Sect. 4, we carry out the dimension reduction and derive the +above-mentioned 1d strain-gradient model. The connection of the 1d model with prior work is +given in Sect. 5. In Sect. 6, we validate the 1d model by making comparisons with finite element +simulations. Finally, concluding remarks are given in Sect. 7. +2. Three-dimensional finite-strain model +We consider a circular cylindrical tube that has a length 2L, inner radius A and outer radius B +in its reference configuration; see Fig. 1(a). The ratio of the outer radius to the length ε “ B{2L is +assumed to be small, thus ε ! 1. The tube deforms axisymmetrically under the combined action of +an internal pressure P and a resultant axial force N, as shown in Fig. 1(b). In terms of cylindrical +coordinates, the current position vector of a representative point is given by +x “ zpZ, Rqez ` rpZ, Rqer, +(2.1) +where pR, Θ, Zq and pr, θ, zq are the coordinates of a representative point before and after defor- +mation, and per, eθ, ezq are the standard basis vectors associated with both pR, Θ, Zq and pr, θ, zq. +The deformation gradient related to (2.1) is given by +F “ r +Reθ b eθ ` zZez b ez ` zRez b er ` rZer b ez ` rRer b er, +(2.2) +where zZ :“ Bz{BZ, zR :“ Bz{BR, etc. +We assume that the tube is made of an incompressible isotropic hyperelastic material, associated +with the strain energy function Wpλ1, λ2, λ3q, where λ1, λ2, λ3 denote the three principal stretches. +Throughout this paper, we identify the indices 1, 2, 3 such that in uniform inflation they coincide +with the θ-, z- and r-directions, respectively. +The total potential energy of the tube is composed of the elastic energy and the load potential, +which reads +E “ +ż L +´L +´ ż B +A +` +wpλ1, λ2q ´ N˚zZ +˘ +2πR dR ´ Pπr2zZ +ˇˇ +R“A +¯ +dZ, +(2.3) +4 + +m +B +G +(a) +m2 +G2 +H2 +(b) +Figure 1: A hyperelastic cylindrical tube of finite thickness in (a) reference (undeformed) configuration and (b) +current configuration. +where wpλ1, λ2q “ Wpλ1, λ2, λ´1 +1 λ´1 +2 q is the reduced strain energy function and N˚ “ N{pπpB2 ´ +A2qq is the resultant axial force per unit cross-sectional area. The elastic model governed by the +energy functional (2.3) will be used as a starting point for the subsequent dimension reduction. +The governing equations for the two unknown functions rpZ, Rq and zpZ, Rq can be derived by +setting the first variation of E to zero, but these equations are not required in the approach that +we follow. +3. Uniform inflation +We now summarise the solution that corresponds to uniform inflation and extension of the tube. +This solution will be referred to as the uniform solution and is indicated by a superposed bar. For +a more detailed derivation, see Haughton & Ogden (1979). +First, incompressibility implies that a uniform solution must necessarily be of the form +¯z “ λZ, +¯r “ +a +a2A2 ` λ´1pR2 ´ A2q, +(3.1) +where λ and a denote the constant axial stretch and azimuthal stretch on the inner surface, respec- +tively. The three principal stretches are simply +¯λ1 “ ¯r +R, +¯λ2 “ λ, +¯λ3 “ d¯r +dR “ ¯λ´1 +1 ¯λ´1 +2 , +(3.2) +and the azimuthal stretch on the outer surface, denoted by b, is given by +b “ ¯λ1|R“B “ +a +a2A2 ` λ´1pB2 ´ A2q +B +. +(3.3) +The three associated principal Cauchy stresses ¯σ11, ¯σ22 and ¯σ33 satisfy the relations +¯σ11 ´ ¯σ33 “ ¯λ1w1, +¯σ22 ´ ¯σ33 “ λw2, +(3.4) +5 + +where w1 “ Bwp¯λ1, ¯λ2q{B¯λ1 and w2 “ Bwp¯λ1, ¯λ2q{B¯λ2. +The only equilibrium equation that is not satisfied automatically is +d¯σ33 +d¯r +“ ¯σ11 ´ ¯σ33 +¯r +“ +¯λ1w1 +¯r +. +(3.5) +On integrating this equation from R “ A to R “ B and making use of the boundary conditions +that ¯σ33|R“A “ ´P and ¯σ33|R“B “ 0, we obtain +P “ Qpa, λq :“ +ż a +b +w1p¯λ1, λq +¯λ2 +1λ ´ 1 d¯λ1, +(3.6) +where the second equation defines the function Qpa, λq and we have made use of the identity +d¯r +¯r “ ´ +d¯λ1 +¯λ1p¯λ2 +1λ ´ 1q, +(3.7) +which can be deduced from (3.1)2. +The overall equilibrium in the axial direction implies +Mpa, λq ´ 1 +2a2P ´ +N +2πA2 “ 0, +(3.8) +where Mpa, λq is given by +Mpa, λq :“ 1 +A2 +ż B +A +λ´1¯σ22R dR “ +ż a +b +p¯λ2 +1 ´ a2qw1p¯λ1, λq ` 2¯λ1λpa2λ ´ 1qw2p¯λ1, λq +2p¯λ2 +1λ ´ 1q2 +d¯λ1. +(3.9) +In view of (2.3), the total potential energy of the uniform deformation (3.1) per unit reference +length, after scaling by 2π, is +Gpa, λq “ +ż B +A +wp¯λ1, λq R dR ´ 1 +2PA2a2λ ´ N +2πλ. +(3.10) +The equilibrium equations (3.6) and (3.8) can also be obtained from BG{Ba “ 0 and BG{Bλ “ 0, +respectively. Once the loads P and N are specified, the deformation parameters a and λ can be +found by solving the equilibrium equations (3.6) and (3.8). +On differentiating the left-hand side of (3.8) with respect to λ, we find that it takes the form +Hw22pa, λq{A ` OpH2q, where H “ B ´ A is the thickness of the tube. We assume that the strong +ellipticity condition is satisfied pointwise which guarantees that w22pa, λq is positive (Knowles & +Sternberg, 1976). This, combined with the implicit function theorem, implies that (3.8) can be +inverted to express λ in terms of a uniquely at least when H is small. We assume that this remains +true for arbitrary H. This enables us to view (3.8) as an implicit equation for λ “ λpaq. We remark +that λ is also dependent on P, but this dependence is not indicated explicitly for notational brevity. +Thus, by definition, λpaq is the solution to the implicit equation +Mpa, λpaqq ´ 1 +2a2P ´ +N +2πA2 “ 0. +(3.11) +6 + +Since λ has been viewed as a function of a, all quantities (except ¯z which also depends on Z) related +to the uniform solution are functions of a and R. For instance, ¯r denotes the function +¯rpa, Rq “ +a +a2A2 ` λpaq´1pR2 ´ A2q. +(3.12) +We denote ¯σ33 by ´qpa, Rq so that +qpa, Rq :“ ´¯σ33 “ +ż ¯λ1 +b +w1p˜λ1, λq +˜λ2 +1λ ´ 1 +d˜λ1. +(3.13) +We also define another function mpa, Rq through +mpa, Rq :“ 1 +R2 +ż B +R +λ´1¯σ22RdR “ +ż ¯λ1 +b +p˜λ2 +1 ´ ¯λ2 +1qw1p˜λ1, λq ` 2˜λ1λp¯λ2 +1λ ´ 1qw2p˜λ1, λq +2p˜λ2 +1λ ´ 1q2 +d˜λ1, +(3.14) +and record the connections +qpa, Aq “ Qpa, λpaqq, +mpa, Aq “ Mpa, λpaqq. +(3.15) +The 1d reduced model to be derived in the next section will be expressed in terms of the two +functions qpa, Rq and mpa, Rq. The integrals in these two functions can be evaluated explicitly for +some commonly used strain energy functions, including the Gent material model that will be used +in our illustrative examples. +4. Derivation of the one-dimensional model +In this section, we apply the dimension reduction methodology proposed by Audoly & Hutchin- +son (2016) to derive a one-dimensional model from the full three-dimensional theory formulated in +Sect. 2. +4.1. Optimal correction +We start our dimension reduction by assuming that all dependent variables related to the +axisymmetric configuration vary slowly in the axial direction. More precisely, it is assumed that +all variables depend on Z through the “far distance” variable +S “ εZ. +(4.1) +Recall that ε is the ratio of the outer radius to the length, which is assumed to be small. +In +particular, we now allow a and λ to depend on S and write a “ apSq, λ “ λpapSqq. Our aim is to +derive a reduced model, an ordinary differential equation, that is satisfied by apSq. We recall that +apSq is the deformed inner radius divided by a constant (i.e. A). +A naive approach would be to use a “ apSq and λ “ λpapSqq to compute the two principal +stretches and then derive the equation satisfied by a “ apSq by minimizing the energy functional +7 + +(2.3). This would yield an equation for apSq that is not self-consistent. The correct way is to allow +for higher-order correction terms by looking for an asymptotic solution of the form +zpZ, Rq “ 1 +ε +ż S +0 +λpapTqq dT ` εv˚pS, Rq ` Opε3q, +rpZ, Rq “ ¯rpapSq, Rq ` ε2u˚pS, Rq ` Opε4q. +(4.2) +We note that the correction terms in zpZ, Rq and rpZ, Rq are of order ε and ε2, respectively. +This is because the Op1q-term in zpZ, Rq and the Opεq-term in rpZ, Rq correspond to a uniform +perturbation and can thus be absorbed into the leading terms. +On substituting (4.2) into (2.2) and truncating at order ε2, we obtain the deformation gradient +F “ +¨ +˚ +˝ +¯r{R ` ε2u˚{R +0 +0 +0 +λpapSqq ` ε2v˚ +S +εv˚ +R +0 +ε¯raa1pSq +¯rR ` ε2u˚ +R +˛ +‹‚, +(4.3) +where the subscripts represent partial differentiation with respect to the indicated variables (in +particular ¯ra “ B¯r{Ba). Consequently, the three principal stretches are given by +λ1 “ ¯λ1 ` ε2 u˚ +R , +λ2 “ ¯λ2 ` ε2´ +v˚ +S ` λp¯r2 +aa1pSq2 ` v˚2 +R q ` 2¯λ3¯raa1pSqv˚ +R +2pλ2 ´ ¯λ3q +¯ +, +(4.4) +where ¯λ1 and ¯λ2 are given by (3.2) but with a and λ replaced by apSq and λpapSqq, respectively. +Substituting (4.4) into (2.3) and expanding to second order, we see that E can be written, in +terms of the un-scaled variables, as +E “ 2π +´ ż L +´L +GpapZq, λpapZqqq dZ ` E2 +¯ +` OpLε3q, +(4.5) +where E2 represents the term of order ε2 and is given by +E2 “ +ż L +´L +´ ż B +A +´ +pw2 ´ N˚qvZ ` w2 +λp¯r2 +aa12 ` v2 +Rq ` 2¯λ3¯raa1vR +2pλ2 ´ ¯λ2 +3q +¯ +R dR +` +ż B +A +w1u dR ´ 1 +2PA2a2vZ|R“A ´ PAaλu|R“A +¯ +dZ. +(4.6) +In the above expression, vpZ, Rq “ εv˚pS, Rq and upZ, Rq “ ε2u˚pS, Rq denote the unscaled dis- +placements, and here and hereafter we write apZq for apSq and so a1 now denotes a1pZq. It is seen +that the only reason for introducing S above is to identify all terms of order ε2 that should be kept +in (4.6). With this task accomplished, the scaled variable S will no longer appear in the subsequent +analysis. Also, w1 “ w1p¯λ1, λq, w2 “ w2p¯λ1, λq in which λ is a function of a and ¯λ1 is a function of +a and R. +8 + +Our formulation in terms of the reduced strain energy function requires the solution (4.2) to +satisfy the incompressibility condition automatically. This can be achieved by eliminating u in +(4.6) with the use of detpF q “ 1 which takes the form +λp¯ruqR ` ¯rp¯rRvZ ´ ¯raa1vRq “ 0. +(4.7) +To this end, we make use of the equilibrium equation (3.5) and write +ż B +A +w1u dR “ λ +ż B +A +¯σ33,R¯ru dR “ λ¯σ33¯ru|B +A ´ λ +ż B +A +¯σ33p¯ruqR dR +“ λPAau|R“A ´ +ż B +A +q¯rp¯rRvZ ´ ¯raa1vRq dR, +(4.8) +where we have replaced ¯σ33 by ´qpa, Rq (cf. (3.13)) and have used (4.7) to eliminate p¯ruqR. +On eliminating u in (4.6) with the use of (4.8), we can recast E2 in the form +E2 “ +ż L +´L +´ ż B +A +` +pλ´1¯σ22 ´ N˚qvZ ` 1 +2ζp¯r2 +aa12 ` v2 +Rq ` ξ¯raa1vR +˘ +R dR +´ 1 +2PA2a2vZ|R“A +¯ +dZ, +(4.9) +where ζ “ λw2{pλ2 ´ ¯λ2 +3q, ξ “ q¯λ1 ` ¯λ3ζ{λ, and we have made use of the connection λw2 ´ q “ ¯σ22 +that follows from (3.4)2 with ¯σ33 “ ´q. Then upon using integration by parts, we obtain +E2 “ +ż L +´L +´ ż B +A +KpR, v, vRq dR ` PA2aa1v|R“A +¯ +dZ +` +´ ż B +A +pλ´1¯σ22 ´ N˚qvR dR ´ 1 +2PA2a2v|R“A +¯ˇˇˇ +Z“L +Z“´L, +(4.10) +where KpR, v, vRq is given by +KpR, v, vRq “ ´pλ´1¯σ22qaa1Rv ` 1 +2Rζp¯r2 +aa12 ` v2 +Rq ` Rξ¯raa1vR. +(4.11) +In the last expression, pλ´1¯σ22qa denotes the partial derivative of λ´1¯σ22 with respect to a with R +fixed. To find the remaining correction field v “ vpZ, Rq, we assume that the leading-order stretch +apZq is prescribed and seek the correction v such that the total potential energy is stationary +(Audoly & Hutchinson, 2016). As a result, the optimal v satisfies the Euler-Lagrange equation and +the boundary conditions +BK +Bv ´ d +dR +´ BK +BvR +¯ +“ 0, +A ď R ď B, +(4.12) +BK +BvR +“ PA2aa1, +R “ A, +(4.13) +BK +BvR +“ 0, +R “ B. +(4.14) +9 + +Solution of the above boundary value problem requires satisfaction of the solvability condition +ż B +A +BK +Bv dR “ ´PA2aa1, +that is +ż B +A +pλ´1¯σ22qaa1R dR “ PA2aa1. +This is automatically satisfied in view of the definition (3.9) for Mpa, λq and the equilibrium con- +dition (3.8). +Written out explicitly, Eqs. (4.12) and (4.14) take the form +d +dRpRζvRq “ ´ +´ +pλ´1¯σ22qaR ` d +dRpRξ¯raq +¯ +a1, +A ď R ď B, +(4.15) +RζvR “ ´Rξ¯raa1, +R “ B. +(4.16) +Integrating (4.15) subject to the boundary condition (4.16) leads to +vR “ cpa, Rqa1pZq, +(4.17) +where cpa, Rq is defined by +cpa, Rq “ ´ ¯ra +¯λ1λ2 ` 1 +Rζ +´ +R2 B +Bampa, Rq ´ ¯r¯raqpa, Rq +¯ +. +(4.18) +Once vR is found, the optimal correction v can be obtained by integrating (4.17) from B to R, +which yields +v “ ´ +ˆż B +R +cpa, Tq dT +˙ +a1pZq, +(4.19) +where we have neglected the function arising from integration since it can be absorbed into λpapZqq. +4.2. One-dimensional energy functional +Substituting the correction function v found in (4.19) back into (4.10), after some simplification +(which is detailed in Appendix A), we obtain the final expression for the energy functional of the +1d model +E1dras “ +ż L +´L +´ +Gpa, λpaqq ` 1 +2Dpaqa1pZq2¯ +dZ ` Cpaqa1pZq|L +´L, +(4.20) +where the gradient moduli D and C are given by +Dpaq “ +ż B +A +Rζp¯r2 +a ´ cpa, Rq2q dR, +(4.21) +Cpaq “ +ż B +A +pλ´1¯σ22 ´ N˚q˜cpa, RqR dR ´ 1 +2PA2a2˜cpa, Aq, +(4.22) +10 + +with ˜cpa, Rq “ ´ +şB +R cpa, Tq dT. +The associated equilibrium equation is obtained by extremizing (4.20) with respect to apZq and +is found to take the form +A2aλpaqpQpa, λpaqq ´ Pq ´ 1 +2D1paqa1pZq2 ´ Dpaqa2pZq “ 0, +(4.23) +where we have used the fact that BG{Bλ “ 0 as it is used to find the implicit relation between λ and +a (see (3.11)). Since Z does not explicitly appear in the integrand of (4.20) due to the translational +invariance of the current problem in Z, by the Beltrami identity, the equilibrium equation (4.23) +admits a first integral of the form +Gpa, λpaqq ´ 1 +2Dpaqa1pZq2 “ constant. +(4.24) +We remark that the variational problem (4.20) is ill-posed due to the presence of the boundary +terms Cpaqa1pZq|L +´L. +This is because the variational structure of the problem is broken when +higher-order terms are dropped. There are two possible ways to get around this issue (Lestringant +& Audoly, 2020a). The first one is to simply ignore the boundary terms, i.e., to set Cpaq “ 0. The +second one is to add an Opε2q-term to apZq so that the boundary terms go away, which is rigorous +but slightly more complex. It has previously been verified in Lestringant & Audoly (2020a) that +the simple and rigorous approaches yield curves that can hardly be distinguished visually in any of +the plots. +To summarize, the second-order nonlinear ordinary differential equation (4.23) is our approxi- +mate 1d model that governs the variation of the inner radius (which is A times apZq) in the axial +direction. Once apZq is determined, the 3d deformation is given by (3.1). We note that the func- +tion Qpa, λpaqq is explicit for most of the commonly used strain energy functions. The only slight +complication is that the function Dpaq is given by an integral; see (4.21), but the functions mpa, Rq, +qpa, Rq, and hence cpa, Rq and the integrand in (4.21) all have explicit expressions for most of the +commonly used strain energy functions. Thus, only one numerical integration is required. This can +easily be implemented on a symbolic manipulation platform such as Mathematica (Wolfram, 1991) +as we shall show later. +5. Connections with previous work +We now demonstrate that the 1d model derived in Sect. 4 can recover the 1d model of Lestringant +& Audoly (2018) for membrane tubes and that of Audoly & Hutchinson (2016) for solid cylinders +under appropriate limits, and it can also reproduce the same weakly nonlinear bulging solution as +that based on the exact 3d theory (Ye et al., 2020). +11 + +5.1. Membrane limit +We first consider the reduction of the 1d model in the membrane limit when the tube thickness +H approaches zero. The general axisymmetric deformation is now described by +r “ rpZq, +θ “ Θ, +z “ zpZq, +(5.1) +and the three principal stretches are given by +λ1 “ r +R, +λ2 “ +a +r12 ` z12, +λ3 “ 1{pλ1λ2q, +(5.2) +where R denotes the constant radius of the mid-surface. The total energy (2.3) reduces to +E “ 2π +ż L +´L +´ +w ´ 1 +2P ˚λ2 +1z1 ´ N˚z1¯ +dZ, +(5.3) +where P ˚ denotes the pressure scaled by H{R. Setting the first variation δE to zero then gives the +governing equations +w1 ´ R +´w2 +λ2 +r1¯1 +´ P ˚λ1z1 “ 0, +(5.4) +w2 +λ2 +z1 ´ 1 +2P ˚λ2 +1 “ N˚. +(5.5) +Under the assumption that |r1| ! 1, we have +λ2 “ z1 ` r12 +2z1 ` ¨ ¨ ¨ . +(5.6) +As an algebraic equation for z1, Eq. (5.5) has an asymptotic solution of the form +z1 “ gpλ1q ` k1pλ1qr12 ` ¨ ¨ ¨ , +(5.7) +where the leading-order term gpλ1q obviously satisfies the algebraic equation +w2pλ1, gpλ1qq ´ 1 +2P ˚λ2 +1 ´ N˚ “ 0, +(5.8) +and the function k1pλ1q can easily be found but is not required. Eq. (5.8) determines gpλ1q uniquely +under the assumption w22 ą 0. +With the use of (5.6) and (5.7), we may expand the integrand in (5.3) to order r12 and obtain +E “ 2π +ż L +´L +´ +wpλ1, gpλ1qq ´ 1 +2P ˚λ2 +1gpλ1q ´ N˚gpλ1q ` 1 +2 +w2pλ1, gpλ1qq +gpλ1q +r12¯ +dZ. +(5.9) +This is the reduced model derived by Lestringant & Audoly (2018). +We now show that our general 1d model (4.20) can recover this 1d model under the limit H Ñ 0. +To this end, we first note that the uniformly deformed configuration is now described by +¯r “ aR, +¯z “ λZ. +(5.10) +12 + +In particular, we have ¯ra “ R. Since qpa, Rq and mpa, Rq involve integrals from R to B, they go to +zero as H Ñ 0. Consequently, the cpa, Rq defined in (4.18) takes the simple form +cpa, Rq “ ´ R +aλ2 . +(5.11) +Taking the limit H Ñ 0 in ζ “ λw2{pλ2 ´ ¯λ2 +3q yields +ζ “ a2λ3w2 +a2λ4 ´ 1. +(5.12) +Substituting (5.11) and (5.12) into (4.20), we obtain +lim +HÑ0 +E1dras +RH +“ +ż L +´L +´ +wpa, λpaqq ´ 1 +2P ˚a2λpaq ´ N˚λpaq ` 1 +2R2 w2pa, λpaqq +λpaq +a1pZq2¯ +dZ. +(5.13) +Note that the modulus Cpaq vanishes in the membrane limit because of the equilibrium in the axial +direction. The integrand on the right-hand side of (5.13) is the same as that on the right-hand side +of (5.9) if we identify λ1, gpλ1q and r1 with apZq, λpaq, and Ra1pZq, respectively. +5.2. Solid cylinder limit +Next we consider the other extreme limit corresponding to A Ñ 0 and P Ñ 0. The uniform +solution takes the form +¯z “ λZ, +¯r “ aR +(5.14) +with a “ λ´1{2. The three principal stretches are +¯λ1 “ ¯λ3 “ λ´1{2, +¯λ2 “ λ. +(5.15) +In particular, we have +w1p¯λ1, ¯λ2q “ 0, +w2p¯λ1, ¯λ2q “ ˆw1pλq, +(5.16) +where ˆwpλq “ Wpλ´1{2, λ, λ´1{2q. It follows from (5.16)1 that qpa, Rq “ 0. Note that the deforma- +tion (5.14) is homogeneous, so (3.14) implies that +mpa, Rq “ A2pB2 ´ R2q +R2pB2 ´ A2qmpa, Aq “ A2pB2 ´ R2q +R2pB2 ´ A2qMpa, λpaqq. +Differentiating this expression with respect to a and noting (3.11), we obtain Bmpa, Rq{Ba “ 0. +Thus cpa, Rq reduces to +cpa, Rq “ ´ R +λ3{2 . +(5.17) +The elastic modulus ζ is easily calculated as +ζ “ λ2 ˆw1pλq +λ3 ´ 1 . +(5.18) +13 + +Substituting (5.17) and (5.18) into (4.20), we obtain +2πE1drλs “ +ż L +´L +´ +πB2 ˆwpλq ` πB4 +16 +ˆw1pλq +λ4 +λ1pZq2 ´ Nλ +¯ +dZ, +(5.19) +where we have made use of the relation a1pZq “ λ1pZq{p2λ3{2q. This recovers the 1d model of +Audoly & Hutchinson (2016) specialized to an incompressible circular cylinder. +5.3. Comparison with exact weakly nonlinear analysis +Finally, we carry out a weakly nonlinear near-critical analysis using our 1d model and compare +the resulting amplitude equation with that obtained by Ye et al. (2020) from the exact 3d theory. +We focus on localized solutions in an infinitely long tube of finite wall thickness. +Denote by a8 the limit of apZq as Z Ñ 8 and λ8 “ λpa8q. It follows from (3.6) and (3.11) +that +P “ Qpa8, λ8q, +N “ 2πA2Fpa8, λ8q, +(5.20) +where Fpa8, λ8q is defined by +Fpa8, λ8q “ Mpa8, λ8q ´ 1 +2a2 +8Qpa8, λ8q. +(5.21) +We look for a localized solution that bifurcates from the uniform solution by writing +apZq “ a8 ` ypZq, +(5.22) +where ypZq is a small perturbation. Substituting (5.22) into the 1d equilibrium equation (4.23) +and expanding in terms of ypZq to quadratic order with the use of (5.20), we obtain +Dpa8qy2pZq “ ωpa8, λ8qypZq ` γpa8, λ8qypZq2, +(5.23) +where the two coefficient functions ωpa, λq and γpa, λq are given by +ωpa, λq “ A2 +2aλ +a2Qλ ` 2Fλ +Ωpa, λq, +(5.24) +γpa, λq “ A2 aλpa2Qa ` 2Faq +Fapa2Qλ ` 2Fλq2 Γpa, λq ` A2ψpa, λqΩpa, λq. +(5.25) +In the above expressions, Qa “ BQpa, λq{Ba, Qλ “ BQpa, λq{Bλ, etc. and Ωpa, λq and Γpa, λq are +defined by +Ωpa, λq “ BQ +Ba +BF +Bλ ´ BQ +Bλ +BF +Ba , +(5.26) +Γpa, λq “ BΩ +Ba +BF +Bλ ´ BΩ +Bλ +BF +Ba , +(5.27) +and ψpa, λq is not written out as it is not required in the weakly nonlinear analysis. +14 + +The solution to the linearized equation of (5.23) changes character when the sign of ωpa8, λ8q +changes. Thus a bifurcation occurs when ωpa8, λ8q “ 0, or equivalently, +Ωpa8, λ8q “ 0. +(5.28) +Note that Qpa8, λ8q and Fpa8, λ8q represent respectively the functional dependence of P and +N on a8 and λ8. Thus the above bifurcation condition is simply the vanishing of the Jacobian +determinant of P and N as functions of a8 and λ8. This is consistent with previous work Fu et al. +(2016) and Yu & Fu (2022). +We consider two typical loading scenarios: either the resultant axial force N or the axial stretch +at infinity λ8 is fixed. The latter case is used to approximate the case of fixed axial length, which +can be realized more easily experimentally or in Abaqus simulations. +Let us first assume that the resultant axial force N “ Nc is fixed, where Nc is the prescribed +axial force. Denote by pacr, λcrq the root of the system of equations +ωpa8, λ8q “ 0, +Fpa8, λ8q “ Nc, +(5.29) +at which the bifurcation occurs according to the previous discussion. In the vicinity of the bifurca- +tion point, the amplitude equation (5.23) reduces to +Dpacrqy2pZq “ ω1pacr, λcrqpa8 ´ acrqypZq ` γpacr, λcrqypZq2, +(5.30) +where the prime on ω denotes d{da8 “ B{Ba8 ` pB{Bλ8qpdλ8{da8q. The above equation admits +a localized solution of the form +ypZq “ ´3ω1pacr, λcrq +2γpacr, λcrq pa8 ´ acrq sech2 ´1 +2 +d +ω1pacr, λcrq +Dpacrq +pa8 ´ acrqZ +¯ +. +(5.31) +On the other hand, the weakly nonlinear amplitude equation derived from the 3d theory (Ye +et al., 2020) takes the form +c2 +1pZq “ λ2 +crk1pa8 ´ acrqc1pZq ` λ2 +crk2c1pZq2, +(5.32) +where c1pZq and ypZq are related by +ypZq “ kc1pZq +(5.33) +with k “ ´2λpaq{λ1paq|a“acr, and k1 and k2 are constants available in Ye et al. (2020). One can +see that (5.30) and (5.32) are identical provided +k1 “ ω1pacr, λcrq +λ2crDpacrq , +k2 “ kγpacr, λcrq +λ2crDpacrq . +(5.34) +We have verified numerically that this is indeed the case, but the current expressions on the right +hand sides of (5.34) are more compact and revealing. +15 + +The case of fixed λ8 can be handled similarly. Let pacr, λcrq be the solution to the system of +equations +ωpa8, λ8q “ 0, +λ8 “ λc, +(5.35) +where λc is a given constant. +In the vicinity of the bifurcation point, the amplitude equation +parallel to (5.30) is of the form +Dpacrqy2pZq “ ω1pacr, λcrqpa8 ´ acrqypZq ` γpacr, λcrqypZq2. +(5.36) +where the prime on ω now signifies B{Ba8. Similar to the previous case, one can verify that the +above amplitude equation is the same as its counterparts in Ye et al. (2020). +6. Comparison with Abaqus simulations +In this section, we demonstrate the power of the 1d model by applying it to investigate localized +bulging in an inflated tube of finite wall thickness in the fully nonlinear regime. Previous studies on +this problem usually treat the tube as a finite length tube, but the problem can be analyzed more +easily and very accurately by assuming the tube to be of infinite length. This assumption only fails +when the tube is very short and when bulging is no longer localized in the axial direction (Wang +& Fu, 2021). The reason is that bulging solutions decay exponentially towards the two ends. Thus +in the following analysis, we shall assume that the tube is effectively infinite and focus on solutions +subject to decaying boundary conditions. This assumption is validated by comparison with Abaqus +simulations based on tubes of finite lengths. We shall consider the two loading scenarios discussed +in Subsection 5.3 and compare the predictions of the 1d model with Abaqus simulations, which +allows us to quantify the accuracy of our 1d model and determine its range of validity. +In all numerical calculations and Abaqus simulations, we use the Gent material model +W “ ´µ +2 Jm ln +´ +1 ´ λ2 +1 ` λ2 +2 ` λ2 +3 ´ 3 +Jm +¯ +, +(6.1) +where µ is the shear modulus and Jm is a material constant. We take µ “ 1 which is equivalent +to scaling all stress variables by µ and Jm “ 97.2 which is typical for rubber. The geometry of the +tube is taken to be H{Rm “ 0.4 and 2L{Rm “ 40, where Rm “ pA ` Bq{2 is the average radius. +In the Abaqus simulations, to ensure that localized bulging occurs in the middle of the tube, a +small section with length 0.1L around the middle point of the tube is weakened by taking its shear +modulus to be 0.9999 times that of the rest of the tube. +The 1d differential equation (4.20) subject to appropriate end conditions (see (6.7) later) can +be solved numerically with the aid of the symbolic computation software Mathematica. Although +the gradient modulus Dpaq involves an integral that cannot be evaluated analytically, this integral +can be defined numerically in Mathematica with the built-in command ?NumericQ and can be +manipulated as elementary functions. Numerically solving the 1d equation is significantly faster +than Abaqus simulations. The 1d equation can typically be solved in a few seconds on a personal +computer. +16 + +6.1. The case of fixed axial force +We first consider the loading scenario whereby the resultant axial force N is fixed. As mentioned +earlier, we assume that the tube is infinitely long and focus on the solution that satisfies the decaying +boundary condition +lim +ZÑ8 apZq “ a8. +(6.2) +A linear analysis shows that the solution to (4.23) satisfying (6.2) decays exponentially as Z Ñ 8. +Thus we have limZÑ8 a1pZq “ 0 automatically. We assume that the bulging solution is symmetric +with respect to Z “ 0 so that a1p0q “ 0. We write λ8 “ λpa8q, a0 “ ap0q and λ0 “ λpap0qq. Since +pa8, λ8q satisfy the equations (3.6) and (3.8), we have +Mpa8, λ8q ´ 1 +2a2 +8Qpa8, λ8q ´ +N +2πA2 “ 0, +(6.3) +Qpa8, λ8q ´ P “ 0. +(6.4) +From the definition of λ0 and the conservation law (4.24), we see that pa0, λ0q satisfies +Mpa0, λ0q ´ 1 +2a2 +0Qpa8, λ8q “ Mpa8, λ8q ´ 1 +2a2 +8Qpa8, λ8q, +(6.5) +Gpa0, λ0q “ Gpa8, λ8q. +(6.6) +Either a8 or P can be taken to be the load parameter. When a8 is specified, one can first obtain +λ8 from (6.3). The associated P is computed according to (6.4). Then by solving Eqs. (6.6) and +(6.5), one obtains the “initial” values a0 and λ0. The localized solution can be found by solving +the initial value problem +A2aλpaqpQpa, λpaqq ´ Pq ´ 1 +2D1paqa1pZq2 ´ Dpaqa2pZq “ 0, +(6.7) +ap0q “ a0, +a1p0q “ 0. +(6.8) +As a first example, fixing the axial force N to be zero, we find from the bifurcation condition (5.29) +that localized bulging takes place at a8 “ acr “ 1.86 with a critical pressure Pcr “ 0.308. As we +trace the bifurcation solution away from the bifurcation point, the pressure drops while the bulge +grows until it has almost reached a maximum amplitude after which the bulge will propagate at a +constant pressure. From Maxwell’s equal-areal rule, the propagation pressure is PM “ 0.197. +Fig. 2 shows the dependence of the pressure on ap0q and the bulging amplitude on a8 based on +Abaqus simulations and use of the 1d model. The bulging solutions given by Abaqus simulations +and the 1d model at the four states marked in Fig. 2(a) are shown in Fig. 3. It is seen that the 1d +solution agrees well with Abaqus simulations in the entire post-bifurcation regime. Remarkably, +the 1d solution remains highly accurate even in the final propagation stage, as shown in Fig. 3(d). +Note also that the Abaqus simulations and 1d calculations are conducted for 2L “ 40Rm and 8, +respectively. This verifies our earlier claim that the tube can effectively be viewed to be infinitely +long. +17 + +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● ● ● ● +● +● +● +● ● ● +● +● +● +● +● ● +● +● +● +1d model +Abaqus simulation +1 +2 +3 +4 +5 +6 +7 a(0) +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +P +a +b +c +d +(a) +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +1d model +Abaqus simulation +1.2 +1.3 +1.4 +1.5 +1.6 +1.7 +1.8 +1.9a∞ +0 +1 +2 +3 +4 +5 +6 +a(0) - a∞ +(b) +Figure 2: Dependence of (a) pressure on ap0q and (b) bulging amplitude on a8 based on Abaqus simulations and the +1d model for fixed N “ 0. +Abaqus simulation +1d model +0 +2 +4 +6 +8 Z +1.6 +1.7 +1.8 +1.9 +2.0 +2.1 +2.2 +2.3 +2.4 +a(Z) +(a) +Abaqus simulation +1d model +0 +2 +4 +6 +8 Z +1.5 +2.0 +2.5 +3.0 +3.5 +a(Z) +(b) +Abaqus simulation +1d model +0 +2 +4 +6 +8 Z +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +a(Z) +(c) +Abaqus simulation +1d model +0 +2 +4 +6 +8 Z +1 +2 +3 +4 +5 +6 +a(Z) +(d) +Figure 3: Bulging solutions given by Abaqus simulations and the 1d model at the four states marked in Fig. 2(a) for +fixed N “ 0: (a) P “ 0.3, (b) P “ 0.25, (c) P “ 0.22, (d) P “ 0.197. +18 + +6.2. The case of fixed ends +Next, we consider the loading scenario whereby the tube is first stretched to a specified length +2ℓ and then its two ends are fixed to prevent further axial displacement (whether the radial dis- +placement is restricted or not at the ends is immaterial since the tube is assumed to be sufficiently +long). In the previous subsection, we have solved the problem for a specified axial force N or +equivalently a specified λ8. For the current problem with a given ℓ, we define λc “ ℓ{L and we +need to find λ8 such that the following condition is satisfied: +ż L +0 +λpapZqq dZ “ λcL. +(6.9) +This can be achieved by a shooting procedure: for each N, we compute the left-hand side using +the procedure outlined in the previous subsection and adjust N such that the left-hand side and +the right-hand side are equal. The procedure may be started by taking λ8 “ λc. However, solving +the present problem by the shooting procedure requires a lot of adjustments by hand due to the +fact that the bulging solution may start to grow after decaying for a range of Z values. To find +solutions for the current case in a more robust way, we use the finite difference method instead. +To implement the finite difference method, we partition the domain r0, Ls using a uniform mesh +Z0, Z1, . . . , Zn with mesh size h “ L{n and coordinate of the j-th grid point given by Zj “ jh. We +use aj to represent the numerical approximation of apZjq. Applying the central difference scheme, +we discretize the differential equation (6.7) as +A2ajλpajqpQpaj, λpajqq ´ Pq ´ 1 +2D1pajq +´aj`1 ´ aj´1 +2h +¯2 +´ Dpajqaj`1 ´ 2aj ` aj´1 +h2 +“ 0, +j “ 1, 2, . . . , n ´ 1. +(6.10) +The left boundary condition is given by +a1p0q “ 0. +(6.11) +We see from (5.23) that the solution to (6.7) subject to (6.2) has the asymptotic behavior +apZq „ a8 ` a1e´κZ +as Z Ñ 8, +(6.12) +where a1 is a constant and +κ “ +d +ωpa8, λ8q +Dpa8q +. +Because of this, we may replace the decaying condition boundary (6.2) by the “soft” asymptotic +condition +a1pLq ` κpapLq ´ a8q “ 0. +(6.13) +19 + +To avoid the loss of accuracy at the two endpoints, we introduce two additional unknowns a´1 and +an`1. Then the left and right boundary conditions yield +a1 ´ a´1 +2h +“ 0, +(6.14) +an`1 ´ an´1 +2h +` κpan ´ a8q “ 0. +(6.15) +Solving for a´1 and an`1 from the above equations, and substituting them into the difference +equations (6.10) at j “ 0 and j “ n, we obtain the discrete boundary conditions with truncation +errors of order h2: +A2a0λpajqpQpa0, λpa0qq ´ Pq ´ 2Dpa0qa1 ´ a0 +h2 +“ 0, +(6.16) +A2anλpanqpQpan, λpanqq ´ Pq ´ 1 +2D1pajqκ2pan ´ a8q2 +´ 2Dpanqan´1 ´ an ´ hκpan ´ a8q +h2 +“ 0. +(6.17) +Finally, the fixed-length restriction (6.9) gives +1 +2λpa0q ` +n´1 +ÿ +j“1 +λpajq ` 1 +2λpanq ´ λcL +h +“ 0. +(6.18) +We use the pressure P as the loading parameter. When P is given, one can first solve (6.4) to +express a8 as a function of λ8. Then N can be viewed as a function of λ8 due to (6.3). It follows +that λpµq and Dpµq also depend on λ8 through their dependence on N. This implicit dependence +should be considered when solving the above algebraic equations. +Setting n to be a sufficiently large number, say n “ 100, and solving the system of nonlinear +algebraic equations consisting of (6.10), (6.16), (6.17) and (6.18) for aj’s and λ8 with a suitable +initial guess, we obtain the finite-difference solution for the present problem. +We may use the +weakly nonlinear solution with fixed λ8 “ λc “ ℓ{L as an initial guess in the near-critical regime +and continue the solution to the fully nonlinear regime by always using the solution at the previous +step as the initial guess for the current step. +When the total length is fixed to be ℓ “ 2L, then initially λ8 “ 2 and localized bulging takes +place at a8 “ acr “ 1.74 with a critical pressure Pcr “ 0.198 according to (5.35). In Fig. 4, we have +shown the dependence of the pressure on ap0q and the bulging amplitude on a8 based on Abaqus +simulations and use of the 1d model. The bulging solutions determined by Abaqus simulations +and the 1d model at the four states indicated in Fig. 4(a) are presented in Fig. 5. It is observed +that the agreement between the 1d model and Abaqus simulations is again excellent in the fully +nonlinear regime. +Finally, Fig. 6 shows the actual variation of P against ap0q predicted by the 1d model when L +is varied and the averaged stretch λc is fixed or λc is varied but L is fixed. These results confirm +the theoretical prediction of Guo et al. (2022) that the right branches of these curves all converge +20 + +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● ● ● +● +● +● ● +● +● +● ● ● +● ● ● +● +● ● +● +● +● +1d model +Abaqus simulation +0 +1 +2 +3 +4 +5 +6 +7 +a(0) +0.00 +0.05 +0.10 +0.15 +0.20 +P +a +b +c +d +(a) +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +● +1d model +Abaqus simulation +1.1 +1.2 +1.3 +1.4 +1.5 +1.6 +1.7 +1.8a∞ +1 +2 +3 +4 +5 +6 +a(0) - a∞ +(b) +Figure 4: Dependence of (a) pressure on ap0q and (b) bulging amplitude on a8 based on Abaqus simulations and +using the 1d model for fixed length ℓ{L “ 2. +Abaqus simulation +1d model +0 +2 +4 +6 +8 +10 +12Z +1.2 +1.4 +1.6 +1.8 +2.0 +2.2 +2.4 +2.6 +2.8 +a(Z) +(a) +Abaqus simulation +1d model +0 +2 +4 +6 +8 +10 +12Z +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +a(Z) +(b) +Abaqus simulation +1d model +0 +2 +4 +6 +8 +10 +12Z +1 +2 +3 +4 +5 +a(Z) +(c) +Abaqus simulation +1d model +0 +2 +4 +6 +8 +10 +12Z +1 +2 +3 +4 +5 +6 +7 +a(Z) +(d) +Figure 5: Bulging solutions based on Abaqus simulations and the 1d model the at the four states indicated in Fig. +4(a) for fixed length ℓ{L “ 2: (a) P “ 0.19, (b) P “ 0.18, (c) P “ 0.173, (d) P “ 0.198. +to a master curve that is independent of L or λc. These curves all terminate at the point where +the axial stress near each end of the tube has become compressive enough so that secondary Euler +buckling or axisymmetric wrinkling becomes possible. +21 + +L=15 +L=20 +L=40 +0 +1 +2 +3 +4 +5 +6 +7 +a(0) +0.12 +0.14 +0.16 +0.18 +0.20 +0.22 +P +(a) +λc=1.5 +λc=2 +λc=2.8 +0 +1 +2 +3 +4 +5 +6 +7 +a(0) +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +P +(b) +Figure 6: Variation of P against ap0q predicted by the 1d model when (a) the total length is fixed with λc “ 2 and +L “ 15, 20 and 40, respectively and (b) L is fixed at 20 and λc “ 1.5, 2 and 2.8, respectively. +7. Conclusion +We have derived a 1d model for the analysis of axisymmetric deformations of an inflated cylin- +drical tube of finite wall thickness, and established its range of validity by comparing its predictions +with those of Abaqus simulations for two typical loading scenarios. The comparison shows that +the 1d model performs extremely well in both the near-critical and fully nonlinear regimes. The +dimension reduction started from three-dimensional finite elasticity theory and is performed in +terms of the energy functional and principal stretches. A key ingredient of the dimension reduc- +tion is the assumption of slow variation of the leading-order solution in the axial direction without +any restriction on its amplitude, which results in a 1d model that is simple but is still capable of +capturing the strain-gradient effect. This is in contrast with the traditional asymptotic analysis +where the leading order solution is assumed to be a small-amplitude perturbation from the primary +deformation. It is because of this difference that the 1d model has a much larger range of validity +than the expansion methods around the bifurcation point. The nonlinearity of the strain is kept +in the 1d model, reflected by the nonlinear potential Gpa, λpaqq and the nonlinear strain gradient +modulus Dpaq. Our expression for the strain gradient coefficient Dpaq is quite simple. For the +Gent material model, Dpaq can be calculated by integrating once. We remark that although the +derivation presented in this work is variational, the 1d model can also be derived by substituting +the asymptotic solution (4.2) into the 3d governing equations and solving the resulting equations +at successive orders. +The 1d model is amenable to asymptotic and numerical solutions. The bifurcation condition and +the weakly nonlinear amplitude equation predicted by the model are exact. In fact, the expressions +(5.24) and (5.25) derived using the 1d model are more compact and more revealing than their +counterparts in Ye et al. (2020). A major advantage of the 1d model is that the entire evolution +process of bulging or necking can be determined using the finite difference method which is more +accessible and much easier to implement than commercial packages such as Abaqus. This advantage +22 + +would become even more significant when other fields such as electric loading and residual stresses +were also present. Such extra fields and new geometries (e.g. axisymmetric necking of a stretched +plate (Wang et al., 2022) ) will be considered in our future studies. +A Mathematica code that produces all the results presented in the paper is available on GitHub +(https://github.com/yfukeele). +Acknowledgments +This work was supported by the National Natural Science Foundation of China (Grant No +12072224) and the Engineering and Physical Sciences Research Council, UK (Grant No EP/W007150/1). +Appendix A. Simplifying the one-dimensional energy functional +Substituting (4.19) into (4.11), we can write the integral of KpR, v, vRq as +ż B +A +KpR, v, vRq dR “ pI1 ` I2 ` I3qa12, +(A.1) +where +I1 “ +ż B +A +pλ´1¯σ22qaR +ż B +R +cpa, Tq dT dR, +(A.2) +I2 “ 1 +2 +ż B +A +Rζp¯r2 +a ` cpa, Rq2q dR, +(A.3) +I3 “ +ż B +A +Rξ¯racpa, Rq dR. +(A.4) +By interchanging the order of integration, we can rewrite I1 as +I1 “ +ż B +A +ż B +R +pλ´1¯σ22qaRcpa, Tq dT dR +“ +ż B +A +ż T +A +pλ´1¯σ22qaRcpa, Tq dR dT +“ +ż B +A +cpa, Tq B +Ba +´ ż T +A +λ´1¯σ22R dR +¯ +dT. +(A.5) +From (3.14), we have +ż T +A +λ´1¯σ22R dR “ A2mpa, Aq ´ T 2mpa, Tq. +(A.6) +Inserting (A.6) into (A.5) and noting (3.15)2 and (3.11), we can simplify I1 as +I1 “ PA2a +ż B +A +cpa, Rq dR ´ +ż B +A +cpa, RqR2 B +Bampa, Rq dR. +(A.7) +23 + +Noting that ξ “ q¯λ1 ` ¯λ3ζ{λ, the integral I3 can be calculated as +I3 “ +ż B +A +R +´ +q¯λ1 ` +¯λ3 +λ ζ +¯ +¯racpa, Rq dR “ +ż B +A +´ +¯r¯raq ` Rζ ¯ra +¯λ1λ2 +¯ +cpa, Rq dR. +(A.8) +Adding up the three integrals, we obtain +ż B +A +KpR, v, vRq dR ` PA2aa1v|R“A “pI1 ` I2 ` I3qa12 ´ PA2aa12 +ż B +A +cpa, Rq dR +“a12 +ż B +A +´ +´ cpa, RqR2 B +Bampa, Rq ` 1 +2Rζp¯r2 +a ` cpa, Rq2q +` +´ +¯r¯raqpa, Rq ` Rζ ¯ra +λ1λ2 +¯ +cpa, Rq +¯ +dR +“a12 +ż B +A +p1 +2Rζp¯r2 +a ` cpa, Rq2q ´ Rζcpa, Rq2q dR +“1 +2a12 +ż B +A +Rζp¯r2 +a ´ cpa, Rq2q dR. +(A.9) +This gives the expression of the coefficient Dpaq announced in (4.21). 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Phys., 73, 1–16. +27 + diff --git a/5NE1T4oBgHgl3EQfBAIU/content/tmp_files/load_file.txt b/5NE1T4oBgHgl3EQfBAIU/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9a7cef606329bfc39c3235a73d286c905922913e --- /dev/null +++ b/5NE1T4oBgHgl3EQfBAIU/content/tmp_files/load_file.txt @@ -0,0 +1,1255 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf,len=1254 +page_content='A one-dimensional model for axisymmetric deformations of an inflated hyperelastic tube of finite wall thickness Xiang Yua,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='˚,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Yibin Fub aSchool of Computer Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Dongguan University of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Dongguan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' China bSchool of Computing and Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Keele University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Staffordshire ST5 5BG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' UK Abstract We derive a one-dimensional (1d) model for the analysis of bulging or necking in an inflated hyper- elastic tube of finite wall thickness from the three-dimensional finite elasticity theory by applying the dimension reduction methodology proposed by Audoly and Hutchinson (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Solids, 97, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The 1d model makes it much easier to characterize fully nonlinear axisymmetric defor- mations of a thick-walled tube using simple numerical schemes such as the finite difference method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The new model recovers the diffuse interface model for analyzing bulging in a membrane tube and the 1d model for investigating necking in a stretched solid cylinder as two limiting cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' It is con- sistent with, but significantly refines, the exact linear and weakly nonlinear bifurcation analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Comparisons with finite element simulations show that for the bulging problem, the 1d model is capable of describing the entire bulging process accurately, from initiation, growth, to propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The 1d model provides a stepping stone from which similar 1d models can be derived and used to study other effects such as anisotropy and electric loading, and other phenomena such as rupture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Keywords: localized bulging;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' necking;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' reduced models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' tubes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' stability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' nonlinear elasticity 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Introduction Hyperelastic tubes are commonly found in various applications ranging from soft robotics (Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 2015, 2020) to energy harvesting (Lu & Suo, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Bucchi & Hearn, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Smith, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' They are also used to model human arteries in order to understand pathological conditions such as aneurysms (Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Alhayani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Demirkoparan & Merodio, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Varatharajan & DasGupta, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Hejazi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Inflation of a hyperelastic tube is one of the few boundary value problems in nonlinear elasticity that have closed-form solutions, and it provides the simplest setup to explain bifurcation, localization, loss of convexity, and “two-phase” deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Thus, understanding this problem is not only important for applications, but may also shed light on other more complicated stability and bifurcation problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' ˚Corresponding author Email address: yuxiang@dgut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='cn (Xiang Yu) Preprint submitted to Journal of the Mechanics and Physics of Solids January 10, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='02845v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='soft] 7 Jan 2023 Simple inflation experiments with commercially available latex rubber tubes show that localized bulging is the dominant deformation form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' For almost all realistic constitutive models for rubber, the pressure versus volume curve has an N shape under the condition of fixed resultant axial force (Green & Adkins, 1960).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This led Yin (1977) and Chater & Hutchinson (1984) to analyze the final observable configuration as that corresponding to a “two-phase” deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The subsequent experimental studies carried out by Kyriakides & Chang (1990, 1991), Pamplona et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (2006) and Goncalves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (2008) have provided a clear picture on how a localized bulge initiates, grows and then propagates under fixed axial force or fixed-ends conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' When the membrane assumption is made, the governing equations for tube inflation can be viewed as a finite-dimensional spatial dynamical system that has two conservation laws/integrals (Pipkin, 1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This realization enabled Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (2008) to demonstrate explicitly how a localized solution initiates as a zero-wave-number mode from the uniform deformation and how it evolves into a “two-phase” state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The stability of bulging solutions and their sensitivity to imperfections have been studied under the same framework (Pearce & Fu, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Fu & Il’ichev, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Fu & Xie, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Fresh analytical insight into the case of fixed ends has also been obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' It is shown that the bifurcation condition for this case corresponds to the axial force reaching a maximum at a fixed pressure (Fu & Il’ichev, 2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' in other words, as pressure is increased, the critical pressure is the value of pressure at which the axial force reaches a maximum when viewed as a function of the axial stretch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Also, in contrast with the case of fixed axial force where the measured pressure approaches a constant value (the propagation pressure), the measured pressure in the case of fixed ends has an N shape where the right branch approaches a master curve that is independent of the pre-axial-stretch or the tube length (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' In some practical applications, however, the tube wall may be of moderate or even large thickness and the membrane model no longer applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' For example, in the context of aneurysm formation, a human artery can be as thick as a quarter of its outer radius (M¨uller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 2008), and fiber- reinforcement also seems to reduce the range of validity of the membrane assumption (Wang & Fu, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Thus, recent studies have begun to consider hyperelastic tubes of finite wall thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (2016) showed that the associated bifurcation condition for localized bulging corresponds to the vanishing of the Jacobian determinant of the internal pressure and resultant axial force as functions of the azimuthal stretch and the axial stretch;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' see also Yu & Fu (2022) for an alternative derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This provides a framework under which additional effects such as rotation (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 2017), double-fiber-reinforcement (Wang & Fu, 2018), bi-laying (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 2019), torsion (Althobaiti, 2022), and surface tension (Emery & Fu, 2021a,b,c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Emery, 2023) can be assessed in a systematic manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (2020) conducted a weakly non-linear analysis and derived the bulging solution explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The analytic predictions were corroborated by numerical simulations and experiments (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' For tubes of finite wall thickness, the equations that govern their axisymmetric deformations are coupled nonlinear partial differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Although analytic solutions can be obtained in 2 the near-critical regime using asymptotic methods (Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 2020), the complexity of the governing equations forbids any further analytic attempts to understand the bulging evolution further away from the bifurcation point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The post-bifurcation behavior in the fully nonlinear regime has so far only been investigated by resorting to Abaqus simulations (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This is not satisfactory since the insight provided by full-scale simulations tends to be limited and there are situations where repeated calculations of the bulging profile are required (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' in the assessment of the rupture potential (Hejazi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' A recent series of studies by Audoly and coworkers has opened the possibility that a 1d reduced model can be derived to describe the fully nonlinear evolution of bulging or necking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' In the first of this series, Audoly & Hutchinson (2016), the authors derived a 1d model for tensile necking localization in a 3d prismatic solid of arbitrary cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The key idea of their derivation is a dimension reduction assuming slow variation in the axial direction that respects self-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' In terms of the language of perturbation analysis, the leading-order solution is almost correct and higher-order terms are only added to restore self-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The method was later applied by Lestrigant and Audoly to obtain a diffuse interface model for the characterization of propagating bulges in membrane tubes (Lestringant & Audoly, 2018) and a 1d model for predicting surface tension-driven necking in soft elastic cylinders (Lestringant & Audoly, 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' It has also been used recently to derive a 1d model for elastic ribbons (Audoly & Neukirch, 2021) and for tape springs (Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The systematic reduction method for deriving 1d strain-gradient models for nonlinear slender structures was further generalized by Lestringant & Audoly (2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' It is worth pointing out that although the 1d models are built on the assumption that localized solutions vary slowly in the longitudinal direction, it is surprisingly accurate, even in the region where the localization is well developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This is illustrated by the numeric examples in the aforementioned work and in the comparative studies by Wang & Fu (2021) and Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This work aims to extend the diffuse interface model of Lestringant & Audoly (2018) for mem- brane tubes to tubes of finite wall thickness, in a similar spirit as the previous work Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (2016) and Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (2020) that extend the bifurcation condition and the weakly nonlinear analysis from membrane tubes to thick-walled tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' In contrast with the case under the membrane assumption where the original governing equations are already one-dimensional, the governing equations for the current case are two-dimensional, and the uniformly inflated deformation is no longer homogeneous since the solution depends on the radial variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' It will be shown that a 1d reduced model can still be derived and simplified to the form E1dras “ ż L ´L ´ Gpa, λpaqq ` 1 2Dpaqa1pZq2¯ dZ ` Cpaqa1pZq|L ´L, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='1) where L is the initial half length, Z is the axial coordinate, apZq is the azimuthal stretch on the inner surface (a constant multiple of the deformed inner radius ) and the expressions for Gpa, λpaqq, Dpaq and Cpaq are given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='10), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='21) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='22), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The first term G in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='1) corresponds to the energy of the uniform deformation, which determines the amplitudes of the two phases in 3 the bulge propagation stage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' the second term accounts for the contribution of the strain gradient to the total energy, which describes how these two phases are connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The Euler-Lagrange equation associated with the energy functional (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='1) is a second-order nonlinear ode for apZq, which is a drastic simplification from the original nonlinear partial differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This 1d model is validated by comparison with finite element simulations, showing excellent agreement with numerical results even for the propagation stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The outline of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 2, we formulate the 3d axisymmetric finite- strain model for a tube of finite wall thickness under inflation and axial stretching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 3, we summarize solutions corresponding to uniform inflation of the tube, making preparation for the subsequent dimension reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 4, we carry out the dimension reduction and derive the above-mentioned 1d strain-gradient model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The connection of the 1d model with prior work is given in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 6, we validate the 1d model by making comparisons with finite element simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Finally, concluding remarks are given in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Three-dimensional finite-strain model We consider a circular cylindrical tube that has a length 2L, inner radius A and outer radius B in its reference configuration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The ratio of the outer radius to the length ε “ B{2L is assumed to be small, thus ε !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The tube deforms axisymmetrically under the combined action of an internal pressure P and a resultant axial force N, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' In terms of cylindrical coordinates, the current position vector of a representative point is given by x “ zpZ, Rqez ` rpZ, Rqer, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='1) where pR, Θ, Zq and pr, θ, zq are the coordinates of a representative point before and after defor- mation, and per, eθ, ezq are the standard basis vectors associated with both pR, Θ, Zq and pr, θ, zq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The deformation gradient related to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='1) is given by F “ r Reθ b eθ ` zZez b ez ` zRez b er ` rZer b ez ` rRer b er, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2) where zZ :“ Bz{BZ, zR :“ Bz{BR, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' We assume that the tube is made of an incompressible isotropic hyperelastic material, associated with the strain energy function Wpλ1, λ2, λ3q, where λ1, λ2, λ3 denote the three principal stretches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Throughout this paper, we identify the indices 1, 2, 3 such that in uniform inflation they coincide with the θ-, z- and r-directions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The total potential energy of the tube is composed of the elastic energy and the load potential, which reads E “ ż L ´L ´ ż B A ` wpλ1, λ2q ´ N˚zZ ˘ 2πR dR ´ Pπr2zZ ˇˇ R“A ¯ dZ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3) 4 m B G (a) m2 G2 H2 (b) Figure 1: A hyperelastic cylindrical tube of finite thickness in (a) reference (undeformed) configuration and (b) current configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' where wpλ1, λ2q “ Wpλ1, λ2, λ´1 1 λ´1 2 q is the reduced strain energy function and N˚ “ N{pπpB2 ´ A2qq is the resultant axial force per unit cross-sectional area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The elastic model governed by the energy functional (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3) will be used as a starting point for the subsequent dimension reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The governing equations for the two unknown functions rpZ, Rq and zpZ, Rq can be derived by setting the first variation of E to zero, but these equations are not required in the approach that we follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Uniform inflation We now summarise the solution that corresponds to uniform inflation and extension of the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This solution will be referred to as the uniform solution and is indicated by a superposed bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' For a more detailed derivation, see Haughton & Ogden (1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' First, incompressibility implies that a uniform solution must necessarily be of the form ¯z “ λZ, ¯r “ a a2A2 ` λ´1pR2 ´ A2q, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='1) where λ and a denote the constant axial stretch and azimuthal stretch on the inner surface, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The three principal stretches are simply ¯λ1 “ ¯r R, ¯λ2 “ λ, ¯λ3 “ d¯r dR “ ¯λ´1 1 ¯λ´1 2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2) and the azimuthal stretch on the outer surface, denoted by b, is given by b “ ¯λ1|R“B “ a a2A2 ` λ´1pB2 ´ A2q B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3) The three associated principal Cauchy stresses ¯σ11, ¯σ22 and ¯σ33 satisfy the relations ¯σ11 ´ ¯σ33 “ ¯λ1w1, ¯σ22 ´ ¯σ33 “ λw2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='4) 5 where w1 “ Bwp¯λ1, ¯λ2q{B¯λ1 and w2 “ Bwp¯λ1, ¯λ2q{B¯λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The only equilibrium equation that is not satisfied automatically is d¯σ33 d¯r “ ¯σ11 ´ ¯σ33 ¯r “ ¯λ1w1 ¯r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5) On integrating this equation from R “ A to R “ B and making use of the boundary conditions that ¯σ33|R“A “ ´P and ¯σ33|R“B “ 0, we obtain P “ Qpa, λq :“ ż a b w1p¯λ1, λq ¯λ2 1λ ´ 1 d¯λ1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6) where the second equation defines the function Qpa, λq and we have made use of the identity d¯r ¯r “ ´ d¯λ1 ¯λ1p¯λ2 1λ ´ 1q, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='7) which can be deduced from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The overall equilibrium in the axial direction implies Mpa, λq ´ 1 2a2P ´ N 2πA2 “ 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8) where Mpa, λq is given by Mpa, λq :“ 1 A2 ż B A λ´1¯σ22R dR “ ż a b p¯λ2 1 ´ a2qw1p¯λ1, λq ` 2¯λ1λpa2λ ´ 1qw2p¯λ1, λq 2p¯λ2 1λ ´ 1q2 d¯λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='9) In view of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3), the total potential energy of the uniform deformation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='1) per unit reference length, after scaling by 2π, is Gpa, λq “ ż B A wp¯λ1, λq R dR ´ 1 2PA2a2λ ´ N 2πλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='10) The equilibrium equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8) can also be obtained from BG{Ba “ 0 and BG{Bλ “ 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Once the loads P and N are specified, the deformation parameters a and λ can be found by solving the equilibrium equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' On differentiating the left-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8) with respect to λ, we find that it takes the form Hw22pa, λq{A ` OpH2q, where H “ B ´ A is the thickness of the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' We assume that the strong ellipticity condition is satisfied pointwise which guarantees that w22pa, λq is positive (Knowles & Sternberg, 1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This, combined with the implicit function theorem, implies that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8) can be inverted to express λ in terms of a uniquely at least when H is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' We assume that this remains true for arbitrary H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This enables us to view (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8) as an implicit equation for λ “ λpaq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' We remark that λ is also dependent on P, but this dependence is not indicated explicitly for notational brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Thus, by definition, λpaq is the solution to the implicit equation Mpa, λpaqq ´ 1 2a2P ´ N 2πA2 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='11) 6 Since λ has been viewed as a function of a, all quantities (except ¯z which also depends on Z) related to the uniform solution are functions of a and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' For instance, ¯r denotes the function ¯rpa, Rq “ a a2A2 ` λpaq´1pR2 ´ A2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='12) We denote ¯σ33 by ´qpa, Rq so that qpa, Rq :“ ´¯σ33 “ ż ¯λ1 b w1p˜λ1, λq ˜λ2 1λ ´ 1 d˜λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='13) We also define another function mpa, Rq through mpa, Rq :“ 1 R2 ż B R λ´1¯σ22RdR “ ż ¯λ1 b p˜λ2 1 ´ ¯λ2 1qw1p˜λ1, λq ` 2˜λ1λp¯λ2 1λ ´ 1qw2p˜λ1, λq 2p˜λ2 1λ ´ 1q2 d˜λ1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='14) and record the connections qpa, Aq “ Qpa, λpaqq, mpa, Aq “ Mpa, λpaqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='15) The 1d reduced model to be derived in the next section will be expressed in terms of the two functions qpa, Rq and mpa, Rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The integrals in these two functions can be evaluated explicitly for some commonly used strain energy functions, including the Gent material model that will be used in our illustrative examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Derivation of the one-dimensional model In this section, we apply the dimension reduction methodology proposed by Audoly & Hutchin- son (2016) to derive a one-dimensional model from the full three-dimensional theory formulated in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Optimal correction We start our dimension reduction by assuming that all dependent variables related to the axisymmetric configuration vary slowly in the axial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' More precisely, it is assumed that all variables depend on Z through the “far distance” variable S “ εZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='1) Recall that ε is the ratio of the outer radius to the length, which is assumed to be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' In particular, we now allow a and λ to depend on S and write a “ apSq, λ “ λpapSqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Our aim is to derive a reduced model, an ordinary differential equation, that is satisfied by apSq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' We recall that apSq is the deformed inner radius divided by a constant (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' A naive approach would be to use a “ apSq and λ “ λpapSqq to compute the two principal stretches and then derive the equation satisfied by a “ apSq by minimizing the energy functional 7 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This would yield an equation for apSq that is not self-consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The correct way is to allow for higher-order correction terms by looking for an asymptotic solution of the form zpZ, Rq “ 1 ε ż S 0 λpapTqq dT ` εv˚pS, Rq ` Opε3q, rpZ, Rq “ ¯rpapSq, Rq ` ε2u˚pS, Rq ` Opε4q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2) We note that the correction terms in zpZ, Rq and rpZ, Rq are of order ε and ε2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This is because the Op1q-term in zpZ, Rq and the Opεq-term in rpZ, Rq correspond to a uniform perturbation and can thus be absorbed into the leading terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' On substituting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2) and truncating at order ε2, we obtain the deformation gradient F “ ¨ ˚ ˝ ¯r{R ` ε2u˚{R 0 0 0 λpapSqq ` ε2v˚ S εv˚ R 0 ε¯raa1pSq ¯rR ` ε2u˚ R ˛ ‹‚, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3) where the subscripts represent partial differentiation with respect to the indicated variables (in particular ¯ra “ B¯r{Ba).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Consequently, the three principal stretches are given by λ1 “ ¯λ1 ` ε2 u˚ R , λ2 “ ¯λ2 ` ε2´ v˚ S ` λp¯r2 aa1pSq2 ` v˚2 R q ` 2¯λ3¯raa1pSqv˚ R 2pλ2 ´ ¯λ3q ¯ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='4) where ¯λ1 and ¯λ2 are given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2) but with a and λ replaced by apSq and λpapSqq, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Substituting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='4) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3) and expanding to second order, we see that E can be written, in terms of the un-scaled variables, as E “ 2π ´ ż L ´L GpapZq, λpapZqqq dZ ` E2 ¯ ` OpLε3q, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5) where E2 represents the term of order ε2 and is given by E2 “ ż L ´L ´ ż B A ´ pw2 ´ N˚qvZ ` w2 λp¯r2 aa12 ` v2 Rq ` 2¯λ3¯raa1vR 2pλ2 ´ ¯λ2 3q ¯ R dR ` ż B A w1u dR ´ 1 2PA2a2vZ|R“A ´ PAaλu|R“A ¯ dZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6) In the above expression, vpZ, Rq “ εv˚pS, Rq and upZ, Rq “ ε2u˚pS, Rq denote the unscaled dis- placements, and here and hereafter we write apZq for apSq and so a1 now denotes a1pZq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' It is seen that the only reason for introducing S above is to identify all terms of order ε2 that should be kept in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' With this task accomplished, the scaled variable S will no longer appear in the subsequent analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Also, w1 “ w1p¯λ1, λq, w2 “ w2p¯λ1, λq in which λ is a function of a and ¯λ1 is a function of a and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 8 Our formulation in terms of the reduced strain energy function requires the solution (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2) to satisfy the incompressibility condition automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This can be achieved by eliminating u in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6) with the use of detpF q “ 1 which takes the form λp¯ruqR ` ¯rp¯rRvZ ´ ¯raa1vRq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='7) To this end, we make use of the equilibrium equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5) and write ż B A w1u dR “ λ ż B A ¯σ33,R¯ru dR “ λ¯σ33¯ru|B A ´ λ ż B A ¯σ33p¯ruqR dR “ λPAau|R“A ´ ż B A q¯rp¯rRvZ ´ ¯raa1vRq dR, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8) where we have replaced ¯σ33 by ´qpa, Rq (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='13)) and have used (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='7) to eliminate p¯ruqR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' On eliminating u in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6) with the use of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8), we can recast E2 in the form E2 “ ż L ´L ´ ż B A ` pλ´1¯σ22 ´ N˚qvZ ` 1 2ζp¯r2 aa12 ` v2 Rq ` ξ¯raa1vR ˘ R dR ´ 1 2PA2a2vZ|R“A ¯ dZ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='9) where ζ “ λw2{pλ2 ´ ¯λ2 3q, ξ “ q¯λ1 ` ¯λ3ζ{λ, and we have made use of the connection λw2 ´ q “ ¯σ22 that follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='4)2 with ¯σ33 “ ´q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Then upon using integration by parts, we obtain E2 “ ż L ´L ´ ż B A KpR, v, vRq dR ` PA2aa1v|R“A ¯ dZ ` ´ ż B A pλ´1¯σ22 ´ N˚qvR dR ´ 1 2PA2a2v|R“A ¯ˇˇˇ Z“L Z“´L, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='10) where KpR, v, vRq is given by KpR, v, vRq “ ´pλ´1¯σ22qaa1Rv ` 1 2Rζp¯r2 aa12 ` v2 Rq ` Rξ¯raa1vR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='11) In the last expression, pλ´1¯σ22qa denotes the partial derivative of λ´1¯σ22 with respect to a with R fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' To find the remaining correction field v “ vpZ, Rq, we assume that the leading-order stretch apZq is prescribed and seek the correction v such that the total potential energy is stationary (Audoly & Hutchinson, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' As a result, the optimal v satisfies the Euler-Lagrange equation and the boundary conditions BK Bv ´ d dR ´ BK BvR ¯ “ 0, A ď R ď B, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='12) BK BvR “ PA2aa1, R “ A, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='13) BK BvR “ 0, R “ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='14) 9 Solution of the above boundary value problem requires satisfaction of the solvability condition ż B A BK Bv dR “ ´PA2aa1, that is ż B A pλ´1¯σ22qaa1R dR “ PA2aa1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This is automatically satisfied in view of the definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='9) for Mpa, λq and the equilibrium con- dition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Written out explicitly, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='12) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='14) take the form d dRpRζvRq “ ´ ´ pλ´1¯σ22qaR ` d dRpRξ¯raq ¯ a1, A ď R ď B, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='15) RζvR “ ´Rξ¯raa1, R “ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='16) Integrating (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='15) subject to the boundary condition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='16) leads to vR “ cpa, Rqa1pZq, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='17) where cpa, Rq is defined by cpa, Rq “ ´ ¯ra ¯λ1λ2 ` 1 Rζ ´ R2 B Bampa, Rq ´ ¯r¯raqpa, Rq ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='18) Once vR is found, the optimal correction v can be obtained by integrating (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='17) from B to R, which yields v “ ´ ˆż B R cpa, Tq dT ˙ a1pZq, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='19) where we have neglected the function arising from integration since it can be absorbed into λpapZqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' One-dimensional energy functional Substituting the correction function v found in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='19) back into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='10), after some simplification (which is detailed in Appendix A), we obtain the final expression for the energy functional of the 1d model E1dras “ ż L ´L ´ Gpa, λpaqq ` 1 2Dpaqa1pZq2¯ dZ ` Cpaqa1pZq|L ´L, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='20) where the gradient moduli D and C are given by Dpaq “ ż B A Rζp¯r2 a ´ cpa, Rq2q dR, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='21) Cpaq “ ż B A pλ´1¯σ22 ´ N˚q˜cpa, RqR dR ´ 1 2PA2a2˜cpa, Aq, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='22) 10 with ˜cpa, Rq “ ´ şB R cpa, Tq dT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The associated equilibrium equation is obtained by extremizing (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='20) with respect to apZq and is found to take the form A2aλpaqpQpa, λpaqq ´ Pq ´ 1 2D1paqa1pZq2 ´ Dpaqa2pZq “ 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='23) where we have used the fact that BG{Bλ “ 0 as it is used to find the implicit relation between λ and a (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='11)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Since Z does not explicitly appear in the integrand of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='20) due to the translational invariance of the current problem in Z, by the Beltrami identity, the equilibrium equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='23) admits a first integral of the form Gpa, λpaqq ´ 1 2Dpaqa1pZq2 “ constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='24) We remark that the variational problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='20) is ill-posed due to the presence of the boundary terms Cpaqa1pZq|L ´L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This is because the variational structure of the problem is broken when higher-order terms are dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' There are two possible ways to get around this issue (Lestringant & Audoly, 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The first one is to simply ignore the boundary terms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', to set Cpaq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The second one is to add an Opε2q-term to apZq so that the boundary terms go away, which is rigorous but slightly more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' It has previously been verified in Lestringant & Audoly (2020a) that the simple and rigorous approaches yield curves that can hardly be distinguished visually in any of the plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' To summarize, the second-order nonlinear ordinary differential equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='23) is our approxi- mate 1d model that governs the variation of the inner radius (which is A times apZq) in the axial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Once apZq is determined, the 3d deformation is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' We note that the func- tion Qpa, λpaqq is explicit for most of the commonly used strain energy functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The only slight complication is that the function Dpaq is given by an integral;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='21), but the functions mpa, Rq, qpa, Rq, and hence cpa, Rq and the integrand in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='21) all have explicit expressions for most of the commonly used strain energy functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Thus, only one numerical integration is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This can easily be implemented on a symbolic manipulation platform such as Mathematica (Wolfram, 1991) as we shall show later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Connections with previous work We now demonstrate that the 1d model derived in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 4 can recover the 1d model of Lestringant & Audoly (2018) for membrane tubes and that of Audoly & Hutchinson (2016) for solid cylinders under appropriate limits, and it can also reproduce the same weakly nonlinear bulging solution as that based on the exact 3d theory (Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Membrane limit We first consider the reduction of the 1d model in the membrane limit when the tube thickness H approaches zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The general axisymmetric deformation is now described by r “ rpZq, θ “ Θ, z “ zpZq, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='1) and the three principal stretches are given by λ1 “ r R, λ2 “ a r12 ` z12, λ3 “ 1{pλ1λ2q, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2) where R denotes the constant radius of the mid-surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The total energy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3) reduces to E “ 2π ż L ´L ´ w ´ 1 2P ˚λ2 1z1 ´ N˚z1¯ dZ, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3) where P ˚ denotes the pressure scaled by H{R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Setting the first variation δE to zero then gives the governing equations w1 ´ R ´w2 λ2 r1¯1 ´ P ˚λ1z1 “ 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='4) w2 λ2 z1 ´ 1 2P ˚λ2 1 “ N˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5) Under the assumption that |r1| !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 1, we have λ2 “ z1 ` r12 2z1 ` ¨ ¨ ¨ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6) As an algebraic equation for z1, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5) has an asymptotic solution of the form z1 “ gpλ1q ` k1pλ1qr12 ` ¨ ¨ ¨ , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='7) where the leading-order term gpλ1q obviously satisfies the algebraic equation w2pλ1, gpλ1qq ´ 1 2P ˚λ2 1 ´ N˚ “ 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8) and the function k1pλ1q can easily be found but is not required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8) determines gpλ1q uniquely under the assumption w22 ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' With the use of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='7), we may expand the integrand in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3) to order r12 and obtain E “ 2π ż L ´L ´ wpλ1, gpλ1qq ´ 1 2P ˚λ2 1gpλ1q ´ N˚gpλ1q ` 1 2 w2pλ1, gpλ1qq gpλ1q r12¯ dZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='9) This is the reduced model derived by Lestringant & Audoly (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' We now show that our general 1d model (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='20) can recover this 1d model under the limit H Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' To this end, we first note that the uniformly deformed configuration is now described by ¯r “ aR, ¯z “ λZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='10) 12 In particular, we have ¯ra “ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Since qpa, Rq and mpa, Rq involve integrals from R to B, they go to zero as H Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Consequently, the cpa, Rq defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='18) takes the simple form cpa, Rq “ ´ R aλ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='11) Taking the limit H Ñ 0 in ζ “ λw2{pλ2 ´ ¯λ2 3q yields ζ “ a2λ3w2 a2λ4 ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='12) Substituting (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='11) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='12) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='20), we obtain lim HÑ0 E1dras RH “ ż L ´L ´ wpa, λpaqq ´ 1 2P ˚a2λpaq ´ N˚λpaq ` 1 2R2 w2pa, λpaqq λpaq a1pZq2¯ dZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='13) Note that the modulus Cpaq vanishes in the membrane limit because of the equilibrium in the axial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The integrand on the right-hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='13) is the same as that on the right-hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='9) if we identify λ1, gpλ1q and r1 with apZq, λpaq, and Ra1pZq, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Solid cylinder limit Next we consider the other extreme limit corresponding to A Ñ 0 and P Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The uniform solution takes the form ¯z “ λZ, ¯r “ aR (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='14) with a “ λ´1{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The three principal stretches are ¯λ1 “ ¯λ3 “ λ´1{2, ¯λ2 “ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='15) In particular, we have w1p¯λ1, ¯λ2q “ 0, w2p¯λ1, ¯λ2q “ ˆw1pλq, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='16) where ˆwpλq “ Wpλ´1{2, λ, λ´1{2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' It follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='16)1 that qpa, Rq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Note that the deforma- tion (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='14) is homogeneous, so (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='14) implies that mpa, Rq “ A2pB2 ´ R2q R2pB2 ´ A2qmpa, Aq “ A2pB2 ´ R2q R2pB2 ´ A2qMpa, λpaqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Differentiating this expression with respect to a and noting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='11), we obtain Bmpa, Rq{Ba “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Thus cpa, Rq reduces to cpa, Rq “ ´ R λ3{2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='17) The elastic modulus ζ is easily calculated as ζ “ λ2 ˆw1pλq λ3 ´ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='18) 13 Substituting (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='17) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='18) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='20), we obtain 2πE1drλs “ ż L ´L ´ πB2 ˆwpλq ` πB4 16 ˆw1pλq λ4 λ1pZq2 ´ Nλ ¯ dZ, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='19) where we have made use of the relation a1pZq “ λ1pZq{p2λ3{2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This recovers the 1d model of Audoly & Hutchinson (2016) specialized to an incompressible circular cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Comparison with exact weakly nonlinear analysis Finally, we carry out a weakly nonlinear near-critical analysis using our 1d model and compare the resulting amplitude equation with that obtained by Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (2020) from the exact 3d theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' We focus on localized solutions in an infinitely long tube of finite wall thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Denote by a8 the limit of apZq as Z Ñ 8 and λ8 “ λpa8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' It follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='11) that P “ Qpa8, λ8q, N “ 2πA2Fpa8, λ8q, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='20) where Fpa8, λ8q is defined by Fpa8, λ8q “ Mpa8, λ8q ´ 1 2a2 8Qpa8, λ8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='21) We look for a localized solution that bifurcates from the uniform solution by writing apZq “ a8 ` ypZq, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='22) where ypZq is a small perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Substituting (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='22) into the 1d equilibrium equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='23) and expanding in terms of ypZq to quadratic order with the use of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='20), we obtain Dpa8qy2pZq “ ωpa8, λ8qypZq ` γpa8, λ8qypZq2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='23) where the two coefficient functions ωpa, λq and γpa, λq are given by ωpa, λq “ A2 2aλ a2Qλ ` 2Fλ Ωpa, λq, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='24) γpa, λq “ A2 aλpa2Qa ` 2Faq Fapa2Qλ ` 2Fλq2 Γpa, λq ` A2ψpa, λqΩpa, λq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='25) In the above expressions, Qa “ BQpa, λq{Ba, Qλ “ BQpa, λq{Bλ, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' and Ωpa, λq and Γpa, λq are defined by Ωpa, λq “ BQ Ba BF Bλ ´ BQ Bλ BF Ba , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='26) Γpa, λq “ BΩ Ba BF Bλ ´ BΩ Bλ BF Ba , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='27) and ψpa, λq is not written out as it is not required in the weakly nonlinear analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 14 The solution to the linearized equation of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='23) changes character when the sign of ωpa8, λ8q changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Thus a bifurcation occurs when ωpa8, λ8q “ 0, or equivalently, Ωpa8, λ8q “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='28) Note that Qpa8, λ8q and Fpa8, λ8q represent respectively the functional dependence of P and N on a8 and λ8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Thus the above bifurcation condition is simply the vanishing of the Jacobian determinant of P and N as functions of a8 and λ8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This is consistent with previous work Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (2016) and Yu & Fu (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' We consider two typical loading scenarios: either the resultant axial force N or the axial stretch at infinity λ8 is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The latter case is used to approximate the case of fixed axial length, which can be realized more easily experimentally or in Abaqus simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Let us first assume that the resultant axial force N “ Nc is fixed, where Nc is the prescribed axial force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Denote by pacr, λcrq the root of the system of equations ωpa8, λ8q “ 0, Fpa8, λ8q “ Nc, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='29) at which the bifurcation occurs according to the previous discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' In the vicinity of the bifurca- tion point, the amplitude equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='23) reduces to Dpacrqy2pZq “ ω1pacr, λcrqpa8 ´ acrqypZq ` γpacr, λcrqypZq2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='30) where the prime on ω denotes d{da8 “ B{Ba8 ` pB{Bλ8qpdλ8{da8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The above equation admits a localized solution of the form ypZq “ ´3ω1pacr, λcrq 2γpacr, λcrq pa8 ´ acrq sech2 ´1 2 d ω1pacr, λcrq Dpacrq pa8 ´ acrqZ ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='31) On the other hand, the weakly nonlinear amplitude equation derived from the 3d theory (Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 2020) takes the form c2 1pZq “ λ2 crk1pa8 ´ acrqc1pZq ` λ2 crk2c1pZq2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='32) where c1pZq and ypZq are related by ypZq “ kc1pZq (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='33) with k “ ´2λpaq{λ1paq|a“acr, and k1 and k2 are constants available in Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' One can see that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='30) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='32) are identical provided k1 “ ω1pacr, λcrq λ2crDpacrq , k2 “ kγpacr, λcrq λ2crDpacrq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='34) We have verified numerically that this is indeed the case, but the current expressions on the right hand sides of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='34) are more compact and revealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 15 The case of fixed λ8 can be handled similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Let pacr, λcrq be the solution to the system of equations ωpa8, λ8q “ 0, λ8 “ λc, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='35) where λc is a given constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' In the vicinity of the bifurcation point, the amplitude equation parallel to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='30) is of the form Dpacrqy2pZq “ ω1pacr, λcrqpa8 ´ acrqypZq ` γpacr, λcrqypZq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='36) where the prime on ω now signifies B{Ba8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Similar to the previous case, one can verify that the above amplitude equation is the same as its counterparts in Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Comparison with Abaqus simulations In this section, we demonstrate the power of the 1d model by applying it to investigate localized bulging in an inflated tube of finite wall thickness in the fully nonlinear regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Previous studies on this problem usually treat the tube as a finite length tube, but the problem can be analyzed more easily and very accurately by assuming the tube to be of infinite length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This assumption only fails when the tube is very short and when bulging is no longer localized in the axial direction (Wang & Fu, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The reason is that bulging solutions decay exponentially towards the two ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Thus in the following analysis, we shall assume that the tube is effectively infinite and focus on solutions subject to decaying boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This assumption is validated by comparison with Abaqus simulations based on tubes of finite lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' We shall consider the two loading scenarios discussed in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3 and compare the predictions of the 1d model with Abaqus simulations, which allows us to quantify the accuracy of our 1d model and determine its range of validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' In all numerical calculations and Abaqus simulations, we use the Gent material model W “ ´µ 2 Jm ln ´ 1 ´ λ2 1 ` λ2 2 ` λ2 3 ´ 3 Jm ¯ , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='1) where µ is the shear modulus and Jm is a material constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' We take µ “ 1 which is equivalent to scaling all stress variables by µ and Jm “ 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2 which is typical for rubber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The geometry of the tube is taken to be H{Rm “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='4 and 2L{Rm “ 40, where Rm “ pA ` Bq{2 is the average radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' In the Abaqus simulations, to ensure that localized bulging occurs in the middle of the tube, a small section with length 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='1L around the middle point of the tube is weakened by taking its shear modulus to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='9999 times that of the rest of the tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The 1d differential equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='20) subject to appropriate end conditions (see (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='7) later) can be solved numerically with the aid of the symbolic computation software Mathematica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Although the gradient modulus Dpaq involves an integral that cannot be evaluated analytically, this integral can be defined numerically in Mathematica with the built-in command ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='NumericQ and can be manipulated as elementary functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Numerically solving the 1d equation is significantly faster than Abaqus simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The 1d equation can typically be solved in a few seconds on a personal computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 16 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The case of fixed axial force We first consider the loading scenario whereby the resultant axial force N is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' As mentioned earlier, we assume that the tube is infinitely long and focus on the solution that satisfies the decaying boundary condition lim ZÑ8 apZq “ a8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2) A linear analysis shows that the solution to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='23) satisfying (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2) decays exponentially as Z Ñ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Thus we have limZÑ8 a1pZq “ 0 automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' We assume that the bulging solution is symmetric with respect to Z “ 0 so that a1p0q “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' We write λ8 “ λpa8q, a0 “ ap0q and λ0 “ λpap0qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Since pa8, λ8q satisfy the equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8), we have Mpa8, λ8q ´ 1 2a2 8Qpa8, λ8q ´ N 2πA2 “ 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3) Qpa8, λ8q ´ P “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='4) From the definition of λ0 and the conservation law (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='24), we see that pa0, λ0q satisfies Mpa0, λ0q ´ 1 2a2 0Qpa8, λ8q “ Mpa8, λ8q ´ 1 2a2 8Qpa8, λ8q, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5) Gpa0, λ0q “ Gpa8, λ8q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6) Either a8 or P can be taken to be the load parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' When a8 is specified, one can first obtain λ8 from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The associated P is computed according to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Then by solving Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5), one obtains the “initial” values a0 and λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The localized solution can be found by solving the initial value problem A2aλpaqpQpa, λpaqq ´ Pq ´ 1 2D1paqa1pZq2 ´ Dpaqa2pZq “ 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='7) ap0q “ a0, a1p0q “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8) As a first example, fixing the axial force N to be zero, we find from the bifurcation condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='29) that localized bulging takes place at a8 “ acr “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='86 with a critical pressure Pcr “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' As we trace the bifurcation solution away from the bifurcation point, the pressure drops while the bulge grows until it has almost reached a maximum amplitude after which the bulge will propagate at a constant pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' From Maxwell’s equal-areal rule, the propagation pressure is PM “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 2 shows the dependence of the pressure on ap0q and the bulging amplitude on a8 based on Abaqus simulations and use of the 1d model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The bulging solutions given by Abaqus simulations and the 1d model at the four states marked in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 2(a) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' It is seen that the 1d solution agrees well with Abaqus simulations in the entire post-bifurcation regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Remarkably, the 1d solution remains highly accurate even in the final propagation stage, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 3(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Note also that the Abaqus simulations and 1d calculations are conducted for 2L “ 40Rm and 8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This verifies our earlier claim that the tube can effectively be viewed to be infinitely long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 17 ● ● ● ● ● ● 1d model Abaqus simulation 1 2 3 4 5 6 7 a(0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='30 P a b c d (a) 1d model Abaqus simulation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='9a∞ 0 1 2 3 4 5 6 a(0) - a∞ (b) Figure 2: Dependence of (a) pressure on ap0q and (b) bulging amplitude on a8 based on Abaqus simulations and the 1d model for fixed N “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Abaqus simulation 1d model 0 2 4 6 8 Z 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='4 a(Z) (a) Abaqus simulation 1d model 0 2 4 6 8 Z 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5 a(Z) (b) Abaqus simulation 1d model 0 2 4 6 8 Z 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5 a(Z) (c) Abaqus simulation 1d model 0 2 4 6 8 Z 1 2 3 4 5 6 a(Z) (d) Figure 3: Bulging solutions given by Abaqus simulations and the 1d model at the four states marked in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 2(a) for fixed N “ 0: (a) P “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3, (b) P “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='25, (c) P “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='22, (d) P “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 18 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The case of fixed ends Next, we consider the loading scenario whereby the tube is first stretched to a specified length 2ℓ and then its two ends are fixed to prevent further axial displacement (whether the radial dis- placement is restricted or not at the ends is immaterial since the tube is assumed to be sufficiently long).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' In the previous subsection, we have solved the problem for a specified axial force N or equivalently a specified λ8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' For the current problem with a given ℓ, we define λc “ ℓ{L and we need to find λ8 such that the following condition is satisfied: ż L 0 λpapZqq dZ “ λcL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='9) This can be achieved by a shooting procedure: for each N, we compute the left-hand side using the procedure outlined in the previous subsection and adjust N such that the left-hand side and the right-hand side are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The procedure may be started by taking λ8 “ λc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' However, solving the present problem by the shooting procedure requires a lot of adjustments by hand due to the fact that the bulging solution may start to grow after decaying for a range of Z values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' To find solutions for the current case in a more robust way, we use the finite difference method instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' To implement the finite difference method, we partition the domain r0, Ls using a uniform mesh Z0, Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' , Zn with mesh size h “ L{n and coordinate of the j-th grid point given by Zj “ jh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' We use aj to represent the numerical approximation of apZjq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Applying the central difference scheme, we discretize the differential equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='7) as A2ajλpajqpQpaj, λpajqq ´ Pq ´ 1 2D1pajq ´aj`1 ´ aj´1 2h ¯2 ´ Dpajqaj`1 ´ 2aj ` aj´1 h2 “ 0, j “ 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' , n ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='10) The left boundary condition is given by a1p0q “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='11) We see from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='23) that the solution to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='7) subject to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2) has the asymptotic behavior apZq „ a8 ` a1e´κZ as Z Ñ 8, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='12) where a1 is a constant and κ “ d ωpa8, λ8q Dpa8q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Because of this, we may replace the decaying condition boundary (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2) by the “soft” asymptotic condition a1pLq ` κpapLq ´ a8q “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='13) 19 To avoid the loss of accuracy at the two endpoints, we introduce two additional unknowns a´1 and an`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Then the left and right boundary conditions yield a1 ´ a´1 2h “ 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='14) an`1 ´ an´1 2h ` κpan ´ a8q “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='15) Solving for a´1 and an`1 from the above equations, and substituting them into the difference equations (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='10) at j “ 0 and j “ n, we obtain the discrete boundary conditions with truncation errors of order h2: A2a0λpajqpQpa0, λpa0qq ´ Pq ´ 2Dpa0qa1 ´ a0 h2 “ 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='16) A2anλpanqpQpan, λpanqq ´ Pq ´ 1 2D1pajqκ2pan ´ a8q2 ´ 2Dpanqan´1 ´ an ´ hκpan ´ a8q h2 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='17) Finally, the fixed-length restriction (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='9) gives 1 2λpa0q ` n´1 ÿ j“1 λpajq ` 1 2λpanq ´ λcL h “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='18) We use the pressure P as the loading parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' When P is given, one can first solve (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='4) to express a8 as a function of λ8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Then N can be viewed as a function of λ8 due to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' It follows that λpµq and Dpµq also depend on λ8 through their dependence on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This implicit dependence should be considered when solving the above algebraic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Setting n to be a sufficiently large number, say n “ 100, and solving the system of nonlinear algebraic equations consisting of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='10), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='16), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='17) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='18) for aj’s and λ8 with a suitable initial guess, we obtain the finite-difference solution for the present problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' We may use the weakly nonlinear solution with fixed λ8 “ λc “ ℓ{L as an initial guess in the near-critical regime and continue the solution to the fully nonlinear regime by always using the solution at the previous step as the initial guess for the current step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' When the total length is fixed to be ℓ “ 2L, then initially λ8 “ 2 and localized bulging takes place at a8 “ acr “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='74 with a critical pressure Pcr “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='198 according to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 4, we have shown the dependence of the pressure on ap0q and the bulging amplitude on a8 based on Abaqus simulations and use of the 1d model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The bulging solutions determined by Abaqus simulations and the 1d model at the four states indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 4(a) are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' It is observed that the agreement between the 1d model and Abaqus simulations is again excellent in the fully nonlinear regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Finally, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 6 shows the actual variation of P against ap0q predicted by the 1d model when L is varied and the averaged stretch λc is fixed or λc is varied but L is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' These results confirm the theoretical prediction of Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (2022) that the right branches of these curves all converge 20 ● ● ● ● ● ● ● ● 1d model Abaqus simulation 0 1 2 3 4 5 6 7 a(0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='20 P a b c d (a) 1d model Abaqus simulation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8a∞ 1 2 3 4 5 6 a(0) - a∞ (b) Figure 4: Dependence of (a) pressure on ap0q and (b) bulging amplitude on a8 based on Abaqus simulations and using the 1d model for fixed length ℓ{L “ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Abaqus simulation 1d model 0 2 4 6 8 10 12Z 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8 a(Z) (a) Abaqus simulation 1d model 0 2 4 6 8 10 12Z 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5 a(Z) (b) Abaqus simulation 1d model 0 2 4 6 8 10 12Z 1 2 3 4 5 a(Z) (c) Abaqus simulation 1d model 0 2 4 6 8 10 12Z 1 2 3 4 5 6 7 a(Z) (d) Figure 5: Bulging solutions based on Abaqus simulations and the 1d model the at the four states indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 4(a) for fixed length ℓ{L “ 2: (a) P “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='19, (b) P “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='18, (c) P “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='173, (d) P “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' to a master curve that is independent of L or λc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' These curves all terminate at the point where the axial stress near each end of the tube has become compressive enough so that secondary Euler buckling or axisymmetric wrinkling becomes possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 21 L=15 L=20 L=40 0 1 2 3 4 5 6 7 a(0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='22 P (a) λc=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5 λc=2 λc=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8 0 1 2 3 4 5 6 7 a(0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='25 P (b) Figure 6: Variation of P against ap0q predicted by the 1d model when (a) the total length is fixed with λc “ 2 and L “ 15, 20 and 40, respectively and (b) L is fixed at 20 and λc “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5, 2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Conclusion We have derived a 1d model for the analysis of axisymmetric deformations of an inflated cylin- drical tube of finite wall thickness, and established its range of validity by comparing its predictions with those of Abaqus simulations for two typical loading scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The comparison shows that the 1d model performs extremely well in both the near-critical and fully nonlinear regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The dimension reduction started from three-dimensional finite elasticity theory and is performed in terms of the energy functional and principal stretches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' A key ingredient of the dimension reduc- tion is the assumption of slow variation of the leading-order solution in the axial direction without any restriction on its amplitude, which results in a 1d model that is simple but is still capable of capturing the strain-gradient effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This is in contrast with the traditional asymptotic analysis where the leading order solution is assumed to be a small-amplitude perturbation from the primary deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' It is because of this difference that the 1d model has a much larger range of validity than the expansion methods around the bifurcation point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The nonlinearity of the strain is kept in the 1d model, reflected by the nonlinear potential Gpa, λpaqq and the nonlinear strain gradient modulus Dpaq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Our expression for the strain gradient coefficient Dpaq is quite simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' For the Gent material model, Dpaq can be calculated by integrating once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' We remark that although the derivation presented in this work is variational, the 1d model can also be derived by substituting the asymptotic solution (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2) into the 3d governing equations and solving the resulting equations at successive orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The 1d model is amenable to asymptotic and numerical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' The bifurcation condition and the weakly nonlinear amplitude equation predicted by the model are exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' In fact, the expressions (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='24) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='25) derived using the 1d model are more compact and more revealing than their counterparts in Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' A major advantage of the 1d model is that the entire evolution process of bulging or necking can be determined using the finite difference method which is more accessible and much easier to implement than commercial packages such as Abaqus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' This advantage 22 would become even more significant when other fields such as electric loading and residual stresses were also present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Such extra fields and new geometries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' axisymmetric necking of a stretched plate (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=', 2022) ) will be considered in our future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' A Mathematica code that produces all the results presented in the paper is available on GitHub (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='com/yfukeele).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant No 12072224) and the Engineering and Physical Sciences Research Council, UK (Grant No EP/W007150/1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' Simplifying the one-dimensional energy functional Substituting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='19) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='11), we can write the integral of KpR, v, vRq as ż B A KpR, v, vRq dR “ pI1 ` I2 ` I3qa12, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='1) where I1 “ ż B A pλ´1¯σ22qaR ż B R cpa, Tq dT dR, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='2) I2 “ 1 2 ż B A Rζp¯r2 a ` cpa, Rq2q dR, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='3) I3 “ ż B A Rξ¯racpa, Rq dR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='4) By interchanging the order of integration, we can rewrite I1 as I1 “ ż B A ż B R pλ´1¯σ22qaRcpa, Tq dT dR “ ż B A ż T A pλ´1¯σ22qaRcpa, Tq dR dT “ ż B A cpa, Tq B Ba ´ ż T A λ´1¯σ22R dR ¯ dT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5) From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='14), we have ż T A λ´1¯σ22R dR “ A2mpa, Aq ´ T 2mpa, Tq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6) Inserting (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='6) into (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='5) and noting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='15)2 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='11), we can simplify I1 as I1 “ PA2a ż B A cpa, Rq dR ´ ż B A cpa, RqR2 B Bampa, Rq dR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='7) 23 Noting that ξ “ q¯λ1 ` ¯λ3ζ{λ, the integral I3 can be calculated as I3 “ ż B A R ´ q¯λ1 ` ¯λ3 λ ζ ¯ ¯racpa, Rq dR “ ż B A ´ ¯r¯raq ` Rζ ¯ra ¯λ1λ2 ¯ cpa, Rq dR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='8) Adding up the three integrals, we obtain ż B A KpR, v, vRq dR ` PA2aa1v|R“A “pI1 ` I2 ` I3qa12 ´ PA2aa12 ż B A cpa, Rq dR “a12 ż B A ´ ´ cpa, RqR2 B Bampa, Rq ` 1 2Rζp¯r2 a ` cpa, Rq2q ` ´ ¯r¯raqpa, Rq ` Rζ ¯ra λ1λ2 ¯ cpa, Rq ¯ dR “a12 ż B A p1 2Rζp¯r2 a ` cpa, Rq2q ´ Rζcpa, Rq2q dR “1 2a12 ż B A Rζp¯r2 a ´ cpa, Rq2q dR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE1T4oBgHgl3EQfBAIU/content/2301.02845v1.pdf'} +page_content='9) This gives the expression of the coefficient Dpaq announced in (4.' metadata={'source': 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b/79E3T4oBgHgl3EQfRwk1/content/tmp_files/2301.04424v1.pdf.txt @@ -0,0 +1,1399 @@ +RIEMANNIAN GEOMETRY AND MOLECULAR SIMILARITY II: +KÄHLER QUANTIZATION +A PREPRINT +Rachael Pirie +School of Natural and Environmental Sciences +Newcastle University +r.pirie2@ncl.ac.uk +Stuart J. Hall +School of Mathematics, Statistics, and Physics +Newcastle University +stuart.hall@ncl.ac.uk +Daniel J. Cole +School of Natural and Environmental Sciences +Newcastle University +daniel.cole@ncl.ac.uk +Thomas Murphy +Department of Mathematics +California State University, Fullerton +tmurphy@fullerton.edu +January 12, 2023 +ABSTRACT +Shape-similarity between molecules is a tool used by chemists for virtual screening, with the goal of +reducing the cost and duration of drug discovery campaigns. This paper reports an entirely novel +shape descriptor as an alternative to the previously described RGMolSA descriptors [1], derived from +the theory of Riemannian geometry and Kähler quantization (KQMolSA). The treatment of a molecule +as a series of intersecting spheres allows us to obtain the explicit Riemannian metric which captures +the geometry of the surface, which can in turn be used to calculate a Hermitian matrix M as a directly +comparable surface representation. The potential utility of this method is demonstrated using a series +of PDE5 inhibitors considered to have similar shape. The method shows promise in its capability to +handle different conformers, and compares well to existing shape similarity methods. The code and +data used to produce the results are available at: https://github.com/RPirie96/KQMolSA. +Keywords Riemannian Geometry · Kähler Quantization · Molecular Shape · Ligand-Based Virtual Screening +1 +Introduction and Summary of Part I +The concept that shared biological activity exists between similar molecules is used widely in drug discovery [2]. +Molecules with known activity can be used as templates to screen large databases for other potential hits. This is +more efficient and allows coverage of a greater area of chemical space than is possible with experimental screening +alone [3]. Estimating similarity between molecules based on their 3D shape has gained popularity due to the +requirement for protein-drug shape complementarity to enable strong binding. However no fixed notion of shape +exists. Instead, comparison relies on mathematical approximation of the molecule’s shape based on its volume, +distribution of atomic distances or surface (most commonly treated as the van der Waals or solvent accessible surface) [4]. +In the accompanying paper [1], the RGMolSA method was presented. +The descriptor developed there ap- +proximates the shape of the molecular surface using a simple nine-element vector containing the surface area and an +approximation to the first eight non-zero eigenvalues of the ordinary Laplacian. The descriptor can be viewed as an +approximation to the Riemannian metric, the underlying mathematical object that describes the shape of a surface. +In this paper we present an entirely different method of approximating the Riemannian metric by using ideas from +the theory of Kähler quantization; we call this method Kähler quantization for Molecular Surface Approximation +(KQMolSA). The theory was originally developed by mathematicians and string theorists in order to give explicit +representations of the shapes of 4-dimensional objects (Calabi–Yau manifolds) that appear in physical theories (see [5] +arXiv:2301.04424v1 [math.DG] 11 Jan 2023 + +Geometry and Molecular Surfaces +A PREPRINT +for the paper that pioneered its use as a numerical technique). In a nutshell, a function called the Kähler potential +is associated to the metric. We then compute something analogous to a Taylor expansion of this function with the +coefficients being stored in a Hermitian matrix. While the matrices themselves do depend upon the precise position and +parameterisation of the molecular surface in three-dimensional space R3, the dependence is easy to calculate. Hence we +can perform our calculations in the ‘quantized’ space of Hermitian matrices and assign a distance between the shapes of +two molecular surfaces this way. The final distance is independent of the position of the molecules and the choices +made in their parameterisations. +1.1 +Summary of Previous Work +As in the accompanying paper [1], our approach begins by treating the molecule as a series of intersecting spheres, with +their radii given by the van der Waals radii of the constituent atoms. The surface is assumed to have a genus of zero, so +any rings (e.g. benzene) are replaced with a single sphere of radius 2.25 Å to facilitate this. The molecular structure +is then defined by the number of spheres N (with each ring counted as a single sphere, and excluding any hydrogen +atoms), the centres ci and radii ri for each sphere and the adjacency matrix T describing intersection of spheres, where +Tij = +� +1 +if spheres i and j intersect +0 +otherwise (or i = j). +The surface area A of the molecule is calculated as the area of each sphere minus the “missing parts" where two spheres +intersect: +A = 2π +� +i +� +�2r2 +i − +� +�ri +� +j +Tij|ri − λij| +� +� +� +� . +(1) +This value is used to re-scale each of the starting constructs such that the surface area of the molecule is equal to that of +a unit sphere (or 4π) to address the observation that Riemannian geometry treats two objects which differ only in size +as having equivalent shape. This re-scaling is accounted for in the final descriptors with some weighting so as not to +dominate the similarity calculation. +From the initial data, a map is constructed to ‘unwrap‘ the surface onto the complex plane C in a process +we refer to as piecewise stereographic projection. This requires an atom to be selected as a starting point from which to +construct our map, which we refer to as the base sphere. This is taken to be the atom closest to the centre of mass by +first finding the centroid of the molecule and then taking the atom with the smallest Euclidean distance from this point. +The Riemannian metric g = Φ∗ +ps(gEuc) induced by the mapping Φps : C → S ⊂ R3 takes the form +g = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +4r2 +B +(1+|z|2)2 (dx2 + dy2) +if z ∈ C +C1 +(|z−A1|2+B1)2 (dx2 + dy2) +if z ∈ D(a1, R1), +C2 +(|z−A2|2+B2)2 (dx2 + dy2) +if z ∈ D(a2, R2), +... +... +CN−1 +(|z−AN−1|2+BN−1)2 (dx2 + dy2) +if z ∈ D(aN−1, RN−1), +(2) +where rB is the radius of the base sphere and +C = C\D(a1, R1) ∪ D(a2, R2) ∪ · · · ∪ D(aN−1, RN−1), +is the complement of the discs D(a1, R1), . . . , D(aN−1, RN−1) which corresponds to the points in the base sphere. +The RGMolSA descriptor uses the explicit form of the Riemannian metric provided by piecewise stereo- +graphic projection to approximate the low-lying eigenfunctions of the Laplacian ∆. In [1], we compared the RGMolSA +descriptor for Sildenafil, Vardenafil and Tadalafil, a series of PDE5 inhibitors that are known to occupy a similar volume +in the binding pocket of their target protein, and thus have similar shape (Figure 1) [6]. Vardenafil is a classic example +of a “me-too" drug, where only a few small modifications have been made to the structure of Sildenafil. As these are +both highly similar chemically, they would be expected to have close to the same shape. Tadalafil on the other hand is +chemically quite different from the other two, but inspection of the molecules in the pocket of PDE5 reveals they occupy +a similar binding pose, and thus would also be expected to have similar shape. In this article, for ease of compari- +son with the previous work [1], we again use these three molecules as the basis for investigating the new shape descriptor. +2 + +Geometry and Molecular Surfaces +A PREPRINT +(a) Sildenafil +Pfizer +First Sold: 1998 +(b) Vardenafil +Bayer +First Sold: 2003 +(c) Tadalafil +Lilly +First Sold: 2003 +Figure 1: PDE5 inhibitors Sildenafil, Vardenafil and Tadalafil of known shape similarity. Tadalafil (different chemical structure, similar shape) is an example of a scaffold +hop from the first in class drug Sildenafil, and offers greatly improved performance, while Vardenafil (a "me-too" follow-up drug) only offers minor improvements. +While RGMolSA was found to give a good description of shape, it has a possible deficiency due to the dependence +of the results on the choice of base sphere, which in turn determines the trial functions for calculating the integrals +used to construct the descriptor. The geometry of the surface near the base sphere is well described, but for atoms +further away a greater number of eigenvalues would be needed for accurate description of the surface. This problem is +greater for larger molecules and can lead to the introduction of numerical errors when the molecule is large enough. +We handled such errors by ignoring any contributions from regions with numerical radii less than 10−9; however, this +forces a somewhat artificial ‘locality’ upon the shape descriptor meaning that it probably only accurately captures the +shape near to the base sphere. +In the following section we outline the theory underpinning the KQMolSA descriptors, that again uses the Riemannian +metric to approximate the geometry of the surface. The resulting descriptors lie in the manifold GL(N, C)/U(N) to +give a global descriptor of molecular geometry with reduced dependence on the starting position. Figure 2 summarises +the steps in computing these, using Sildenafil as an example. While the descriptor itself does depend upon the choices +made and the position of the surface within R3, this is easily accounted for within the space GL(N, C)/U(N). This +makes computing the ‘distance’ between the shape descriptors particularly straightforward. +2 +The Mathematics of Kähler Quantization +2.1 +Overview of the Theory +We should say immediately that the theory of Kähler quantization is far too advanced to be able to detail in the current +paper. For readers with sufficient mathematical background, a good account (and the original account of its use as +a numerical technique) is given in [5]. An exposition, aimed at readers with a general scientific background, of the +mathematical theory is currently being written by two of the authors [7]. +The theory is concerned with the geometry of complex manifolds (shapes that locally look like Cn); any sur- +face that sits in R3 is a complex manifold as it locally looks like a copy of the complex numbers C (i.e. n = 1). More +concretely, we will be concerned with the surfaces that are topologically equivalent to S2; in the language of complex +manifolds, the sphere is often referred to as the Riemann Sphere and denoted CP1. The restriction on the topology +of the surface is justified by the fact that chemists do not expect any activity in the centre of rings occurring in most +3 + +N +N +HN +N +N +NN +HN +N +N +N +NH +N +N +N +010000Geometry and Molecular Surfaces +A PREPRINT +Figure 2: Key steps involved in the computation of the KQMolSA surface descriptor for Sildenafil (a PDE5 inhibitor). +drug-like molecules. The exceptions to this are macrocyclic molecules (those with large rings of more than 12 atoms) +where genuine activity occurs in the centre of the ring. Such molecules are therefore excluded from comparison by both +methods proposed. +The natural class of functions to work with when dealing with complex manifolds are those that are com- +plex differentiable, often called holomorphic functions. We consider a general complex manifold X; unfortunately, +if the manifold X is compact, the only holomorphic functions f : X → C are constant. Thus we cannot hope to +understand X simply by studying the holomorphic functions on X. A generalisation of the notion of a holomorphic +function is that of a section of a holomorphic line bundle L with base X. For readers familiar with the theory, a function +is a section of the trivial bundle. A line bundle is positive if there is a Hermitian metric h on L with positive curvature. +A foundational result of Kodaira [8] says that if the line bundle L is positive then for large enough k the tensor power +Lk, has a lot of holomorphic sections. In fact, the space of all such sections, denoted H0(Lk), is a complex vector +space of dimension that has order O(kn) as k → ∞. +The curvature of a positively curved Hermitian metric h gives rise to an object called a Kähler form, ω, +which in turn gives rise to a Riemannian metric g (the mathematical object being used in [1] to describe shape). It +turns out that the set of all positively curved Hermitian metrics on a line bundle L can be identified with the set of all +real-valued functions ϕ : X → R that satisfy, in some local coordinate z, the ∂ ¯∂-equation +√ +−1∂ ¯∂ϕ = ω − ω0 +where ω is the Kähler form of the metric and ω0 is a fixed reference Kähler form. We will give more detail on the +differential operators ∂ and ¯∂ in Section 2.3; in particular, we will explain that in the molecular surface setting, the +∂ ¯∂-equation is really just the familiar Poisson equation in the plane. The function ϕ is called a Kähler potential for +ω. The associated potential is not unique but any two differ by a constant; this does not affect the metric which is +constructed by taking two derivatives of the potential. However, we will see that the addition of a constant to a potential +will have the affect of scaling the Hermitian matrix we produce as a shape descriptor by a positive real number and we +will be required to find the ‘optimal’ rescaling in our distance calculation. +To summarise, what we have for a positive Hermitian line bundle (L, h) → X are: +• a Kähler form ω and a Kähler potential ϕ : X → R, +• a complex vector space H0(Lk). +4 + +e.g. Sildenafil +Space Filling Model +Replace Rings, Base Sphere (Grey) +- AiilD +01 +Map to Complex Plane +Surface Area +Matrix of Levels +2r? +ifz εc +(1 + [z/2)2 +C1 +2.16 + 0j +-2.44 + 0.86j +2.41 - 1.93j 1 +ifz E D(ai,R1) +F(z) = +zje-kF(z)V-1dzΛdz +M +-2.44 - 0.86j +3.01 + 0j +(lz - A1/2 + B1)2 +-3.48 + 1.22j +MI +-3.48 - 1.22j +: +2.41 + 1.93j +4.41 + 0j +Cn-1 +ifz E D(an-1,Rn-1) +(Iz - An-1/2 + Bn-1)2 +Riemannian Metric +Construct Hermitian Matrix +Hermitian Matrix Shape DescriptorGeometry and Molecular Surfaces +A PREPRINT +What Kähler quantization amounts to is relating the geometry described by the Kähler potentials (an infinite dimensional +space of functions) to the finite dimensional complex vector space H0(Lk). This theme occurs throughout numerical +analysis and shape description, for example in the theories of Fourier analysis, spherical harmonics, Taylor series, all of +which produce a finite-dimensional vector space out of some infinite-dimensional set of functions. +2.2 +Quantization and Tian’s Theorem +The data (L, h) → X allows for a natural L2-inner product on the vector space of sections H0(Lk). Given sections +s1, s2 ∈ H0(Lk), we compute +⟨s1, s2⟩ := +� +X +hk(s1, s2)ωn +n! , +where hk is the Hermitian metric induced on Lk by h, and ωn/n! is the volume element produced by the Kähler form. +It is this inner product that is the quantization of the data (L, h) → X. The space of all (Hermitian) inner products on a +complex N-dimensional vector space can be thought of as GL(N; C)/U(N). This is a negatively curved symmetric +space and has a natural notion of distance on it; it is this distance that we will use to measure shape similarity (see +Section 2.5). +To recover the geometry defined by (L, h) → X from the quantization, we choose a basis {sj} of the vec- +tor space H0(Lk) which gives rise to the matrix representation of the inner product +Mij := ⟨si, sj⟩. +If we let v be the vector of sections +v = (s1, s2, . . . sN) , +then we can define a Kähler potential (recalling that the sections are locally defined holomorphic functions) ˜ϕ by +˜ϕ := −1 +k log +� +v∗M−1v +� +. +Theorem 2.1 (Tian, [9]). Let (X, L, h) be a complex manifold with holomorphic line bundle L and positively curved +Hermitian metric h with curvature ω. If we produce another Kähler form +�ω = ω0 + +√ +−1∂ ¯∂ ˜ϕ, +then +∥ω − �ω∥C0 = O(k−2). +Paraphrasing this theorem, we can say any Kähler form coming from a Kähler potential ϕ can be well approximated by +the Kähler form coming from the ‘algebraic’ function �ϕ. If we pick local complex coordinates z1, z2, . . . , zn then the +term v∗M−1v is just a power series in the coordinates. In the case of a molecular surface, we will have something like a +polynomial. This is the sense in which the function �ϕ is similar to a truncated Taylor series for the original function ϕ. +The theorem then says that this series really does converge. +Tian’s Theorem is stated for smooth metrics (those where one can take an arbitrary number of derivatives of +the Kähler potential ϕ); in practice (see Section 2.3), we will be working with metrics where the potentials are in +C2(X), that is twice continuously differentiable. The theory of approximating such metrics algebraically has not +been written down but we will demonstrate that we get a method that does produce meaningful shape comparisons. +We expect that, suitably adapted to this setting, something like Tian’s Theorem is still true; for example, the case of +potentials with lower regularity is discussed in [10]. +2.3 +Implementation in Practice +As mentioned already, in practice we take X = CP1 the Riemann sphere and the line bundle to be the anticanonical +bundle K∗ +CP1 = O(2). The Kähler form ω, can be explicitly constructed from the Riemannian metric g, and in the +coordinates furnished by the piecewise stereographic projection map Φps, we can use the form of the metric (2) to get +ω = F(z) +√ +−1dz ∧ dz, +5 + +Geometry and Molecular Surfaces +A PREPRINT +where F : C → R+ is the ‘metric function’ given by +F(z) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +2r2 +B +(1+|z|2)2 +if z ∈ C, +C1 +(|z−A1|2+B1)2 +if z ∈ D(a1, R1), +C2 +(|z−A2|2+B2)2 +if z ∈ D(a2, R2), +... +... +CN−1 +(|z−AN−1|2+BN−1)2 +if z ∈ D(aN−1, RN−1). +(3) +Note we have replaced, in the metric g, the real symmetric 2-tensor dx2 + dy2 with the antisymmetric form +(√−1/2)dz ∧ d¯z, where dz = dx + √−1dy and d¯z = dx − √−1dy. +To find the Kähler potential ϕ : C → R, we solve the ‘∂∂-equation’ +ω = +√ +−1∂∂ϕ. +If we consider the complex differential operators +∂ +∂z = 1 +2 +� ∂ +∂x − +√ +−1 ∂ +∂y +� +and +∂ +∂z = 1 +2 +� ∂ +∂x + +√ +−1 ∂ +∂y +� +, +then the ∂∂-equation is equivalent to solving the Poisson equation +∂2ϕ +∂z∂z = 1 +4∆Eucϕ = F, +where ∆Euc is the usual 2-dimensional Laplacian. We can solve the Poisson problem explicitly to find ϕ. The solution +can be thought of as having two parts: a ‘local’ part that is found by simply observing that +∂2 +∂z∂z +�C log(|z − A|2 + B) +B +� += +C +(|z − A|2 + B)2 , +and a ‘correction term’, named thus as the term is needed to ensure the function is in C2(C). The correction term is a +linear combination of functions of the form +log(|αz + β|2), +where we get one term for each sphere. As each of the correction terms is a harmonic function, that is +∆ log(|αz + β|2) = 0, +the addition of the correction terms is still a solution of the Poisson equation. It would appear the correction terms +are singular at the points z = −β/α; however, these points always lie outside the disc where the function takes this +particular form. We record the form of the potential as a theorem and refer the reader to the appendix (Section 5) for a +derivation of the solution. +Theorem 2.2 (Form of Kähler potential). Let g be of the form Equation (2). In the region associated to the ith sphere, +the Kähler potential can be written +ϕ(z) = Ci +Bi +log(|z − Ai|2 + Bi) + +N +� +j=1 +Kij log(|αijz + βij|2), +where K ∈ M N×N(R), and α, β ∈ M N×N(C). +The matrices K, α, and β in the previous theorem are easily calculated from the geometric data associated to the +molecule and so it is straightforward to describe the Kähler potential explicitly. +The space of global sections H0(O(2k)) ∼= C2k+1 can be identified with the span of the functions +⟨1, z, z2, . . . , z2k⟩. +Thus the shape descriptor associated to the surface is the (2k + 1) × (2k + 1) Hermitian matrix M where (considering +indices that run from 0 to 2k) +Mij = +�� +C +zizje−kϕF(z) +√ +−1dz ∧ dz. +(4) +6 + +Geometry and Molecular Surfaces +A PREPRINT +2.4 +Computing the Relevant Integrals +A naïve numerical calculation of the integrals described by Equation (4) gives rise to two obvious problems: +firstly, the domain of integration is unbounded (being the whole complex plane C); secondly, the domains and +values describing the metric and the Kähler potential ϕ could become so small that numerical instabilities start +to dominate the contribution of the associated atom. +The second problem has been discussed as a limitation +in the approximation of the spectrum of the Laplacian [1]. In this paper, we exploit the fact that the automor- +phism group of CP1 is the group of Möbius transformations, PSL(2, C); we can use elements of this group to +ensure the coordinates we perform calculations in are always in a numerically controlled region (here we use a unit disc). +Put more concretely, let m ∈ {1, 2, . . . , N} index the mth sphere making up the molecular surface, then +there is an element Tm ∈ PSL(2, C) that maps the unit disc +D = {z ∈ C | |z| < 1}, +onto the region D(am, Rm) from Equation (2). We note that if the mth sphere has level l, then the pre-image of the +regions corresponding to level (l + 1) spheres which intersect the mth sphere will describe certain discs properly +contained in D. Hence the contribution of the mth sphere to the matrix described by Equation (4) is given by +�� +D− ˆ +D +(Tm(w))i(Tm(w))je−kϕ(Tm(w))F(Tm(w)) dTm(w) ∧ dTm(w), +(5) +where ˆD represents the union of the discs corresponding to the next level spheres intersecting the mth sphere. In practice, +we account for these higher-level spheres by assigning the value 0 to the volume form F(Tm(w)) dTm(w) ∧ dTm(w) +whenever w ∈ ˆD (note this produces a jump discontinuity in the volume form). Numerical calculation of integrals +of the form of Equation (5) is done by splitting into an angular and radial direction and then performing successive +applications of the trapezium rule; we choose a radial step size corresponding to nr = 15 integration points and an +angular step size corresponding to taking nθ = 10 points. This seems to achieve a reasonable accuracy; for example, +one can check the area integral for a given integration scheme. We have also determined that the distance between +shape descriptors does not seem to be significantly changed by taking smaller step sizes (Section 3.1). +2.5 +Finding the Distance Between Shape Descriptors +Given two positive definite Hermitian matrices M1, M2, such as those generated by Equation (4), there are innumerable +ways of defining a notion of distance between such matrices. With regards to the theory of Kähler quantization, it is +natural to consider M1, M2 as two Hermitian inner products on the fixed complex vector space H0(O(2k)). This space +is naturally seen as the manifold GL(2k + 1; C)/U(2k + 1). An inner product is specified by declaring a particular +basis to be orthonormal; any basis conjugate under the action of U(2k + 1) defines the same inner product. This +space has a natural distance on it; one characterisation of this distance is that shortest paths (geodesics) are given +by one-parameter subgroups of GL(2k + 1; C), that is by paths of matrices of the form exp(tA) where A is some +(2k + 1) × (2k + 1) complex matrix. +More explicitly, if {v1, v2, . . . , v2k+1} is a basis of H0(O(2k))such that both inner products are represented +by diagonal matrices +M1 = Diag +� +eλ1, eλ2, . . . , eλ2k+1� +, +M2 = Diag (eµ1, eµ2, . . . , eµ2k+1) , +then +d(M1, M2) = k− 3 +2 +� +� +� +� +2k+1 +� +i=1 +(λi − µi)2. +(6) +The factor of k−3/2 ensures that the distances stabilise as k → ∞ (see Theorem 1.1 in [11]). It will be useful to consider +the following more compact form for the distance +d(M1, M2) = k− 3 +2 +� +� +� +� +2k+1 +� +i=1 +(log(ηi))2, +(7) +where {ηi} are the eigenvalues of the matrix M−1 +1 M2. +7 + +Geometry and Molecular Surfaces +A PREPRINT +It is a well-known fact that the automorphism group of the Riemann sphere CP1 is the group of Möbius +transformations PSL(2, C). Roughly speaking, the subgroup PSU(2) ⊂ PSL(2, C) corresponds to rotations of the +original surface and the remaining maps correspond to reparameterisations that preserve the complex structure. If +ϖ ∈ PSL(2, C) is an automorphism of the form +ϖ(z) = αz + β +γz + δ , +then ϖ also acts on the vector space H0(O(2k)). In representation theoretic terms, this action is the representation +induced on Sym2k(C2) by the standard representation of SL(2, C). If we denote the element of SL(2k+1, C) by ϑ(ϖ) +(see [12], Lemma 8) and the original shape descriptor computed in the z-coordinate by M, then the shape descriptor +computed in the ϖ(z)-coordinate will be +(ϑ(ϖ))∗ M (ϑ(ϖ)) . +As mentioned in Section 2, the fact that the Kähler potential is only defined up to the addition of a constant means we +can also scale the Hermitian matrix M by a positive constant. Hence our calculation of distance between two shape +descriptors M1 and M2 becomes the concrete problem of minimising, over (p, ϑ) ∈ R × SL(2, C), +ζ(p, ϑ) = +2k+1 +� +i=1 +(log(ηi))2, +where {ηi} are the eigenvalues of the matrix M−1 +1 ep (ϑ(ϖ))∗ M2 (ϑ(ϖ)). +It is easy to see that the value of p at a critical point of ζ is independent of the element ϑ. +Elementary +calculus yields that the value of p is given by +p = − +1 +2k + 1 +2k+1 +� +i=1 +log(˜ηi), +where {˜ηi} are the eigenvalues of the matrix M−1 +1 M2. As the matrix (ϑ(ϖ)) has unit determinant, the value of p +does not depend up the SL(2, C) action on the Hermitian matrix M2. We thus reduce the distance calculation to a +minimisation over the six-dimensional Lie group SL(2, C). +Note that the distance between the shape descriptors given by Equation (6) is the distance between the molecular shapes +after they have been re-scaled to have area 4π. Hence the distance between two molecular surfaces S1 and S2 should +include a component to reflect the difference in area between S1 and S2. As we are interested in producing a similarity +score rather than a distance between two inputs, we do not take this point up further in the article. Our initial attempts at +creating a similarity score are detailed in the subsequent section. +The remaining minimisation over SL(2, C) is done by parameterising a generic matrix by the 6 real variables x1, ...x6 +and taking +ϖ(x1, x2, . . . , x6) = +� +x1 + √−1x2 +x3 + √−1x4 +x5 + √−1x6 +∗ +� +, +where ∗ is chosen to ensure det(ϖ) = 1. To perform the minimisation, we use algorithms that do not require the +input of a gradient vector, such as Nelder–Mead or Powell methods [13]. These are implemented using off-the-shelf +packages in SciPy [14]. We found that for k = 1 there was very little difference between the results for either +method; the minimisation algorithm converges to produce a robust distance value. For k = 2 the minimisation +methods appear to be a little less stable and occasionally did not converge. One way around this was to use the +element of SL(2, C) found by the k = 1 minimisation as the initial guess for the k = 2 step (otherwise the identity +matrix was used). We anticipate that one might be able to improve this process; for example, by computing the +gradient of the function to be minimised explicitly and then using this in an algorithm such as conjugate gradient descent. +One further consideration in implementing the distance measure between two matrices was in shape descrip- +tors for k > 2 (and for k = 2 in some cases), where numerical instability exists within the method. Occasionally +non-positive definite matrices are produced, that cannot be compared using the above approach. As Hermitian matrices +that differ only by scale can be considered equivalent, such cases have been treated by scaling one matrix by a factor of +10, 100 or 1000 as needed in order to bring the eigenvalues into the range required for consideration with Python. +8 + +Geometry and Molecular Surfaces +A PREPRINT +3 +Initial Case Study: Phosphodiesterase 5 (PDE5) Inhibitors +3.1 +Tuning the Parameters nr and nθ +To determine the effect of varying the parameters nr and nθ (Section 2.4) on the quality of the shape descriptors +produced, we considered three sets of parameters: nr = 200 and nθ = 100; nr = 50 and nθ = 25; nr = 15 and +nθ = 10. The distances produced between the descriptor for each set and the area returned during the computation of +the relevant integrals (which should be ∼ 12.57 for an accurate descriptor, as constrained by the choice of scaling the +surface area to 4π) are reported here for Sildenafil (Table 1), Vardenafil (Table 2) and Tadalafil (Table 3). +Table 1: Computed distances between descriptors of Sildenafil generated using different values of nr and nθ for k = 1. The area reported is that returned by the +integration step. +(200, 100), area = 12.59 +(50, 25), area = 12.62 +(15, 10), area = 12.62 +(200, 100) +- +0.032 +0.032 +(50, 25) +0.038 +- +0.040 +(15, 10) +0.038 +0.040 +- +Table 2: Computed distances between descriptors of Vardenafil generated using different values of nr and nθ for k = 1. The area reported is that returned by the +integration step. +(200, 100) area = 12.57 +(50, 25), area = 12.58 +(15, 10), area = 12.58 +(200, 100) +- +0.005 +0.005 +(50, 25) +0.005 +- +0.004 +(15, 10) +0.005 +0.004 +- +Table 3: Computed distances between descriptors of Tadalafil generated using different values of nr and nθ for k = 1. The area reported is that returned by the +integration step. +(200, 100), area = 14.32 +(50, 25), area = 14.37 +(15, 10), area = 14.37 +(200, 100) +- +0.003 +0.003 +(50, 25) +0.003 +- +0.001 +(15, 10) +0.003 +0.001 +- +As these distances are small in each case, there is no significant loss of quality when the number of points considered is +reduced. The areas for both Sildenafil and Vardenafil are also close to 12.57, indicating high quality descriptors. The +area for Tadalafil is overestimated slightly, however this is due to an issue with the replacement of the rings for motifs +with a 5-membered ring between two other rings rather than the choice of nr and nθ. Similar results were observed for +the consideration of k = 2. As the quality is unaffected, the minimum parameters of nr = 15 and nθ = 10 were used +in the final descriptors to increase the speed of calculation. +3.2 +Constructing a Similarity Score +In order to facilitate familiar comparison of molecules, we wish to construct a similarity score rather than simply taking +the distance between two matrices. In chemoinformatics, this score typically takes a value between 0 (no similarity) +and 1 (identical) [4]. To achieve this we take the inverse distance, and account for size by taking the ratio of two surface +areas. Equation 8 gives the similarity score between two molecular surfaces S1 and S2, +score(S1, S2) = x(Amin/Amax) + y +1 +1 + d(M1, M2), +(8) +where Amin is the smaller of the two surface areas, and Amax is the larger, in order to give a score bounded by 0 and 1. +We therefore need to choose an appropriate set of weights x and y such that x + y = 1, and x < 0.5, to ensure the +shape is the primary contributor to the score. +Table 4 gives the resulting similarity scores for pairwise comparison of the PDE5 inhibitors. In all three cases, +the similarity increases with increasing contribution from the surface area term as expected. The increase for +Sildenafil-Vardenafil is only small, while for Tadalafil there is a greater effect of including the area. Final weights of +x = 0.3 and y = 0.7 were selected to balance the contribution of the surface area without it dominating over the shape +contribution. The PDE5 inhibitors were selected for tuning due to their known similarity, however further refinement of +9 + +Geometry and Molecular Surfaces +A PREPRINT +Table 4: Similarity scores for the PDE5 inhibitors for surface area weights ranging from 0 to 0.5. +x +y +Sildenafil-Vardenafil +Sildenafil-Tadalafil +Vardenafil-Tadalafil +0 +1 +0.884 +0.286 +0.275 +0.1 +0.9 +0.892 +0.340 +0.328 +0.2 +0.8 +0.900 +0.394 +0.380 +0.3 +0.7 +0.908 +0.449 +0.432 +0.4 +0.6 +0.916 +0.503 +0.485 +0.5 +0.5 +0.924 +0.557 +0.537 +these parameters with a larger set of examples may be required for full scale virtual screening. +3.3 +Investigating Variation in 3D Conformers +As discussed in the previous work [1], consideration of the different orientations a molecule can adopt (known as +conformers) is important when using 3D shape descriptors. Conformers of the same molecule should theoretically have +scores in the range 0.7 < score < 1, as high self-similarity is expected (scores above 0.7 in chemoinformatics), while +retaining the ability to distinguish between them. +As with RGMolSA, two small sets of 10 conformers of the PDE5 inhibitors are used to investigate how KQ- +MolSA regards different conformers. One set contains 10 random conformers, in which we would expect slightly +more variance, while the other has 10 low energy conformers, for which higher similarity is expected. Both sets +were produced using the ETKDG algorithm [15] with energy optimisation using the MMFF94 force field [16], both +implemented in RDKit [17]. The minimum, maximum and average shape similarity as well as the average RMSD +(which compares conformers based on their atomic positions) for each set are given in Figure 3. The full set of RMSD +and shape similarity comparisons are available in the Supporting Data. +The RMSD and shape similarity for each set are compared in the swarm plots shown in Figure 4. For k = 1, generally +high similarity was observed, with some scores for the random conformers of Tadalafil falling slightly below 0.7. +Greater variation is observed for k = 2, where some conformer pairs have scores below 0.6. This reduction in similarity +is expected for k = 2 as the descriptors represent a more detailed approximation to the original surface than those +for k = 1 and hence will be more sensitive to differences in the geometry. However, the similarity scores obtained +were on the whole lower than for RGMolSA, where the similarity between most conformer pairs is greater than 0.8 [1]. +For the random sets, the similarity between conformers showed more variation than for RGMolSA, where clusters of +similar conformers were observed. While KQMolSA does handle conformers well, RGMolSA appears to do a better +job of this, due to the insensitivity to surface deformation of the spectrum of the Laplace–Beltrami operator. For virtual +screening, this consideration of conformers as similar negates the need for a pre-alignment step prior to shape similarity +calculation, and may allow molecules that can deform to fit in the binding pocket to be identified as potential hits, where +these would otherwise be classified as the wrong shape by methods that depend on atomic coordinates. +3.4 +Comparison to Existing Methods +The PDE5 inhibitor series was also used to investigate how well KQMolSA compares to the previous work, and to +other open source shape similarity methods. Table 5 provides the shape-similarity scores observed between the PDE5 +inhibitors for KQMolSA (for k = 1 and k = 2), RGMolSA [1], USRCAT [18, 17], Shape-It [19] and MolSG [20]. A +2D representation, in the form of the 1024-bit Morgan fingerprint using radius 3, is also included. Each descriptor uses +a similarity score between 0 (different) and 1 (identical). +Table 5: Comparison of the work presented here (KQMolSA) to the previous work (RGMolSA) [1] and existing atomic-distance [18], atomic-centred [19] and molecular +surface based [20] descriptors. In all cases the similarity scores given are bound by 0 (no similarity) and 1 (identical). +KQMolSA +(k=1) +KQMolSA +(k=2) +RGMolSA +USRCAT +Shape-It +MolSG +Morgan +Fingerprint +Sildenafil- +Vardenafil +0.907 +0.652 +0.903 +0.384 +0.388 +0.704 +0.667 +Sildenafil- +Tadalafil +0.449 +0.482 +0.809 +0.269 +0.278 +0.746 +0.201 +Vardenafil- +Tadalafil +0.432 +0.470 +0.725 +0.291 +0.353 +0.887 +0.209 +10 + +Geometry and Molecular Surfaces +A PREPRINT +(a) k = 1 +(b) k = 2 +Figure 3: Overlay of the most and least shape-similar conformers of Sildenafil, Vardenafil and Tadalafil and the average shape similarity and RMSD for each set for (a) +k = 1 and (b) k = 2. On average the conformers display a high degree of self-similarity despite the variance in atom-position similarity. +As discussed in the prequel to this paper, as Sildenafil and Vardenafil are close structural analogues they should display +both high shape and fingerprint similarity. As Tadalafil is known to occupy a similar volume in PDE5 compared to +the other inhibitors, we’d expect high shape similarity scores also, but lower 2D similarity. One conformer of each +molecule is considered for simplicity. +As for RGMolSA, Sildenafil and Vardenafil are scored as highly similar, with a score of 0.907 (k += 1). +However Tadalafil is not scored as highly, and for KQMolSA would be classed as dissimilar if the typical threshold +of 0.7 was used. Lower similarity is observed for k = 2, which is expected as discussed previously. The similarity +score for k = 2 has a small dependence on the order of comparison (A compared to B yields a score which may +differ at the second decimal place from B compared to A, Table 6). This is due to the distance calculation involving +a numerical minimisation procedure rather than an exact expression, but this will have no practical implications in +chemoinformatics applications. Both proposed methods (RGMolSA and KQMolSA) perform well in this simple study, +with a higher predicted similarity for Sildenafil and Vardenafil than all the other 3D methods, and a more intuitive +ordering of the relative similarity measures than MolSG. However, a full scale benchmarking study will be required to +verify their performance. +11 + +Geometry and Molecular Surfaces +A PREPRINT +(a) RMS Similarity +(b) Shape Similarity (k = 1) +(c) Shape Similarity (k = 2) +Figure 4: Swarm plots of the RMSD (in Å) and shape similarity for our set of conformers highlight the general trend that different conformers are classed as having +similar shape, despite significant variance in their atomic positions. Conformers with RMSD less than 1 Å are considered similar, while those over 3 Å have significant +differences. +Table 6: Similarity scores for the PDE5 inhibitors for k=2 highlighting the dependence on the order of comparison. +Sildenafil +Vardenafil +Tadalafil +Sildenafil +- +0.652 +0.462 +Vardenafil +0.648 +- +0.470 +Tadalafil +0.482 +0.470 +- +3.5 +Similarity to Potential Decoys +As for RGMolSA, we also wanted to check how the method handles molecules that should be classed as genuinely +different from the PDE5 inhibitor molecules. We therefore present a comparison to four other molecules (Figure 5): +Arginine (supplement) which has a lower molecular weight, but similar general shape (a long chain of spheres); +Lymecycline (antibiotic), with a higher molecular weight and a four-ring motif potentially giving part of the molecule a +similar shape to Sildenafil; Diflorasone (topical corticosteroid), which has a similar molecular weight and four rings, but +has a different therapeutic target/indication and S-octylglutathione (oligopeptide), which again has similar molecular +weight, but no rings and the potential for similarity due to the branching in the centre of the molecule. +The results of this comparison are presented in Figure 6. Most of the scores obtained for both k = 1 and k = 2 fall +significantly below the typical threshold of 0.7 for similarity, and as such these molecules would be classed as genuinely +different and likely inactive against PDE5. The exception is the comparison between Tadalafil and Diflorasone, where a +higher score of 0.74 (k = 1) is obtained. Due to the similarity between their structures (both contain a motif of 4 fused +rings), we would expect to see some similarity between the two. Inspection by eye of both the space filling model and +surface of the two molecules also suggests they do have genuinely similar shapes (Figure 7). These were also classed as +potentially similar by RGMolSA (similarity of 0.872). +4 +Conclusion +We have outlined the theory underpinning an entirely novel shape descriptor, +Mij = +�� +C +zizje−kϕF(z) +√ +−1dz ∧ dz, +(9) +12 + +····· +Sildenafil Random +Sildenafil Low Energy +Vardenafil Random +Vardenafil Low Energy +： +Tadalafil Random +Tadalafil Low Energy +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Root Mean Square Deviation··…··… +Sildenafil Random +：8 +Sildenafil Low Energy +Vardenafil Random +（ +8 +Vardenafil Low Energy +Tadalafil Random +Tadalafil Low Energy +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Shape SimilaritySildenafil Random +Sildenafil Low Energy +Vardenafil Random +Vardenafil Low Energy +Tadalafil Random +Tadalafil Low Energy +0.6 +0.7 +0.8 +0.9 +1.0 +Shape SimilarityGeometry and Molecular Surfaces +A PREPRINT +Arginine +Lymecycline +Diflorasone +S-Octylglutathione +Figure 5: Chemical structures of potential decoy molecules. +the (2k + 1) × (2k + 1) Hermitian matrix which captures the geometry of the molecular surface. The distance between +two such matrix representations is then given as +d(M1, M2) = k− 3 +2 +� +� +� +� +2k+1 +� +i=1 +(λi − µi)2. +(10) +An overall similarity score of 1 for identical molecules and 0 for no similarity is then obtained as +score(S1, S2) = 0.3(Amin/Amax) + 0.7 +1 +1 + d(M1, M2). +(11) +As with the previously reported work, the capabilities of KQMolSA were investigated using a series of PDE5 inhibitors +known to have similar shape. The method generally handles conformers well, with similarity scores generally higher +than 0.7. The scores obtained were higher for k = 1 than k = 2, which is expected due to the greater detail leading to +more sensitivity to changes in geometry. The insensitivity to deformation of the surface lead to RGMolSA outperforming +KQMolSA in this area. KQMolSA performs relatively well compared to existing methods, identifying Sildenafil +and Vardenafil as highly similar, but assigning lower similarity scores to Tadalafil. This small study suggests that +RGMolSA might still perform better, but a full retrospective benchmarking study is required to confirm this. Compared +to RGMolSA, KQMolSA does have the advantage of a lower dependence on the choice of base sphere. There may +therefore be some instances where the use of KQMolSA is more appropriate despite its seemingly poorer performance, +for example in the consideration of long chain molecules with few rings, where numerical errors are often observed for +RGMolSA. Comparison to a set of potential decoy molecules yielded low scores for all except comparison of Tadalafil +to Diflorasone, which were also classed as similar by RGMolSA. Inspection by eye of both the space filling and surface +models of the molecules suggests that this assignment is reasonable, as they look similar in shape. Identification of such +similarity evidences the potential for scaffold hopping by these methods. +Whilst the above tests suggest that the matrix M does give a promising description of molecular shape, the method +does have some drawbacks, primarily in the calculation of the distance between two descriptors. While the notion of +the distance between two Hermitian inner products (represented by the matrices M1 and M2) is well understood, the +calculation of the distance between molecular surfaces requires the distance between a point on an SL(2, C)-orbit to be +minimised. Despite the use of existing optimised minimisation algorithms, this process is still quite slow, depending +on the extent of the required minimisation, and further does not guarantee that the global minimum has been found. +This step typically takes a few seconds per pair, compared to a near-instantaneous calculation for RGMolSA. Further +refinement of this step would be required for use of the method in screening ultra-large chemical libraries as part of a +drug discovery pipeline. +13 + +NH2 +H2N +N +HO +NH2OH +N +H +H +N +N +OH +OH +OH +HO +OH +NH2HO +OH +OHOH +NH +s +N +HO +H +NH2Geometry and Molecular Surfaces +A PREPRINT +Figure 6: KQMolSA similarity (for k = 1 and k = 2) of four ‘different’ molecules (blue) to the PDE5 inhibitor test series (red). The overlay of the structures was +computed using Open3DAlign [21] +Of course, there are many other ways of measuring the distance between two Hermitian matrices. +One +might hope that some form of machine learning, trained on an appropriate data set, might discern other useful +geometries on the space of descriptors. +The method also contains numerical instability above k = 2 (and for k = 2 in a few instances), producing Hermitian +matrices that are not positive definite. As Hermitian matrices differing only by a scale factor can be considered +equivalent, we have handled such cases by scaling one matrix by a factor of 10-1000 to bring the eigenvalues into the +range of Python’s numerical tolerance. +Along with addressing these issues, both of the methods proposed could be further improved through the consideration +of pharmacorphoric features, such as aromatic rings, hydrogen bond donors and acceptors, alongside the shape. As +these features are important for binding, this may lead to improved predictions compared to the consideration of shape +alone. As for RGMolSA, there would also be scope to investigate the use the Hermitian matrix descriptors produced by +KQMolSA as a feature descriptor in machine learning. +5 +Appendix: Finding the Kähler potential +Before giving the proof of the form of the Kähler potential, we dispense with a small technical point. From the point of +view of describing the Kähler form ω via +√ +−1∂ ¯∂ϕ = ω, +the Kähler potential ϕ is only locally defined and adding any function H satisfying √−1∂ ¯∂H = 0 +will also define a Kähler potential for ω. In our setting where the underlying complex manifold is CP1 and we are using +the standard coordinate z, we can add any harmonic function H : C → R to obtain a valid Kähler potential. +14 + +Arginina +Lymecyclina +Diforaaone +S-octylglubthiona +K=1:0.469 +k=1:0.513 +k=1:0.367 +k=1:0.467 +k=2: 0.413 +k=2: 0.411 +k=2: 0.3B9 +k=2: 0.378 +sildenfl +K=1:0.493 +K=1:0.535 +K=1:0.354 +K=1:0.467 +K=2:0.445 +K=2:0.45 +k=2: 0.377 +K=2:0.378 +Vardanafil +K=1:0.282 +k=1:0.347 +K=1:0.74 +K=1:0.421 +k=2: 0.315 +K=2: 0.339 +k=2: 0.614 +k=2: 0.4 +dalahlGeometry and Molecular Surfaces +A PREPRINT +(a) Tadalafil - Space Filling Model +(b) Diflorasone - Space Filling Model +(c) Tadalafil - Surface +(d) Diflorasone - Surface +Figure 7: Comparison by eye of both the space filling model and the surface of Tadalafil and Diflorasone highlights their similarity. +However, in Kähler Quantization, the potential ϕ actually describes a global object, the Hermitian metric h +on the line bundle L. This means that the functions +h(zj, zj) = e−kϕ(z)|z|2j, +are defined over whole sphere CP1. In particular, they extend to functions over the point at infinity. For example the +round metric has Kähler potential ϕ = −2 log(|z|2 + 1) and so, if we add a harmonic function H we require +|z|4k +(1 + |z|2)2k e−kH +to be bounded. The Liouville Theorem then implies H must be constant. +Theorem 5.1 (Form of Kähler potential). Let ω be a Kähler metric of the form given by Equation (3). If we denote the +region corresponding to the ith sphere as Ri ⊂ C, then the Kähler potential potential ϕ, which satisfies √−1∂∂ϕ = ω, +is of the form +ϕ(z) = Ci +Bi +log(|z − Ai|2 + Bi) + +N +� +j=1 +Kij log(|αijz + βij|2), +where K ∈ M N×N(R), and α, β ∈ M N×N(C). +Proof. The proof is by induction on the number of spheres N. For N = 1 the metric ω is the round metric and we can +take K = 0. Adding a new sphere to the surface changes the metric by adding a new region Rk which is a disc where +the metric takes the form +ω(z)|Rk = +Ck +(|z − Ak|2 + Bk)2 +√ +−1dz ∧ dz. +15 + +Geometry and Molecular Surfaces +A PREPRINT +We can map Rk to the unit disc about the origin by a Möbius transformation M in such a way that, in the coordinate of +the unit disc, the metric is given by +�ω(w) = +� +� +� +� +� +F(w)√−1dw ∧ dw +if +|w| > 1, +κ +(|w|2 + ε)2 +√−1dw ∧ dw +if +|w| ≤ 1, +for some function F : C → R and constants κ, ε ∈ R. +We solve the ¯∂-equation using the Dolbeault method; for a compactly supported1 continuous function H : C → C, +ψ(w) = +1 +2π√−1 +�� +C +H(p) +p − wdp ∧ dp, +solves ∂ψ = H(w)dw. We split the integral according to the form of the metric and consider +ψ(w) = +1 +2π√−1 +�� +D +κ +(|p|2 + ε)2(p − w)dp ∧ dp + +1 +2π√−1 +�� +C\D +F(p) +p − wdp ∧ dp. +To compute the first integral we use the Cauchy–Pompeiu integral formula and the fact that +κ +(|p|2 + ε)2 = ∂ +∂p +� (κ/ε)p +(|p|2 + ε) +� +, +to give +1 +2π√−1 +�� +D +κ +(|p|2 + ε)2(p − w)dp ∧ dp = +� +� +� +� +� +� +� +� +� +� (κ/ε)w +(|w|2 + ε) +� +− +1 +2π√−1 +� +∂D +(κ/ε)p +(|p|2 + ε)(p − w)dp +if +|w| < 1, +− +1 +2π√−1 +� +∂D +(κ/ε)p +(|p|2 + ε)(p − w)dp +if +|w| > 1. +The contour integral +1 +2π√−1 +� +∂D +(κ/ε)p +(|p|2 + B)(p − w)dp, +can be easily computed using the Cauchy Residue Formula and this yields +1 +2π√−1 +� +∂D +(κ/ε)p +(|p|2 + ε)(p − w)dp = +� +0 +if |w| < 1, +− (κ/ε) +(1+ε)w +if |w| > 1. +Finally, we arrive at +1 +2π√−1 +�� +D +κ +(|p|2 + ε)2(p − w)dp ∧ dp = +� � +(κ/ε)w +|w|2+ε +� +if |w| < 1, +(κ/ε) +(1+ε)w +if |w| > 1. +To compute the second integral, we again split the domain and consider +1 +2π√−1 +�� +C\D +F(p) +p − wdp ∧ dp = +1 +2π√−1 +�� +C +F(p) +p − wdp ∧ dp − +1 +2π√−1 +�� +D +F(p) +p − wdp ∧ dp. +The integral +S(w) = +1 +2π√−1 +�� +C +F(p) +p − wdp ∧ dp, +is a solution to +∂S +∂w = F(w). +1Our function is not compactly supported but we could cut off at an arbitrary radius to produce such a function. +16 + +Geometry and Molecular Surfaces +A PREPRINT +In the unit disc D, F has the form +F(w) = +˜κ +(|w|2 + ˜ε)2 , +where ˜κ and ˜ε are positive constants. Hence +ψ(w) = +� +� +� +� +� +� +� +� +� +S(w) + +� (κ/ε) +|w|2 + ε − +(˜κ/˜ε) +|w|2 + ˜ε +� +w +if +|w| < 1, +S(w) + +�(κ/ε) +1 + ε − (˜κ/˜ε) +1 + ˜ε +� +w−1 +if +|w| > 1, +solves dw ∧ ∂ψ = �ω(w). +If Q(w) is a Kähler potential for F(w)√−1dw ∧ dw then +�ϕ(w) = +� +� +� +Q(w) + (κ/ε) log(|w|2 + ε) − (˜κ/˜ε) log(|w|2 + ˜ε) − K +if +|w| < 1, +Q(w) + +�(κ/ε) +1 + ε − (˜κ/˜ε) +1 + ˜ε +� +log(|w|2) +if +|w| > 1, +where +K = (κ/ε) log(1 + ε) − (˜κ/˜ε) log(1 + ˜ε), +is a Kähler potential for �ω. Pulling back the function �ϕ via the Möbius transformation +M(z) = αz + β +γz + δ +we see +ϕk(z) = +� +� +� +� +� +� +� +� +� +� +� +Q +�αz + β +γz + δ +� ++ (κ/ε) log +����� +αz + β +γz + δ +���� +2 ++ ε +� +− K +if +z ∈ Rk +Q +�αz + β +γz + δ +� ++ +�(κ/ε) +1 + ε − (˜κ/˜ε) +1 + ˜ε +� +log +����� +αz + β +γz + δ +���� +2� +if +z ̸∈ Rk +is a Kähler potential for the metric which is singular at at the point z = −δ/γ. We can replace the Q-term by the +appropriate function for the previous ϕ and then add the appropriate multiple of log(|γz + δ|2) to produce a Kähler +potential of the appropriate form. +6 +Acknowledgements +The authors acknowledge support from an EPSRC Doctoral Training Partnership studentship (grant EP/R51309X/1), +the Alan Turing Institute Enrichment Scheme (R.P.), and a UKRI Future Leaders Fellowship (grant MR/T019654/1) +(D.J.C.). S.J.H. would like to thank Dr R. L. Hall for his interest and for useful conversations about the project. T.M. +would like to thank University of California, Irvine for their hospitality whilst some of the work on this paper was +completed. +References +[1] Daniel J. Cole, Stuart J. Hall, and Rachael Pirie. Riemannian geometry and molecular surfaces I: Spectrum of the +Laplacian, (preprint), 2022. +[2] Mark A. Johnson and Gerald M. Maggiora. Concepts and Applications of Molecular Similarity. 1990. +[3] Sumudu P. Leelananda and Steffen Lindert. Computational methods in drug discovery. Beilstein J. Org. Chem., +12:2694–2718, 2016. +[4] Ashutosh Kumar and Kam Y. J. Zhang. Advances in the development of shape similarity methods and their +application in drug discovery. Front. Chem., 6:1–21, 2018. +[5] S. K. Donaldson. Some numerical results in complex differential geometry. Pure Appl. Math. Q., 5(2):571–618, +2009. +17 + +Geometry and Molecular Surfaces +A PREPRINT +[6] Ann E. Cleves and Ajay N. Jain. Effects of inductive bias on computational evaluations of ligand-based modelling +and on drug discovery. J. Comput. Aided Mol. Des., 22(3):147–159, 2008. +[7] Stuart J. Hall and Thomas Murphy. Kähler geometry of molecular surfaces, in preparation. +[8] Phillip Griffiths and Joseph Harris. Principles of algebraic geometry. Pure and Applied Mathematics. A Wiley- +Interscience Publication. New York etc.: John Wiley & Sons. XII, 813 p. £ 29.60; $ 58.00 (1978)., 1978. +[9] Gang Tian. On a set of polarized Kähler metrics on algebraic manifolds. J. Differ. Geom., 32(1):99–130, 1990. +[10] Robert Berman and Julien Keller. Bergman geodesics. In Complex Monge-Ampère equations and geodesics in the +space of Kähler metrics. Lecture notes, pages 283–302. Berlin: Springer, 2012. +[11] Xiuxiong Chen and Song Sun. Space of Kähler metrics (V)—Kähler quantization. In Metric and differential +geometry, volume 297 of Progr. Math., pages 19–41. Birkhäuser/Springer, Basel, 2012. +[12] Yoshinori Hashimoto. Quantisation of extremal Kähler metrics. J. Geom. Anal., 31(3):2970–3028, 2021. +[13] William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery. Numerical recipes. Cambridge +University Press, Cambridge, third edition, 2007. The art of scientific computing. +[14] Pauli Virtanen et al. SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python. Nature Methods, +17:261–272, 2020. +[15] Sereina Riniker and Gregory A. Landrum. Better informed distance geometry: Using what we know to improve +conformation generation. J. Chem. Inf. Model, 55(12):2562–2574, 2015. +[16] Paolo Tosco, Nikolaus Stiefl, and Gregory Landrum. Bringing the mmff force field to the rdkit: Implementation +and validation. J.Cheminformatics, 6(1), 2014. +[17] Greg Landrum. Rdkit: Open-source cheminformatics software. Version 2021.09.1. +[18] Adrian M Schreyer and Tom Blundell. USRCAT: Real-time ultrafast shape recognition with pharmacophoric +constraints. J. Cheminform., 4:1489–1495, 2012. +[19] Jonatan Taminau, Gert Thijs, and Hans De Winter. Pharao: Pharmacophore alignment and optimization. J. Mol. +Graph, 27(2):161–169, 2008. +[20] Matthew P. Seddon, David A. Cosgrove, Martin J. Packer, and Valerie J. Gillet. Alignment-free molecular shape +comparison using spectral geometry: The framework. J. Chem. Inf. Model, 59:98–116, 2019. +[21] Paolo Tosco, Thomas Balle, and Fereshteh Shiri. Open3dalign: an open-source software aimed at unsupervised +ligand alignment. Journal of Computer-Aided Molecular Design, 25(8):777–783, 2011. +18 + diff --git a/79E3T4oBgHgl3EQfRwk1/content/tmp_files/load_file.txt b/79E3T4oBgHgl3EQfRwk1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..847c818ff06e5719ed848ab15f04990737838f33 --- /dev/null +++ b/79E3T4oBgHgl3EQfRwk1/content/tmp_files/load_file.txt @@ -0,0 +1,706 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf,len=705 +page_content='RIEMANNIAN GEOMETRY AND MOLECULAR SIMILARITY II: KÄHLER QUANTIZATION A PREPRINT Rachael Pirie School of Natural and Environmental Sciences Newcastle University r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='pirie2@ncl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='uk Stuart J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Hall School of Mathematics, Statistics, and Physics Newcastle University stuart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='hall@ncl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='uk Daniel J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Cole School of Natural and Environmental Sciences Newcastle University daniel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='cole@ncl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='uk Thomas Murphy Department of Mathematics California State University, Fullerton tmurphy@fullerton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='edu January 12, 2023 ABSTRACT Shape-similarity between molecules is a tool used by chemists for virtual screening, with the goal of reducing the cost and duration of drug discovery campaigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' This paper reports an entirely novel shape descriptor as an alternative to the previously described RGMolSA descriptors [1], derived from the theory of Riemannian geometry and Kähler quantization (KQMolSA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The treatment of a molecule as a series of intersecting spheres allows us to obtain the explicit Riemannian metric which captures the geometry of the surface, which can in turn be used to calculate a Hermitian matrix M as a directly comparable surface representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The potential utility of this method is demonstrated using a series of PDE5 inhibitors considered to have similar shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The method shows promise in its capability to handle different conformers, and compares well to existing shape similarity methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The code and data used to produce the results are available at: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='com/RPirie96/KQMolSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Keywords Riemannian Geometry · Kähler Quantization · Molecular Shape · Ligand-Based Virtual Screening 1 Introduction and Summary of Part I The concept that shared biological activity exists between similar molecules is used widely in drug discovery [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Molecules with known activity can be used as templates to screen large databases for other potential hits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' This is more efficient and allows coverage of a greater area of chemical space than is possible with experimental screening alone [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Estimating similarity between molecules based on their 3D shape has gained popularity due to the requirement for protein-drug shape complementarity to enable strong binding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' However no fixed notion of shape exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Instead, comparison relies on mathematical approximation of the molecule’s shape based on its volume, distribution of atomic distances or surface (most commonly treated as the van der Waals or solvent accessible surface) [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' In the accompanying paper [1], the RGMolSA method was presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The descriptor developed there ap- proximates the shape of the molecular surface using a simple nine-element vector containing the surface area and an approximation to the first eight non-zero eigenvalues of the ordinary Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The descriptor can be viewed as an approximation to the Riemannian metric, the underlying mathematical object that describes the shape of a surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' In this paper we present an entirely different method of approximating the Riemannian metric by using ideas from the theory of Kähler quantization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' we call this method Kähler quantization for Molecular Surface Approximation (KQMolSA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The theory was originally developed by mathematicians and string theorists in order to give explicit representations of the shapes of 4-dimensional objects (Calabi–Yau manifolds) that appear in physical theories (see [5] arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='04424v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='DG] 11 Jan 2023 Geometry and Molecular Surfaces A PREPRINT for the paper that pioneered its use as a numerical technique).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' In a nutshell, a function called the Kähler potential is associated to the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' We then compute something analogous to a Taylor expansion of this function with the coefficients being stored in a Hermitian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' While the matrices themselves do depend upon the precise position and parameterisation of the molecular surface in three-dimensional space R3, the dependence is easy to calculate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Hence we can perform our calculations in the ‘quantized’ space of Hermitian matrices and assign a distance between the shapes of two molecular surfaces this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The final distance is independent of the position of the molecules and the choices made in their parameterisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='1 Summary of Previous Work As in the accompanying paper [1], our approach begins by treating the molecule as a series of intersecting spheres, with their radii given by the van der Waals radii of the constituent atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The surface is assumed to have a genus of zero, so any rings (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' benzene) are replaced with a single sphere of radius 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='25 Å to facilitate this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The molecular structure is then defined by the number of spheres N (with each ring counted as a single sphere, and excluding any hydrogen atoms), the centres ci and radii ri for each sphere and the adjacency matrix T describing intersection of spheres, where Tij = � 1 if spheres i and j intersect 0 otherwise (or i = j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The surface area A of the molecule is calculated as the area of each sphere minus the “missing parts" where two spheres intersect: A = 2π � i � �2r2 i − � �ri � j Tij|ri − λij| � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' (1) This value is used to re-scale each of the starting constructs such that the surface area of the molecule is equal to that of a unit sphere (or 4π) to address the observation that Riemannian geometry treats two objects which differ only in size as having equivalent shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' This re-scaling is accounted for in the final descriptors with some weighting so as not to dominate the similarity calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' From the initial data, a map is constructed to ‘unwrap‘ the surface onto the complex plane C in a process we refer to as piecewise stereographic projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' This requires an atom to be selected as a starting point from which to construct our map, which we refer to as the base sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' This is taken to be the atom closest to the centre of mass by first finding the centroid of the molecule and then taking the atom with the smallest Euclidean distance from this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The Riemannian metric g = Φ∗ ps(gEuc) induced by the mapping Φps : C → S ⊂ R3 takes the form g = � � � � � � � � � � � � � � � � � � � � � � � 4r2 B (1+|z|2)2 (dx2 + dy2) if z ∈ C C1 (|z−A1|2+B1)2 (dx2 + dy2) if z ∈ D(a1, R1), C2 (|z−A2|2+B2)2 (dx2 + dy2) if z ∈ D(a2, R2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' CN−1 (|z−AN−1|2+BN−1)2 (dx2 + dy2) if z ∈ D(aN−1, RN−1), (2) where rB is the radius of the base sphere and C = C\\D(a1, R1) ∪ D(a2, R2) ∪ · · · ∪ D(aN−1, RN−1), is the complement of the discs D(a1, R1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' , D(aN−1, RN−1) which corresponds to the points in the base sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The RGMolSA descriptor uses the explicit form of the Riemannian metric provided by piecewise stereo- graphic projection to approximate the low-lying eigenfunctions of the Laplacian ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' In [1], we compared the RGMolSA descriptor for Sildenafil, Vardenafil and Tadalafil, a series of PDE5 inhibitors that are known to occupy a similar volume in the binding pocket of their target protein, and thus have similar shape (Figure 1) [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Vardenafil is a classic example of a “me-too" drug, where only a few small modifications have been made to the structure of Sildenafil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' As these are both highly similar chemically, they would be expected to have close to the same shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Tadalafil on the other hand is chemically quite different from the other two, but inspection of the molecules in the pocket of PDE5 reveals they occupy a similar binding pose, and thus would also be expected to have similar shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' In this article, for ease of compari- son with the previous work [1], we again use these three molecules as the basis for investigating the new shape descriptor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' 2 Geometry and Molecular Surfaces A PREPRINT (a) Sildenafil Pfizer First Sold: 1998 (b) Vardenafil Bayer First Sold: 2003 (c) Tadalafil Lilly First Sold: 2003 Figure 1: PDE5 inhibitors Sildenafil, Vardenafil and Tadalafil of known shape similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Tadalafil (different chemical structure, similar shape) is an example of a scaffold hop from the first in class drug Sildenafil, and offers greatly improved performance, while Vardenafil (a "me-too" follow-up drug) only offers minor improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' While RGMolSA was found to give a good description of shape, it has a possible deficiency due to the dependence of the results on the choice of base sphere, which in turn determines the trial functions for calculating the integrals used to construct the descriptor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The geometry of the surface near the base sphere is well described, but for atoms further away a greater number of eigenvalues would be needed for accurate description of the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' This problem is greater for larger molecules and can lead to the introduction of numerical errors when the molecule is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' We handled such errors by ignoring any contributions from regions with numerical radii less than 10−9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' however, this forces a somewhat artificial ‘locality’ upon the shape descriptor meaning that it probably only accurately captures the shape near to the base sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' In the following section we outline the theory underpinning the KQMolSA descriptors, that again uses the Riemannian metric to approximate the geometry of the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The resulting descriptors lie in the manifold GL(N, C)/U(N) to give a global descriptor of molecular geometry with reduced dependence on the starting position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Figure 2 summarises the steps in computing these, using Sildenafil as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' While the descriptor itself does depend upon the choices made and the position of the surface within R3, this is easily accounted for within the space GL(N, C)/U(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' This makes computing the ‘distance’ between the shape descriptors particularly straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' 2 The Mathematics of Kähler Quantization 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='1 Overview of the Theory We should say immediately that the theory of Kähler quantization is far too advanced to be able to detail in the current paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' For readers with sufficient mathematical background, a good account (and the original account of its use as a numerical technique) is given in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' An exposition, aimed at readers with a general scientific background, of the mathematical theory is currently being written by two of the authors [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The theory is concerned with the geometry of complex manifolds (shapes that locally look like Cn);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' any sur- face that sits in R3 is a complex manifold as it locally looks like a copy of the complex numbers C (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' n = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' More concretely, we will be concerned with the surfaces that are topologically equivalent to S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' in the language of complex manifolds, the sphere is often referred to as the Riemann Sphere and denoted CP1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The restriction on the topology of the surface is justified by the fact that chemists do not expect any activity in the centre of rings occurring in most 3 N N HN N N NN HN N N N NH N N N 010000Geometry and Molecular Surfaces A PREPRINT Figure 2: Key steps involved in the computation of the KQMolSA surface descriptor for Sildenafil (a PDE5 inhibitor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' drug-like molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The exceptions to this are macrocyclic molecules (those with large rings of more than 12 atoms) where genuine activity occurs in the centre of the ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Such molecules are therefore excluded from comparison by both methods proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The natural class of functions to work with when dealing with complex manifolds are those that are com- plex differentiable, often called holomorphic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' We consider a general complex manifold X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' unfortunately, if the manifold X is compact, the only holomorphic functions f : X → C are constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Thus we cannot hope to understand X simply by studying the holomorphic functions on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' A generalisation of the notion of a holomorphic function is that of a section of a holomorphic line bundle L with base X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' For readers familiar with the theory, a function is a section of the trivial bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' A line bundle is positive if there is a Hermitian metric h on L with positive curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' A foundational result of Kodaira [8] says that if the line bundle L is positive then for large enough k the tensor power Lk, has a lot of holomorphic sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' In fact, the space of all such sections, denoted H0(Lk), is a complex vector space of dimension that has order O(kn) as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The curvature of a positively curved Hermitian metric h gives rise to an object called a Kähler form, ω, which in turn gives rise to a Riemannian metric g (the mathematical object being used in [1] to describe shape).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' It turns out that the set of all positively curved Hermitian metrics on a line bundle L can be identified with the set of all real-valued functions ϕ : X → R that satisfy, in some local coordinate z, the ∂ ¯∂-equation √ −1∂ ¯∂ϕ = ω − ω0 where ω is the Kähler form of the metric and ω0 is a fixed reference Kähler form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' We will give more detail on the differential operators ∂ and ¯∂ in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' in particular, we will explain that in the molecular surface setting, the ∂ ¯∂-equation is really just the familiar Poisson equation in the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The function ϕ is called a Kähler potential for ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The associated potential is not unique but any two differ by a constant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' this does not affect the metric which is constructed by taking two derivatives of the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' However, we will see that the addition of a constant to a potential will have the affect of scaling the Hermitian matrix we produce as a shape descriptor by a positive real number and we will be required to find the ‘optimal’ rescaling in our distance calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' To summarise, what we have for a positive Hermitian line bundle (L, h) → X are: a Kähler form ω and a Kähler potential ϕ : X → R, a complex vector space H0(Lk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' 4 e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Sildenafil Space Filling Model Replace Rings, Base Sphere (Grey) AiilD 01 Map to Complex Plane Surface Area Matrix of Levels 2r?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' ifz εc (1 + [z/2)2 C1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='16 + 0j 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='44 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='86j 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='41 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='93j 1 ifz E D(ai,R1) F(z) = zje-kF(z)V-1dzΛdz M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='44 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='86j 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='01 + 0j (lz - A1/2 + B1)2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='48 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='22j MI 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='48 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='22j : 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='41 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='93j 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='41 + 0j Cn-1 ifz E D(an-1,Rn-1) (Iz - An-1/2 + Bn-1)2 Riemannian Metric Construct Hermitian Matrix Hermitian Matrix Shape DescriptorGeometry and Molecular Surfaces A PREPRINT What Kähler quantization amounts to is relating the geometry described by the Kähler potentials (an infinite dimensional space of functions) to the finite dimensional complex vector space H0(Lk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' This theme occurs throughout numerical analysis and shape description, for example in the theories of Fourier analysis, spherical harmonics, Taylor series, all of which produce a finite-dimensional vector space out of some infinite-dimensional set of functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='2 Quantization and Tian’s Theorem The data (L, h) → X allows for a natural L2-inner product on the vector space of sections H0(Lk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Given sections s1, s2 ∈ H0(Lk), we compute ⟨s1, s2⟩ := � X hk(s1, s2)ωn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' , where hk is the Hermitian metric induced on Lk by h, and ωn/n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' is the volume element produced by the Kähler form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' It is this inner product that is the quantization of the data (L, h) → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The space of all (Hermitian) inner products on a complex N-dimensional vector space can be thought of as GL(N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' C)/U(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' This is a negatively curved symmetric space and has a natural notion of distance on it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' it is this distance that we will use to measure shape similarity (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' To recover the geometry defined by (L, h) → X from the quantization, we choose a basis {sj} of the vec- tor space H0(Lk) which gives rise to the matrix representation of the inner product Mij := ⟨si, sj⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' If we let v be the vector of sections v = (s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' sN) , then we can define a Kähler potential (recalling that the sections are locally defined holomorphic functions) ˜ϕ by ˜ϕ := −1 k log � v∗M−1v � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='1 (Tian, [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Let (X, L, h) be a complex manifold with holomorphic line bundle L and positively curved Hermitian metric h with curvature ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' If we produce another Kähler form �ω = ω0 + √ −1∂ ¯∂ ˜ϕ, then ∥ω − �ω∥C0 = O(k−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Paraphrasing this theorem, we can say any Kähler form coming from a Kähler potential ϕ can be well approximated by the Kähler form coming from the ‘algebraic’ function �ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' If we pick local complex coordinates z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' , zn then the term v∗M−1v is just a power series in the coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' In the case of a molecular surface, we will have something like a polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' This is the sense in which the function �ϕ is similar to a truncated Taylor series for the original function ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The theorem then says that this series really does converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Tian’s Theorem is stated for smooth metrics (those where one can take an arbitrary number of derivatives of the Kähler potential ϕ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' in practice (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='3), we will be working with metrics where the potentials are in C2(X), that is twice continuously differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The theory of approximating such metrics algebraically has not been written down but we will demonstrate that we get a method that does produce meaningful shape comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' We expect that, suitably adapted to this setting, something like Tian’s Theorem is still true;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' for example, the case of potentials with lower regularity is discussed in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='3 Implementation in Practice As mentioned already, in practice we take X = CP1 the Riemann sphere and the line bundle to be the anticanonical bundle K∗ CP1 = O(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The Kähler form ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' can be explicitly constructed from the Riemannian metric g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' and in the coordinates furnished by the piecewise stereographic projection map Φps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' we can use the form of the metric (2) to get ω = F(z) √ −1dz ∧ dz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' 5 Geometry and Molecular Surfaces A PREPRINT where F : C → R+ is the ‘metric function’ given by F(z) = � � � � � � � � � � � � � � � � � � � � � � � 2r2 B (1+|z|2)2 if z ∈ C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' C1 (|z−A1|2+B1)2 if z ∈ D(a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' R1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' C2 (|z−A2|2+B2)2 if z ∈ D(a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' R2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' CN−1 (|z−AN−1|2+BN−1)2 if z ∈ D(aN−1, RN−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' (3) Note we have replaced, in the metric g, the real symmetric 2-tensor dx2 + dy2 with the antisymmetric form (√−1/2)dz ∧ d¯z, where dz = dx + √−1dy and d¯z = dx − √−1dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' To find the Kähler potential ϕ : C → R, we solve the ‘∂∂-equation’ ω = √ −1∂∂ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' If we consider the complex differential operators ∂ ∂z = 1 2 � ∂ ∂x − √ −1 ∂ ∂y � and ∂ ∂z = 1 2 � ∂ ∂x + √ −1 ∂ ∂y � , then the ∂∂-equation is equivalent to solving the Poisson equation ∂2ϕ ∂z∂z = 1 4∆Eucϕ = F, where ∆Euc is the usual 2-dimensional Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' We can solve the Poisson problem explicitly to find ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The solution can be thought of as having two parts: a ‘local’ part that is found by simply observing that ∂2 ∂z∂z �C log(|z − A|2 + B) B � = C (|z − A|2 + B)2 , and a ‘correction term’, named thus as the term is needed to ensure the function is in C2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The correction term is a linear combination of functions of the form log(|αz + β|2), where we get one term for each sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' As each of the correction terms is a harmonic function, that is ∆ log(|αz + β|2) = 0, the addition of the correction terms is still a solution of the Poisson equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' It would appear the correction terms are singular at the points z = −β/α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' however, these points always lie outside the disc where the function takes this particular form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' We record the form of the potential as a theorem and refer the reader to the appendix (Section 5) for a derivation of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='2 (Form of Kähler potential).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Let g be of the form Equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' In the region associated to the ith sphere, the Kähler potential can be written ϕ(z) = Ci Bi log(|z − Ai|2 + Bi) + N � j=1 Kij log(|αijz + βij|2), where K ∈ M N×N(R), and α, β ∈ M N×N(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The matrices K, α, and β in the previous theorem are easily calculated from the geometric data associated to the molecule and so it is straightforward to describe the Kähler potential explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The space of global sections H0(O(2k)) ∼= C2k+1 can be identified with the span of the functions ⟨1, z, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' , z2k⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Thus the shape descriptor associated to the surface is the (2k + 1) × (2k + 1) Hermitian matrix M where (considering indices that run from 0 to 2k) Mij = �� C zizje−kϕF(z) √ −1dz ∧ dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' (4) 6 Geometry and Molecular Surfaces A PREPRINT 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='4 Computing the Relevant Integrals A naïve numerical calculation of the integrals described by Equation (4) gives rise to two obvious problems: firstly, the domain of integration is unbounded (being the whole complex plane C);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' secondly, the domains and values describing the metric and the Kähler potential ϕ could become so small that numerical instabilities start to dominate the contribution of the associated atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The second problem has been discussed as a limitation in the approximation of the spectrum of the Laplacian [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' In this paper, we exploit the fact that the automor- phism group of CP1 is the group of Möbius transformations, PSL(2, C);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' we can use elements of this group to ensure the coordinates we perform calculations in are always in a numerically controlled region (here we use a unit disc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Put more concretely, let m ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' , N} index the mth sphere making up the molecular surface, then there is an element Tm ∈ PSL(2, C) that maps the unit disc D = {z ∈ C | |z| < 1}, onto the region D(am, Rm) from Equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' We note that if the mth sphere has level l, then the pre-image of the regions corresponding to level (l + 1) spheres which intersect the mth sphere will describe certain discs properly contained in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Hence the contribution of the mth sphere to the matrix described by Equation (4) is given by �� D− ˆ D (Tm(w))i(Tm(w))je−kϕ(Tm(w))F(Tm(w)) dTm(w) ∧ dTm(w), (5) where ˆD represents the union of the discs corresponding to the next level spheres intersecting the mth sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' In practice, we account for these higher-level spheres by assigning the value 0 to the volume form F(Tm(w)) dTm(w) ∧ dTm(w) whenever w ∈ ˆD (note this produces a jump discontinuity in the volume form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Numerical calculation of integrals of the form of Equation (5) is done by splitting into an angular and radial direction and then performing successive applications of the trapezium rule;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' we choose a radial step size corresponding to nr = 15 integration points and an angular step size corresponding to taking nθ = 10 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' This seems to achieve a reasonable accuracy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' for example, one can check the area integral for a given integration scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' We have also determined that the distance between shape descriptors does not seem to be significantly changed by taking smaller step sizes (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='5 Finding the Distance Between Shape Descriptors Given two positive definite Hermitian matrices M1, M2, such as those generated by Equation (4), there are innumerable ways of defining a notion of distance between such matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' With regards to the theory of Kähler quantization, it is natural to consider M1, M2 as two Hermitian inner products on the fixed complex vector space H0(O(2k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' This space is naturally seen as the manifold GL(2k + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' C)/U(2k + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' An inner product is specified by declaring a particular basis to be orthonormal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' any basis conjugate under the action of U(2k + 1) defines the same inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' This space has a natural distance on it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' one characterisation of this distance is that shortest paths (geodesics) are given by one-parameter subgroups of GL(2k + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' C), that is by paths of matrices of the form exp(tA) where A is some (2k + 1) × (2k + 1) complex matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' More explicitly, if {v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' , v2k+1} is a basis of H0(O(2k))such that both inner products are represented by diagonal matrices M1 = Diag � eλ1, eλ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' , eλ2k+1� , M2 = Diag (eµ1, eµ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' , eµ2k+1) , then d(M1, M2) = k− 3 2 � � � � 2k+1 � i=1 (λi − µi)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' (6) The factor of k−3/2 ensures that the distances stabilise as k → ∞ (see Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='1 in [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' It will be useful to consider the following more compact form for the distance d(M1, M2) = k− 3 2 � � � � 2k+1 � i=1 (log(ηi))2, (7) where {ηi} are the eigenvalues of the matrix M−1 1 M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' 7 Geometry and Molecular Surfaces A PREPRINT It is a well-known fact that the automorphism group of the Riemann sphere CP1 is the group of Möbius transformations PSL(2, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Roughly speaking, the subgroup PSU(2) ⊂ PSL(2, C) corresponds to rotations of the original surface and the remaining maps correspond to reparameterisations that preserve the complex structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' If ϖ ∈ PSL(2, C) is an automorphism of the form ϖ(z) = αz + β γz + δ , then ϖ also acts on the vector space H0(O(2k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' In representation theoretic terms, this action is the representation induced on Sym2k(C2) by the standard representation of SL(2, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' If we denote the element of SL(2k+1, C) by ϑ(ϖ) (see [12], Lemma 8) and the original shape descriptor computed in the z-coordinate by M, then the shape descriptor computed in the ϖ(z)-coordinate will be (ϑ(ϖ))∗ M (ϑ(ϖ)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' As mentioned in Section 2, the fact that the Kähler potential is only defined up to the addition of a constant means we can also scale the Hermitian matrix M by a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Hence our calculation of distance between two shape descriptors M1 and M2 becomes the concrete problem of minimising, over (p, ϑ) ∈ R × SL(2, C), ζ(p, ϑ) = 2k+1 � i=1 (log(ηi))2, where {ηi} are the eigenvalues of the matrix M−1 1 ep (ϑ(ϖ))∗ M2 (ϑ(ϖ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' It is easy to see that the value of p at a critical point of ζ is independent of the element ϑ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Elementary calculus yields that the value of p is given by p = − 1 2k + 1 2k+1 � i=1 log(˜ηi), where {˜ηi} are the eigenvalues of the matrix M−1 1 M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' As the matrix (ϑ(ϖ)) has unit determinant, the value of p does not depend up the SL(2, C) action on the Hermitian matrix M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' We thus reduce the distance calculation to a minimisation over the six-dimensional Lie group SL(2, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Note that the distance between the shape descriptors given by Equation (6) is the distance between the molecular shapes after they have been re-scaled to have area 4π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Hence the distance between two molecular surfaces S1 and S2 should include a component to reflect the difference in area between S1 and S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' As we are interested in producing a similarity score rather than a distance between two inputs, we do not take this point up further in the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Our initial attempts at creating a similarity score are detailed in the subsequent section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The remaining minimisation over SL(2, C) is done by parameterising a generic matrix by the 6 real variables x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='x6 and taking ϖ(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' , x6) = � x1 + √−1x2 x3 + √−1x4 x5 + √−1x6 ∗ � , where ∗ is chosen to ensure det(ϖ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' To perform the minimisation, we use algorithms that do not require the input of a gradient vector, such as Nelder–Mead or Powell methods [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' These are implemented using off-the-shelf packages in SciPy [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' We found that for k = 1 there was very little difference between the results for either method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' the minimisation algorithm converges to produce a robust distance value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' For k = 2 the minimisation methods appear to be a little less stable and occasionally did not converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' One way around this was to use the element of SL(2, C) found by the k = 1 minimisation as the initial guess for the k = 2 step (otherwise the identity matrix was used).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' We anticipate that one might be able to improve this process;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' for example, by computing the gradient of the function to be minimised explicitly and then using this in an algorithm such as conjugate gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' One further consideration in implementing the distance measure between two matrices was in shape descrip- tors for k > 2 (and for k = 2 in some cases), where numerical instability exists within the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Occasionally non-positive definite matrices are produced, that cannot be compared using the above approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' As Hermitian matrices that differ only by scale can be considered equivalent, such cases have been treated by scaling one matrix by a factor of 10, 100 or 1000 as needed in order to bring the eigenvalues into the range required for consideration with Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' 8 Geometry and Molecular Surfaces A PREPRINT 3 Initial Case Study: Phosphodiesterase 5 (PDE5) Inhibitors 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='1 Tuning the Parameters nr and nθ To determine the effect of varying the parameters nr and nθ (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='4) on the quality of the shape descriptors produced, we considered three sets of parameters: nr = 200 and nθ = 100;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' nr = 50 and nθ = 25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' nr = 15 and nθ = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The distances produced between the descriptor for each set and the area returned during the computation of the relevant integrals (which should be ∼ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='57 for an accurate descriptor, as constrained by the choice of scaling the surface area to 4π) are reported here for Sildenafil (Table 1), Vardenafil (Table 2) and Tadalafil (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Table 1: Computed distances between descriptors of Sildenafil generated using different values of nr and nθ for k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The area reported is that returned by the integration step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' (200, 100), area = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='59 (50, 25), area = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='62 (15, 10), area = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='62 (200, 100) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='032 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='032 (50, 25) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='038 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='040 (15, 10) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='038 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='040 Table 2: Computed distances between descriptors of Vardenafil generated using different values of nr and nθ for k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The area reported is that returned by the integration step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' (200, 100) area = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='57 (50, 25), area = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='58 (15, 10), area = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='58 (200, 100) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='005 (50, 25) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='004 (15, 10) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='004 Table 3: Computed distances between descriptors of Tadalafil generated using different values of nr and nθ for k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The area reported is that returned by the integration step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' (200, 100), area = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='32 (50, 25), area = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='37 (15, 10), area = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='37 (200, 100) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='003 (50, 25) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='001 (15, 10) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='001 As these distances are small in each case, there is no significant loss of quality when the number of points considered is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The areas for both Sildenafil and Vardenafil are also close to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='57, indicating high quality descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The area for Tadalafil is overestimated slightly, however this is due to an issue with the replacement of the rings for motifs with a 5-membered ring between two other rings rather than the choice of nr and nθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Similar results were observed for the consideration of k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' As the quality is unaffected, the minimum parameters of nr = 15 and nθ = 10 were used in the final descriptors to increase the speed of calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='2 Constructing a Similarity Score In order to facilitate familiar comparison of molecules, we wish to construct a similarity score rather than simply taking the distance between two matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' In chemoinformatics, this score typically takes a value between 0 (no similarity) and 1 (identical) [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' To achieve this we take the inverse distance, and account for size by taking the ratio of two surface areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Equation 8 gives the similarity score between two molecular surfaces S1 and S2, score(S1, S2) = x(Amin/Amax) + y 1 1 + d(M1, M2), (8) where Amin is the smaller of the two surface areas, and Amax is the larger, in order to give a score bounded by 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' We therefore need to choose an appropriate set of weights x and y such that x + y = 1, and x < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='5, to ensure the shape is the primary contributor to the score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Table 4 gives the resulting similarity scores for pairwise comparison of the PDE5 inhibitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' In all three cases, the similarity increases with increasing contribution from the surface area term as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The increase for Sildenafil-Vardenafil is only small, while for Tadalafil there is a greater effect of including the area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Final weights of x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='3 and y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='7 were selected to balance the contribution of the surface area without it dominating over the shape contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The PDE5 inhibitors were selected for tuning due to their known similarity, however further refinement of 9 Geometry and Molecular Surfaces A PREPRINT Table 4: Similarity scores for the PDE5 inhibitors for surface area weights ranging from 0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' x y Sildenafil-Vardenafil Sildenafil-Tadalafil Vardenafil-Tadalafil 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='884 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='286 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='275 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='892 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='340 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='328 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='394 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='380 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='908 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='449 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='432 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='916 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='503 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='485 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='924 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='557 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='537 these parameters with a larger set of examples may be required for full scale virtual screening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='3 Investigating Variation in 3D Conformers As discussed in the previous work [1], consideration of the different orientations a molecule can adopt (known as conformers) is important when using 3D shape descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Conformers of the same molecule should theoretically have scores in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='7 < score < 1, as high self-similarity is expected (scores above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='7 in chemoinformatics), while retaining the ability to distinguish between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' As with RGMolSA, two small sets of 10 conformers of the PDE5 inhibitors are used to investigate how KQ- MolSA regards different conformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' One set contains 10 random conformers, in which we would expect slightly more variance, while the other has 10 low energy conformers, for which higher similarity is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Both sets were produced using the ETKDG algorithm [15] with energy optimisation using the MMFF94 force field [16], both implemented in RDKit [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The minimum, maximum and average shape similarity as well as the average RMSD (which compares conformers based on their atomic positions) for each set are given in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The full set of RMSD and shape similarity comparisons are available in the Supporting Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The RMSD and shape similarity for each set are compared in the swarm plots shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' For k = 1, generally high similarity was observed, with some scores for the random conformers of Tadalafil falling slightly below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Greater variation is observed for k = 2, where some conformer pairs have scores below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' This reduction in similarity is expected for k = 2 as the descriptors represent a more detailed approximation to the original surface than those for k = 1 and hence will be more sensitive to differences in the geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' However, the similarity scores obtained were on the whole lower than for RGMolSA, where the similarity between most conformer pairs is greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='8 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' For the random sets, the similarity between conformers showed more variation than for RGMolSA, where clusters of similar conformers were observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' While KQMolSA does handle conformers well, RGMolSA appears to do a better job of this, due to the insensitivity to surface deformation of the spectrum of the Laplace–Beltrami operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' For virtual screening, this consideration of conformers as similar negates the need for a pre-alignment step prior to shape similarity calculation, and may allow molecules that can deform to fit in the binding pocket to be identified as potential hits, where these would otherwise be classified as the wrong shape by methods that depend on atomic coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='4 Comparison to Existing Methods The PDE5 inhibitor series was also used to investigate how well KQMolSA compares to the previous work, and to other open source shape similarity methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Table 5 provides the shape-similarity scores observed between the PDE5 inhibitors for KQMolSA (for k = 1 and k = 2), RGMolSA [1], USRCAT [18, 17], Shape-It [19] and MolSG [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' A 2D representation, in the form of the 1024-bit Morgan fingerprint using radius 3, is also included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Each descriptor uses a similarity score between 0 (different) and 1 (identical).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Table 5: Comparison of the work presented here (KQMolSA) to the previous work (RGMolSA) [1] and existing atomic-distance [18], atomic-centred [19] and molecular surface based [20] descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' In all cases the similarity scores given are bound by 0 (no similarity) and 1 (identical).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' KQMolSA (k=1) KQMolSA (k=2) RGMolSA USRCAT Shape-It MolSG Morgan Fingerprint Sildenafil- Vardenafil 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='907 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='652 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='903 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='384 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='388 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='704 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='667 Sildenafil- Tadalafil 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='449 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='482 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='809 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='269 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='278 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='746 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='201 Vardenafil- Tadalafil 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='432 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='470 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='725 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='291 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='353 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='887 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='209 10 Geometry and Molecular Surfaces A PREPRINT (a) k = 1 (b) k = 2 Figure 3: Overlay of the most and least shape-similar conformers of Sildenafil, Vardenafil and Tadalafil and the average shape similarity and RMSD for each set for (a) k = 1 and (b) k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' On average the conformers display a high degree of self-similarity despite the variance in atom-position similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' As discussed in the prequel to this paper, as Sildenafil and Vardenafil are close structural analogues they should display both high shape and fingerprint similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' As Tadalafil is known to occupy a similar volume in PDE5 compared to the other inhibitors, we’d expect high shape similarity scores also, but lower 2D similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' One conformer of each molecule is considered for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' As for RGMolSA, Sildenafil and Vardenafil are scored as highly similar, with a score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='907 (k = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' However Tadalafil is not scored as highly, and for KQMolSA would be classed as dissimilar if the typical threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='7 was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Lower similarity is observed for k = 2, which is expected as discussed previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The similarity score for k = 2 has a small dependence on the order of comparison (A compared to B yields a score which may differ at the second decimal place from B compared to A, Table 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' This is due to the distance calculation involving a numerical minimisation procedure rather than an exact expression, but this will have no practical implications in chemoinformatics applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Both proposed methods (RGMolSA and KQMolSA) perform well in this simple study, with a higher predicted similarity for Sildenafil and Vardenafil than all the other 3D methods, and a more intuitive ordering of the relative similarity measures than MolSG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' However, a full scale benchmarking study will be required to verify their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' 11 Geometry and Molecular Surfaces A PREPRINT (a) RMS Similarity (b) Shape Similarity (k = 1) (c) Shape Similarity (k = 2) Figure 4: Swarm plots of the RMSD (in Å) and shape similarity for our set of conformers highlight the general trend that different conformers are classed as having similar shape, despite significant variance in their atomic positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Conformers with RMSD less than 1 Å are considered similar, while those over 3 Å have significant differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Table 6: Similarity scores for the PDE5 inhibitors for k=2 highlighting the dependence on the order of comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Sildenafil Vardenafil Tadalafil Sildenafil 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='652 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='462 Vardenafil 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='648 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='470 Tadalafil 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='482 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='470 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='5 Similarity to Potential Decoys As for RGMolSA, we also wanted to check how the method handles molecules that should be classed as genuinely different from the PDE5 inhibitor molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' We therefore present a comparison to four other molecules (Figure 5): Arginine (supplement) which has a lower molecular weight, but similar general shape (a long chain of spheres);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Lymecycline (antibiotic), with a higher molecular weight and a four-ring motif potentially giving part of the molecule a similar shape to Sildenafil;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Diflorasone (topical corticosteroid), which has a similar molecular weight and four rings, but has a different therapeutic target/indication and S-octylglutathione (oligopeptide), which again has similar molecular weight, but no rings and the potential for similarity due to the branching in the centre of the molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The results of this comparison are presented in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Most of the scores obtained for both k = 1 and k = 2 fall significantly below the typical threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='7 for similarity, and as such these molecules would be classed as genuinely different and likely inactive against PDE5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The exception is the comparison between Tadalafil and Diflorasone, where a higher score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='74 (k = 1) is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Due to the similarity between their structures (both contain a motif of 4 fused rings), we would expect to see some similarity between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Inspection by eye of both the space filling model and surface of the two molecules also suggests they do have genuinely similar shapes (Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' These were also classed as potentially similar by RGMolSA (similarity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='872).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' 4 Conclusion We have outlined the theory underpinning an entirely novel shape descriptor, Mij = �� C zizje−kϕF(z) √ −1dz ∧ dz, (9) 12 ····· Sildenafil Random Sildenafil Low Energy Vardenafil Random Vardenafil Low Energy ： Tadalafil Random Tadalafil Low Energy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='0 Root Mean Square Deviation··…··… Sildenafil Random ：8 Sildenafil Low Energy Vardenafil Random （ 8 Vardenafil Low Energy Tadalafil Random Tadalafil Low Energy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='00 Shape SimilaritySildenafil Random Sildenafil Low Energy Vardenafil Random Vardenafil Low Energy Tadalafil Random Tadalafil Low Energy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='0 Shape SimilarityGeometry and Molecular Surfaces A PREPRINT Arginine Lymecycline Diflorasone S-Octylglutathione Figure 5: Chemical structures of potential decoy molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' the (2k + 1) × (2k + 1) Hermitian matrix which captures the geometry of the molecular surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The distance between two such matrix representations is then given as d(M1, M2) = k− 3 2 � � � � 2k+1 � i=1 (λi − µi)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' (10) An overall similarity score of 1 for identical molecules and 0 for no similarity is then obtained as score(S1, S2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='3(Amin/Amax) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='7 1 1 + d(M1, M2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' (11) As with the previously reported work, the capabilities of KQMolSA were investigated using a series of PDE5 inhibitors known to have similar shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The method generally handles conformers well, with similarity scores generally higher than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The scores obtained were higher for k = 1 than k = 2, which is expected due to the greater detail leading to more sensitivity to changes in geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The insensitivity to deformation of the surface lead to RGMolSA outperforming KQMolSA in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' KQMolSA performs relatively well compared to existing methods, identifying Sildenafil and Vardenafil as highly similar, but assigning lower similarity scores to Tadalafil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' This small study suggests that RGMolSA might still perform better, but a full retrospective benchmarking study is required to confirm this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Compared to RGMolSA, KQMolSA does have the advantage of a lower dependence on the choice of base sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' There may therefore be some instances where the use of KQMolSA is more appropriate despite its seemingly poorer performance, for example in the consideration of long chain molecules with few rings, where numerical errors are often observed for RGMolSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Comparison to a set of potential decoy molecules yielded low scores for all except comparison of Tadalafil to Diflorasone, which were also classed as similar by RGMolSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Inspection by eye of both the space filling and surface models of the molecules suggests that this assignment is reasonable, as they look similar in shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Identification of such similarity evidences the potential for scaffold hopping by these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Whilst the above tests suggest that the matrix M does give a promising description of molecular shape, the method does have some drawbacks, primarily in the calculation of the distance between two descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' While the notion of the distance between two Hermitian inner products (represented by the matrices M1 and M2) is well understood, the calculation of the distance between molecular surfaces requires the distance between a point on an SL(2, C)-orbit to be minimised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Despite the use of existing optimised minimisation algorithms, this process is still quite slow, depending on the extent of the required minimisation, and further does not guarantee that the global minimum has been found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' This step typically takes a few seconds per pair, compared to a near-instantaneous calculation for RGMolSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Further refinement of this step would be required for use of the method in screening ultra-large chemical libraries as part of a drug discovery pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' 13 NH2 H2N N HO NH2OH N H H N N OH OH OH HO OH NH2HO OH OHOH NH s N HO H NH2Geometry and Molecular Surfaces A PREPRINT Figure 6: KQMolSA similarity (for k = 1 and k = 2) of four ‘different’ molecules (blue) to the PDE5 inhibitor test series (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The overlay of the structures was computed using Open3DAlign [21] Of course, there are many other ways of measuring the distance between two Hermitian matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' One might hope that some form of machine learning, trained on an appropriate data set, might discern other useful geometries on the space of descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The method also contains numerical instability above k = 2 (and for k = 2 in a few instances), producing Hermitian matrices that are not positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' As Hermitian matrices differing only by a scale factor can be considered equivalent, we have handled such cases by scaling one matrix by a factor of 10-1000 to bring the eigenvalues into the range of Python’s numerical tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Along with addressing these issues, both of the methods proposed could be further improved through the consideration of pharmacorphoric features, such as aromatic rings, hydrogen bond donors and acceptors, alongside the shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' As these features are important for binding, this may lead to improved predictions compared to the consideration of shape alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' As for RGMolSA, there would also be scope to investigate the use the Hermitian matrix descriptors produced by KQMolSA as a feature descriptor in machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' 5 Appendix: Finding the Kähler potential Before giving the proof of the form of the Kähler potential, we dispense with a small technical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' From the point of view of describing the Kähler form ω via √ −1∂ ¯∂ϕ = ω, the Kähler potential ϕ is only locally defined and adding any function H satisfying √−1∂ ¯∂H = 0 will also define a Kähler potential for ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' In our setting where the underlying complex manifold is CP1 and we are using the standard coordinate z, we can add any harmonic function H : C → R to obtain a valid Kähler potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' 14 Arginina Lymecyclina Diforaaone S-octylglubthiona K=1:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='469 k=1:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='513 k=1:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='367 k=1:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='467 k=2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='413 k=2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='411 k=2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='3B9 k=2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='378 sildenfl K=1:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='493 K=1:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='535 K=1:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='354 K=1:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='467 K=2:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='445 K=2:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='45 k=2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='377 K=2:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='378 Vardanafil K=1:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='282 k=1:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='347 K=1:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='74 K=1:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='421 k=2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='315 K=2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='339 k=2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='614 k=2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='4 dalahlGeometry and Molecular Surfaces A PREPRINT (a) Tadalafil - Space Filling Model (b) Diflorasone - Space Filling Model (c) Tadalafil - Surface (d) Diflorasone - Surface Figure 7: Comparison by eye of both the space filling model and the surface of Tadalafil and Diflorasone highlights their similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' However, in Kähler Quantization, the potential ϕ actually describes a global object, the Hermitian metric h on the line bundle L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' This means that the functions h(zj, zj) = e−kϕ(z)|z|2j, are defined over whole sphere CP1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' In particular, they extend to functions over the point at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' For example the round metric has Kähler potential ϕ = −2 log(|z|2 + 1) and so, if we add a harmonic function H we require |z|4k (1 + |z|2)2k e−kH to be bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The Liouville Theorem then implies H must be constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='1 (Form of Kähler potential).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Let ω be a Kähler metric of the form given by Equation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' If we denote the region corresponding to the ith sphere as Ri ⊂ C, then the Kähler potential potential ϕ, which satisfies √−1∂∂ϕ = ω, is of the form ϕ(z) = Ci Bi log(|z − Ai|2 + Bi) + N � j=1 Kij log(|αijz + βij|2), where K ∈ M N×N(R), and α, β ∈ M N×N(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The proof is by induction on the number of spheres N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' For N = 1 the metric ω is the round metric and we can take K = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Adding a new sphere to the surface changes the metric by adding a new region Rk which is a disc where the metric takes the form ω(z)|Rk = Ck (|z − Ak|2 + Bk)2 √ −1dz ∧ dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' 15 Geometry and Molecular Surfaces A PREPRINT We can map Rk to the unit disc about the origin by a Möbius transformation M in such a way that, in the coordinate of the unit disc, the metric is given by �ω(w) = � � � � � F(w)√−1dw ∧ dw if |w| > 1, κ (|w|2 + ε)2 √−1dw ∧ dw if |w| ≤ 1, for some function F : C → R and constants κ, ε ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' We solve the ¯∂-equation using the Dolbeault method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' for a compactly supported1 continuous function H : C → C, ψ(w) = 1 2π√−1 �� C H(p) p − wdp ∧ dp, solves ∂ψ = H(w)dw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' We split the integral according to the form of the metric and consider ψ(w) = 1 2π√−1 �� D κ (|p|2 + ε)2(p − w)dp ∧ dp + 1 2π√−1 �� C\\D F(p) p − wdp ∧ dp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' To compute the first integral we use the Cauchy–Pompeiu integral formula and the fact that κ (|p|2 + ε)2 = ∂ ∂p � (κ/ε)p (|p|2 + ε) � , to give 1 2π√−1 �� D κ (|p|2 + ε)2(p − w)dp ∧ dp = � � � � � � � � � � (κ/ε)w (|w|2 + ε) � − 1 2π√−1 � ∂D (κ/ε)p (|p|2 + ε)(p − w)dp if |w| < 1, − 1 2π√−1 � ∂D (κ/ε)p (|p|2 + ε)(p − w)dp if |w| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The contour integral 1 2π√−1 � ∂D (κ/ε)p (|p|2 + B)(p − w)dp, can be easily computed using the Cauchy Residue Formula and this yields 1 2π√−1 � ∂D (κ/ε)p (|p|2 + ε)(p − w)dp = � 0 if |w| < 1, − (κ/ε) (1+ε)w if |w| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Finally, we arrive at 1 2π√−1 �� D κ (|p|2 + ε)2(p − w)dp ∧ dp = � � (κ/ε)w |w|2+ε � if |w| < 1, (κ/ε) (1+ε)w if |w| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' To compute the second integral, we again split the domain and consider 1 2π√−1 �� C\\D F(p) p − wdp ∧ dp = 1 2π√−1 �� C F(p) p − wdp ∧ dp − 1 2π√−1 �� D F(p) p − wdp ∧ dp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' The integral S(w) = 1 2π√−1 �� C F(p) p − wdp ∧ dp, is a solution to ∂S ∂w = F(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' 1Our function is not compactly supported but we could cut off at an arbitrary radius to produce such a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' 16 Geometry and Molecular Surfaces A PREPRINT In the unit disc D, F has the form F(w) = ˜κ (|w|2 + ˜ε)2 , where ˜κ and ˜ε are positive constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Hence ψ(w) = � � � � � � � � � S(w) + � (κ/ε) |w|2 + ε − (˜κ/˜ε) |w|2 + ˜ε � w if |w| < 1, S(w) + �(κ/ε) 1 + ε − (˜κ/˜ε) 1 + ˜ε � w−1 if |w| > 1, solves dw ∧ ∂ψ = �ω(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' If Q(w) is a Kähler potential for F(w)√−1dw ∧ dw then �ϕ(w) = � � � Q(w) + (κ/ε) log(|w|2 + ε) − (˜κ/˜ε) log(|w|2 + ˜ε) − K if |w| < 1, Q(w) + �(κ/ε) 1 + ε − (˜κ/˜ε) 1 + ˜ε � log(|w|2) if |w| > 1, where K = (κ/ε) log(1 + ε) − (˜κ/˜ε) log(1 + ˜ε), is a Kähler potential for �ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Pulling back the function �ϕ via the Möbius transformation M(z) = αz + β γz + δ we see ϕk(z) = � � � � � � � � � � � Q �αz + β γz + δ � + (κ/ε) log ����� αz + β γz + δ ���� 2 + ε � − K if z ∈ Rk Q �αz + β γz + δ � + �(κ/ε) 1 + ε − (˜κ/˜ε) 1 + ˜ε � log ����� αz + β γz + δ ���� 2� if z ̸∈ Rk is a Kähler potential for the metric which is singular at at the point z = −δ/γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' We can replace the Q-term by the appropriate function for the previous ϕ and then add the appropriate multiple of log(|γz + δ|2) to produce a Kähler potential of the appropriate form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' 6 Acknowledgements The authors acknowledge support from an EPSRC Doctoral Training Partnership studentship (grant EP/R51309X/1), the Alan Turing Institute Enrichment Scheme (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' ), and a UKRI Future Leaders Fellowship (grant MR/T019654/1) (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' would like to thank Dr R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Hall for his interest and for useful conversations about the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' would like to thank University of California, Irvine for their hospitality whilst some of the work on this paper was completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' References [1] Daniel J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' Cole, Stuart J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} 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25(8):777–783, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} +page_content=' 18' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E3T4oBgHgl3EQfRwk1/content/2301.04424v1.pdf'} diff --git a/7dE0T4oBgHgl3EQffQBI/content/tmp_files/2301.02401v1.pdf.txt b/7dE0T4oBgHgl3EQffQBI/content/tmp_files/2301.02401v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f6435078a5524d4b0d368e693f649658d2873fcf --- /dev/null +++ b/7dE0T4oBgHgl3EQffQBI/content/tmp_files/2301.02401v1.pdf.txt @@ -0,0 +1,1848 @@ +You Truly Understand What I Need +: Intellectual and Friendly Dialogue Agents grounding +Knowledge and Persona +Jungwoo Lim1, Myunghoon Kang1∗, Yuna Hur1∗, Seungwon Jung1∗, Jinsung Kim1∗, +Yoonna Jang1, Dongyub Lee3, Hyesung Ji2, Donghoon Shin2, +Seungryong Kim1§ and Heuiseok Lim1§ +1Korea University, 2Dialogue Tech Division, NCSOFT, 3Naver Corporation +{wjddn803,chaos8527,yj72722,redlion0929,jin62304,seungryong_kim,limhseok}@korea.ac.kr, +{hyesung84,dhshin}@ncsoft.com, dongyub.lee@navercorp.com +Abstract +To build a conversational agent that interacts +fluently +with +humans, +previous +studies +blend knowledge or personal profile into the +pre-trained language model. However, the +model that considers knowledge and persona +at the same time is still limited, leading to +hallucination and a passive way of using +personas. We propose an effective dialogue +agent that grounds external knowledge and +persona simultaneously. The agent selects +the proper knowledge and persona to use for +generating the answers with our candidate +scoring implemented with a poly-encoder. +Then, our model generates the utterance with +lesser hallucination and more engagingness +utilizing +retrieval +augmented +generation +with +knowledge-persona +enhanced +query. +We conduct experiments on the persona- +knowledge chat and achieve state-of-the-art +performance in grounding and generation +tasks on the automatic metrics. Moreover, +we validate the answers from the models +regarding +hallucination +and +engagingness +through human evaluation and qualitative +results. We show our retriever’s effectiveness +in extracting relevant documents compared +to +the +other +previous +retrievers, +along +with the comparison of multiple candidate +scoring +methods. +Code +is +available +at +https://github.com/dlawjddn803/INFO +1 +Introduction +To +build +an +ultimate +conversational +agent +that interacts with humans fluently, previous +studies provide generative neural network-based +models (Sordoni et al., 2015; Vinyals and Le, +2015). Although the answers generated from those +models are plausible, they lack informativeness +and engagingness resulting in bland responses +compared to humans (Li et al., 2016; Gao et al., +* +Equal Contributors +§ Corresponding author +Dialogue +Human: Is it in England? +Machine: No, it is actually in Scotland where you are going. +Human: Where in Scotland? +Human’s Persona +I will travel through North Ayrshire. +I am going to Scotland. +I like history. +I am interested in architecture. +I love to garden. +Ground Truth Knowledge +Eglinton Castle was a large Gothic castellated mansion in +Kilwinning, North Ayrshire, Scotland. +Predicted Answers +BARTbase +It is in Scotland, which is a place you love. +BARTlarge +It is in Scotland. in Scotland. in Scotland. in +Ground Truth Response +It is in North Ayrshire so you could visit when you travel through. +Table 1: Example of the generated answers from a +typical generative model, i.e., BART. We can find that +BARTbase uses different persona sentence which has +not appeared human’s personal profiles resulting in +hallucinated answer. Also, BARTlarge generates less +engaging answers by making use of the knowledge only +to answer the question. Both generated responses are in +the situation of hallucination and are less engaging. +2018). However, for knowledgeable and attractive +conversation, people usually provide informative +replies by considering the background of the person +whom they are talking to. Towards a human-like +manner of dialogue, Ghazvininejad et al. (2018) +and Dinan et al. (2018) introduce the knowledge- +grounded conversation for the knowledgeable +and informative responses, whereas Zhang et al. +(2018a) suggest the persona-grounded dialogue for +the personalized responses to the users. +To improve the machine’s answer with the +external knowledge base, one injects the factual +knowledge into the parameters of the language +model (Raffel et al., 2020; Roberts et al., 2020). +Despite the models’ capability of utilizing external +knowledge implicitly, they produce “hallucinations” +in the responses (Marcus, 2020). The hallucination +arXiv:2301.02401v1 [cs.CL] 6 Jan 2023 + +in the dialogue involves the situation where +the generated output contradicts the reference +knowledge. Also, it includes the situation when +the generated output cannot be confirmed from the +knowledge source (Ji et al., 2022). To mitigate these +hallucinated answers, hybrid models employing +parametric memory with non-parametric (i.e., +retrieval-based) memory are introduced to directly +access external memories, leading the source to be +inspected and interpreted (Karpukhin et al., 2020; +Petroni et al., 2020; Lewis et al., 2020b). +On the other hand, Zhang et al. (2018a) suggest +persona-chat dialogues with the corresponding +personal profiles of each interlocutor to avoid +general and monotonous answers from the machine. +Though See et al. (2019); Liu et al. (2020) show +comparable quality in generating personalized +conversation, the generated utterances merely +confirm each interlocutor’s persona resulting +in a passive manner of speaking such as “I +have four children”. In addition, the incoherent +topics of the dialogues lead to shallow levels +of conversation between the interlocutors. To +elaborate on this chit-chat conversation supported +by external knowledge, Jang et al. (2022) presents +a novel persona-knowledge chat with a generative +model that considers persona information and +world knowledge altogether. Despite obtaining +the knowledge and persona when generating the +answers, the generative models’ responses still +exhibit both hallucination and lesser engagingness +as in Table 1. +In this paper, we propose INFO (Intellectual +and Friendly dialOg agents) that responds with +external knowledge and persona simultaneously. +Owing to the enhanced capturing relevancy +between the context and each candidate set, +the knowledge selector and persona selector for +the grounding task are implemented with the +poly-encoder. To alleviate hallucinated responses +from the model, we adopt retrieval-augmented +generation (RAG) (Lewis et al., 2020b) by +utilizing non-parametric memory and parametric +generator in addition to the enhanced input +query. By injecting predicted sources as input +to the retrieved-augmented generator, our model +maintains consistency between grounding and +generation while training. Therefore, our model +generates more knowledgeable and engaging +answers in an active manner with less hallucination. +We show that INFO achieves the highest +scores on both grounding and generation tasks in +empirical experiments. Also, we compare diverse +candidate scoring modules including bi-encoder, +cross-encoder, and poly-encoder and demonstrate +their effect on generation. We additionally conduct +experiments to show the effectiveness of the +retriever module compared to sparse and dense +retrievers. The qualitative results and human +evaluation are also presented to validate our +model’s capability to generate human-like answers. +Our contributions are as follows: +• We propose the model that grounds persona +information and external knowledge with +lesser hallucination and adequate utilization of +persona in an active manner simultaneously. +• Our approach suggests that the generated +responses from the model are interpretable +regarding what the model refers to while +generating. +• We show that INFO achieves the SoTA +performance in all of the automatic metrics +and demonstrate its comparable quality with +human evaluation and qualitative analysis. +2 +Related Works +2.1 +Knowledge Grounded Conversation +To let the neural network models ground external +knowledge and generate informative answers, +Ghazvininejad et al. (2018) suggests a data- +driven neural conversational agent that provides +knowledgeable +answers. +Also, +Dinan +et +al. +(2018) introduces open-domain dialogue where +the two speakers are talking with Wikipedia +knowledge. To inject the external knowledge +into the pre-trained language model efficiently, +Raffel et al. (2020); Roberts et al. (2020) +success in equipping the knowledge into the +parameters and show comparable performance +in open-domain question and answering tasks. +However, the approach is not capable of expand +or revise their inherent knowledge and provides +hallucination (Marcus, 2020). To overcome the +limitations, Lewis et al. (2020b) combines a +pre-trained parametric model and non-parametric +memory for the open-domain question and +answering to reduce hallucination. Since their non- +parametric memory can be updated without extra +pre-training, revising knowledge is more efficient. +Furthermore, it is found that a retrieval-augmented + +Figure 1: Overview of our method. U is the input comprises dialogue history and knowledge snippet, and cand +denotes each candidate from the grounding tasks. The grounding score is obtained through the dot product +operation with the representation of input context Udial and candidate at. The predicted sources convert into the +knowledge-persona enhanced query (KPEQ) with dialogue history and KPEQ is fed into the retrieval-augmented +generator to generate the responses. +generator also reduces hallucination in knowledge- +grounded conversation as well (Shuster et al., +2021), and a similar approach recently achieves +outstanding performance in knowledge-grounded +conversation (Paranjape et al., 2021). +2.2 +Persona Grounded Conversation +In order to alleviate bland and general answers +with consistent personality, Zhang et al. (2018a) +constructs a persona-chat dataset. In the dataset, +the two interlocutors chat with the persona +profile sentences. Along with this dataset, Zhang +et al. (2018a) introduces the model with a +profile memory network by considering the +dialogue history to perform attention over the +persona. They enlarge the persona-chat dataset +with Reddit corpus, and pre-trained the model +with these dataset. After that, they fine-tune pre- +trained model on the persona-chat (Mazare et al., +2018). Also, Liu et al. (2020) trains a receiver +to reinforce the mutual persona understanding +between interlocutors, and Wolf et al. (2019) utilize +pre-trained models (Radford et al., 2019) to build +personalized dialogue agents. +2.3 +Encoders for Sentence Scoring +There exist diverse encoder structures for sentence +scoring. Bi-encoder scores the relevance between +sentences by feeding context and candidates into +separate encoders. An example of bi-encoders +are memory networks +(Zhang et al., 2018a), +transformer memory networks (Dinan et al., +2018), LSTM (Lowe et al., 2015). Since bi- +encoder calculates with cached encoded sentence +representations, it is relatively fast in computation. +However, the bi-encoder has a limitation of +capturing mutual information between context +and candidates. Cross-encoder, on the other hand, +scores by aligning context and candidates in +one sequence. A type of cross-encoders is a +sequential matching network that is based on +deep matching networks (Yang et al., 2018) and +gated self-attention (Zhang et al., 2018b). Although +using a cross-encoder can achieve rich interaction +between the sentences within the encoder, the +problem of slow processing still remains. To +exploit both benefits of each model, poly-encoder +adopts attention mechanism into the bi-encoder +architecture and shows satisfactory performances +as cross-encoder with fast inference time (Humeau +et al., 2019). For the enhanced representation of +grounding knowledge and persona, we employ a +poly-encoder as a selector for each grounding task. +3 +Method +To generate more knowledgeable and engaging +dialogue, we introduce our conversational model +that grounds external knowledge and persona +information as in Figure 1. We first encode the +input with the pre-trained language model, and +then choose the proper knowledge and persona +from the given candidates for each selector. We +employ poly-encoder +(Humeau et al., 2019) +as knowledge selector and persona selector to +exploit its enhanced capability of capturing +relevance between candidate set and context (i.e., +dialogue history). Then, the predicted persona +and knowledge are aligned into one sequence + +KPEQ +Retriever(Non-Parametric) +Poly-encoder + Knowledge Selector + Document Index +Uaial +O- + Score + Poly- +U +Z1-08 +encoder +Attention +Z7777 +Z1-03 +Z2-02 +Uaial +Uaal +acand +Knowledge +Z1-01 +Candidate +Z2-09 +C +CM +Dialogue +Z2-05 +Z2-07 +Attention +Attention +Candidate Aggregator +Persona +Candidate +个 +↑ +h1 +hn +h2 +a2 +a1 +aT +Persona Selector +↑ +↑ +Persona +Marginalize +Generator +Generated +Context Encoder +Candidate Encoder + Poly- +Level +(Parametric) +Answer +↑ +T +↑ + encoder +Indicator +U + candi +cand2 +candTto the dialogue history for consistency between +grounding and generation. The sequence is defined +as a knowledge-persona enhanced query (KPEQ), +then it feeds into the retriever-augmented generator +(RAG). The generator then extracts the relevant +paragraphs to refer from the knowledge index to +reduce hallucination. +3.1 +Input Construction +The +given +dialogue +is +notated +as +{(uhm +1 , umc +1 ), ...(uhm +o +, umc +o )}, +where +o +is +the +number of rounds. uhm and umc indicate the +utterances of human and machines, respectively. +We first take o-th round dialogue history, except +for the final machine’s reply umc +o , for the initial +input for the model. We define the clue of the +dialogue as knowledge snippet clk to inform the +machine of which topic the user is interested in. +The knowledge snippet is the name of the landmark +that the user encounters, which is given topic from +the dialogue. We then align the dialogue history +and knowledge snippet into the one sequence for +the model input as U = {uhm +1 , umc +1 , ...uhm +o +, clk}. +3.2 +Model Components +3.2.1 +Poly-Encoder Based Candidate Scoring +For knowledge and persona grounding tasks, we +suggest poly-encoder-based candidate scoring to +leverage the capability of capturing the semantic +similarities between the context input and the +candidates. It is employed to select proper sources +to be used when generating the utterance. When +the context input U comes in, we compute +the grounding scores of each candidate utilizing +the embeddings of context input and encoded +candidates in the poly-encoder. The grounding +score is used to select the most suitable source(s) in +the knowledge selector and persona selector, which +will be introduced in the following Section 3.2.2 +and 3.2.3. +In poly-encoder architecture (Humeau et al., +2019), candidates are fed into the candidate +encoder and denoted as {a1, ..., aT } where T is +the number of candidates in the set. Each candidate +embedding at is the first output of the candidate +encoder, which is represented by the transformer +model. After encoding candidates, the context +input (i.e., dialogue history) is embedded with +a separate context encoder. Unlike the candidate +encoder, the context encoder embeds the dialogue +into multiple vectors through M context codes +{c1, ...cM}, which are learned for capturing diverse +aspects of a given context rather than using one +embedding. Each context code is used to extract +U m +dial by attending over all the previous layer’s +output as follows. +U m +dial = +� +j +wcm +j hj +(1) +Note that the h1, ..., hn is the output of the pre- +trained language model and n is the number of +tokens in the input. The weights are computed as +(wcm +1 , ..., wcm +n ) = softmax(cm · h1, ..., cm · hn). +Then, the final attention proceeds between +the global features of the input and a given +candidate. In other words, the final dialogue feature +Udial is obtained by aggregating each dialogue +feature U m +dial, while gaining richer interactions with +context codes as in Equation 2. +Udial = +� +m +wmU m +dial, +(2) +where +w1, ..., wM +can +be +obtained +from +softmax(at · U 1 +dial, ..., at · U M +dial). +The final predicted candidate is chosen based +on the highest score that is acquired from the dot +product operation as (Udial · at). +3.2.2 +Knowledge Selector (KS) +We build a knowledge selector for the knowledge +grounding task, employing poly-encoder-based +candidate scoring. When the grounding scores are +produced from the candidate scoring module, the +label with the highest score is selected as the +predicted knowledge. +The knowledge loss LKG for the knowledge +grounding task is computed with cross-entropy +loss (Brier et al., 1950) as in Equation 3. +LKG = − +� +j +klj · log ˆ +klj, +(3) +klj is the ground-truth label from the knowledge +candidates of the j-th example. +3.2.3 +Persona Selector (PS) +We also implement a persona selector for +the persona grounding task. Since multiple +personas can be chosen to generate the responses, +consideration of one or more persona sentences +are needed. Similar to the knowledge selector, +we assign the grounding score to each persona + +candidate with the candidate scoring module as +in Equation 1 and 2. +When the scores of each candidate are computed +from the candidate scoring module, then the +persona level indicator classifies which the number +of the persona should be selected with the [CLS] +token of the model input U. After predicting the +level of persona-engagingness, we pick persona +sentences to be grounded according to the number +predicted. For example, if the persona level +indicator predicts 2, then top-2 persona sentences +are chosen in the persona grounding task. The +selected persona sentence(s) are marked as 1 +otherwise, 0. We use binary cross-entropy loss for +persona grounding as in Equation 4. +LPG = +− +� +j +plj · log ˆ +plj + (1 − plj) · log(1 − ˆ +plj) +(4) +Note that plj is the ground-truth label from the +knowledge candidates of the j-th example. +3.2.4 +Query-Enhanced Generator +Following the works of Lewis et al. (2020b), +we exploit the retrieval augmented generation’s +capability to reduce hallucination and access the +memory directly. For a consistent way of training +while solving grounding and generation tasks, +we reconstruct the query that feeds into the +retriever. When the knowledge and persona are +predicted from each selector, we aggregate them +with dialogue history into one sequence. Then, the +final query is denoted as KPEQ = {U; ˆP; ˆK} and +defined as a knowledge-persona enhanced query. ˆP +and ˆK are predicted persona and knowledge from +each candidate set, respectively. +The retriever rη aims to search top-K latent +paragraphs with the KPEQ. We utilize a pre- +trained dense passage retriever (DPR) (Karpukhin +et +al., +2020) +trained +on +natural +question +dataset (Kwiatkowski et al., 2019) which has +parametric memory and bi-encoder architecture to +retrieve a latent document embedding following +Lewis et al. (2020b) : +rη(z|KPEQ) ∝ exp(d(z)⊤q(KPEQ)), +(5) +where d(·) is an embedding from a document +encoder and q(·) is a representation from query +encoder, both implemented with BERTbase. z +denotes the list of document. +With the relevant paragraphs from the retriever, +we employ RAG-Token architecture as the +generator to borrow its strength of predicting +each target token based on top-K different +paragraphs. Since RAG-Sequence, which has a +different architecture to RAG-Token, uses the same +document from the retriever to predict each token +as depicted in Equation 6, the result may opt to +depend on the retrieved document (Lewis et al., +2020a). The two different versions of RAGs (Lewis +et al., 2020b) are as follows: +SRS(y|x) ≈ +� +z∈top-k(p(·|x)) +rη(z|x) +N +� +i +gθ(yi|x, z, y1:i−1) (6) +SRT(y|x) ≈ +N +� +i +� +z∈top-k(p(·|x)) +rη(z|x)gθ(yi|x, z, y1:i−1), +(7) +where SRS indicates our method with RAG- +Sequence architecture and SRT denotes ours with +the RAG-Token model. x is a token of KPEQ and +yi is a single token from the ground truth responses. +Also, z is a retrieved paragraph from the retriever +and N is the maximum sequence length. +The SRT generator g(·) marginalizes the loss +from different paragraphs when generating answers. +In detail, the generator outputs a distribution +for the next token for each document before +marginalizing as in Equation 7 where η denotes +the parameter of the retriever, and θ indicates the +parameter of the generator. After that, the generator +repeats the process with the following output +token. Finally, the SRT aims to generate the next +token following an auto-regressive manner with a +standard beam search. In other words, the model +minimizes the negative marginal log-likelihood for +each input/output pair (KPEQj, yj). The language +model loss is formulated as : +LS = − +� +j +logp(yj|KPEQj) +(8) +3.3 +Final Objectives +We then train the full model in the multi-tasking +manner. The full objectives of the model is +indicated as Equation 9. +L = λKGLKG + λPGLPG + λSLS +(9) + +Models +Generation +Grounding (Acc.) +chrF++ +BLEU +R-1 +R-2 +R-L +BERTScore +Persona +Knowledge +GPT2small +28.73 +11.43 +36.58 +19.44 +32.62 +88.56 +67.44 +69.59 +GPT2medium +30.12 +12.31 +38.29 +21.17 +34.12 +88.92 +67.44 +72.42 +BARTbase +29.77 +11.99 +36.24 +19.73 +32.13 +88.35 +67.45 +72.18 +BARTlarge +30.69 +11.91 +36.57 +19.83 +32.05 +88.10 +67.44 +71.01 +INFO (SRS) +51.33 +29.36 +53.36 +40.36 +51.16 +92.00 +82.70 +99.24 +INFO (SRT ) +53.29 +31.46 +58.26 +42.35 +53.06 +92.29 +80.87 +99.22 +Table 2: Main results on the official validation set. SRS denotes our method with RAG-Sequence architecture and +SRT indicates the model with RAG-Token model as generator. The models are evaluated by generation metrics, +including chrF++, BLEU, ROUGE-1 (R-1), ROUGE-2 (R-2), ROUGE-L (R-L), and BERTScore. +We control the proportion of each task and we +set λKG, λPG, and λS as 1:1:5 for the experiments, +respectively. We find the value of each λ with +manual search. +4 +Experiments +4.1 +Experiment Details +Dataset +FoCus (Jang et al., 2022) is the dataset +for customized dialogue benchmark, where each +conversation is directly grounded with knowledge +and persona. The dataset includes knowledge- +aware dialogue with personal profiles between +humans and machines. There are 12,484 dialogues +about 5,152 knowledge sources from Wikipedia +and 32,855 persona sentences. To validate the +knowledge grounding capability and customized +dialogue generation, we evaluate our method +with the official FoCus validation set for the +effectiveness of experiments since the result from +the official test set can be tested only through the +leaderboard*. +Experimental Setup +For each candidate scoring +module, we implement poly-encoder (Humeau +et al., 2019) with BERTlarge, and the number of +context codes is 16. For the dialogue generation, we +implement our method with Hugging Face (Wolf +et al., 2020) and use facebook/rag-token-nq as +the backbone model. We use the same architecture +of retriever and generator from RAG along +with the decoding and leverage our knowledge +index for non-parametric query-document ranking +with FAISS library (Johnson et al., 2019). The +knowledge index consists of the paragraphs from +the given Wikipedia knowledge entitled with the +name of the given landmark. We set learning rate +as 6.25e-6 with AdamW (Kingma and Ba, 2014) +*https://codalab.lisn.upsaclay.fr/competitions/3754 +for the optimization. The batch size is set as 32, +and the number of dialogue history is 1. The whole +model was trained for three epochs on RTX A6000 +GPU and took 8 hours per one epoch. +Baselines +We implement the baselines from +previous study (Jang et al., 2022) and we conduct +experiments with GPT-2 (Radford et al., 2019) and +BART (Lewis et al., 2020a) as well. For a fair +comparison, we demonstrate the results on GPT- +2small, which has 12 layers, and BARTbase, which +has 6 encoders and 6 decoder layers. Also, GPT- +2medium contains 24 layers of the decoder, and +BARTlarge possesses 12 layers for each encoder +and decoder. +4.2 +Automatic Evaluation +We show the main results on the FoCus dataset +with automatic metrics in grounding and generation +tasks. The official metrics for the benchmark are +chrF++ (Popovi´c, 2017), BLEU (Papineni et al., +2002), ROUGE-1, ROUGE-2, and ROUGE-L (Lin, +2004). To consider the semantic similarity score +for each token between candidate and reference +sentences using contextual representation, we +additionally adopt BERTscore (Zhang* et al., +2020). For grounding task, we used accuracy for +both knowledge and persona grounding, and F1 +score for the persona grounding. +In Table 2, it is found that our method shows +substantial improvements in all the metrics from +generation to grounding compared to the baselines. +Especially, the performances of INFO increase +over 18% at least regarding the generation metrics +except for BERTScore. Furthermore, our model +achieves remarkable success in persona and +knowledge accuracy. Unlike the performance in +other generation metrics, SRS demonstrates better +persona accuracy than SRT . This result might be + +Model +Generation +Grounding +chrF++ +BLEU +R-1 +R-2 +R-L +BERTScore +Persona +(Acc.) +Persona +(F1) +Knowledge +(Acc.) +SRT +Bi-encoder +51.83 +29.51 +56.35 +40.80 +51.37 +91.86 +88.10 +38.20 +99.18 +Cross-encoder +49.90 +27.18 +53.57 +38.25 +49.29 +91.52 +87.09 +35.32 +99.49 +Poly-encoder +53.29 +31.46 +58.26 +42.35 +53.06 +92.29 +80.87 +39.56 +99.22 +Table 3: Performances comparison between the encoding modules for grounding tasks +attributed to the architecture of the generator, which +is more applicable to sentence classification tasks +such as persona grounding. The official test result is +also demonstrated in Appendix A, but BERTscore +is missing due to the unreleased ground truth. +4.3 +Human Evaluation +We conduct a human evaluation to validate +the responses from our model through Amazon +Mturk services†. The assessment criteria are +fluency, adequacy, provenance, engagingness, and +hallucination. In specific, provenance is the level of +utilization of the ground truth knowledge into the +responses, whereas engagingness means how much +the answers are persona-related. Also, hallucination +indicates whether the answer contradicts the +persona and knowledge or cannot be verified +from the source content. We randomly chose 50 +dialogues from the official test set, and three +workers were allocated to evaluate each dialogue +generated by our model and baselines. We asked +the workers to rank the answers according to each +criterion following Cho and May (2020). Rank is +scaled from 1 to 5, and the lower number is mapped +to the better quality except for hallucination. The +agreement between the annotators is calculated +with Fleiss’ Kappa coefficient and is 0.4185 +indicating fair agreement. The relations between +the annotators hardly exist since we collect the +results from the Amazon Mturk workers. +As in Table 4, INFO surpasses BARTbase, +BARTlarge, GPT-2small and GPT-2medium in all +of the criteria. INFO achieves the highest rank in +adequacy, fluency, and provenance and generates +a more human-like response than other generative +models. Also, the workers ranked our model the +lowest when they were asked to rank the responses +in the most hallucinated order. Thus, it can be found +that INFO generates more engaging and fewer +hallucination utterances with respect to the human. +The distribution of the rank per each criterion is +illustrated in Appendix B. +†https://www.mturk.com/ +Models +Avg. Rank +Ad. ↓ +Fl. ↓ +Prov. ↓ +Eng. ↓ +Hall. ↑ +GPT-2small +3.57 +3.41 +3.58 +3.46 +2.49 +GPT-2medium +3.11 +3.10 +3.04 +3.25 +3.02 +BARTbase +3.43 +3.29 +3.47 +3.22 +2.45 +BARTlarge +3.31 +3.63 +3.29 +3.44 +2.69 +INFO (Ours) +1.57 +1.57 +1.62 +1.63 +4.35 +Table 4: Human evaluation. The value in the table +is the average rank of the each model’s response. +The +abbreviation +Ad. +Fl. +Prov. +Eng. +and +Hall +denote adequacy, fluency, provenance, engaginess, and +hallucination, respectively. +5 +Results and Analysis +5.1 +Variants on Candidate Scoring Module +To validate the poly-encoder as a candidate +scoring module, we apply diverse candidate scoring +modules, including the bi-encoder and cross- +encoder. From the results in Table 3, we can find +that the poly-encoder outperforms in the generation +task. In the grounding task, SRT with cross-encoder +scoring shows improved accuracy on grounding +persona and knowledge. The result seems to be +SRT with bi-encoder and cross-encoder are better +than that with poly-encoder. However, the F1 +score of INFO is higher than the two candidate +scoring modules implying that low accuracy in +persona is due to the tendency of active use on the +persona in poly-encoder while the other two models +opt to predict not to use persona sentence. The +results suggest that the high accuracy of persona +not always guarantees the engagingness in the +dialogue. +5.2 +Comparison on other Retrievers +We show that INFO is effective in retrieving +knowledge compared to other sparse and dense +retrievers. We retrieve the knowledge from our +knowledge index built with Wikipedia paragraphs. +We utilize TF-IDF (Joachims, 1996), and deep +passage retrieval (DPR) (Karpukhin et al., 2020). +In the case of TF-IDF, we set the sum of query + +and knowledge tokens less than or equal to +512, which is the maximum sequence length of +DPR and INFO. We use bert-base-uncased +as the tokenizer. For DPR, we extract less than +40 knowledge using TF-IDF due to memory +limitations. We first retrieve the five paragraphs +related to the query that comprises knowledge +snippet, dialogue history, predicted knowledge +candidate, and selected persona sentences. In Table +5, we find that the retriever we used outperforms +compared to the TF-IDF and DPR in all the +metrics, including BERTscore. The results imply +that INFO’s retriever is suitable for extracting +similar paragraphs rather than other retrievers. +Model +chrF++ +BLEU +R-1 +R-2 +R-L +BERTScore +TF-IDF +19.91 +3.52 +13.91 +9.96 +12.43 +51.54 +DPR +20.57 +3.86 +12.44 +6.55 +10.20 +47.48 +INFO +26.36 +7.40 +15.48 +12.18 +14.32 +53.14 +Table 5: Comparison with other retrievers +5.3 +Effect of Selectors on Generation +We measure each selector module’s effect on +the generation task by changing the query which +feds into the retriever on a validation set. The +experimental results are shown in Table 6, where +GTK, GTP represents ground truth knowledge and +persona. Although the query that comprises the +ground truth source shows the highest scores, INFO +demonstrates comparable results on the generation +task. From the result where the performance +increase of INFO + GTP is larger than that of +INFO + GTK about 2.8%p, we can identify that +our persona selector still has more space to achieve +its maximum level. +Query +chrF++ +BLEU +R-1 +R-2 +R-L +BERTScore +INFO (RT) +53.29 +31.46 +58.26 +42.35 +53.06 +92.29 ++GTK +53.35 +31.56 +58.31 +42.55 +53.18 +92.29 ++GTP +56.19 +34.39 +61.61 +45.46 +56.01 +92.79 ++GTK+GTP +56.40 +34.60 +61.88 +45.64 +56.16 +92.84 +Table +6: +Comparison +between +the +generation +performances based on the variants of query with +ground truth knowledge and persona. Note that all the +performance is evaluated with the official validation +set. +5.4 +Qualitative Analysis +In Table 7, an example from the predicted results +is illustrated. In the case of BARTlarge, and GPT- +2medium, the responses only reflect the ground +Given Landmark +Finding Nemo Submarine Voyage +Dialogue +Human: What area of the park is this ride in? +Machine: This ride is located in the Tomorrowland area of Disneyland. +Human: Has this ride always been about Finding Nemo? +Human’s Persona +I’ve never been to California. +My favorite cartoon is Finding Nemo. +I would like to visit Disneyland. +My favorite color is yellow. +I enjoy swimming. +Ground Truth Knowledge (Grounding) +Based on the characters and settings of the 2003 Disney · Pixar, Finding +Nemo, it is a re-theming of the classic Submarine Voyage attraction that +operated from 1959 to 1998. +Retrieved Knowledge (Generation) +The original Submarine Voyage was built in 1959 as part of the then new +Tomorrowland... +In 2008, Finding Nemo Submarine Voyage received an award for +outstanding achievement from the Themed Entertainment Association. +Finding Nemo Submarine Voyage is ... which opened on June 11, 2007. ... +it is a re-theming of the classic Submarine Voyage attraction that operated +from 1959 to 1998. +On July 15, 2005, two days before the 50th Anniversary of Disneyland, +the Finding Nemo Submarine Voyage was officially announced at the +new Turtle Talk... +The attraction reuses the eight original 1959 Submarine Voyage through +Liquid Space attraction... +Predicted Answers +BARTbase +Yes, that’s right. You’re a fan of the “Fantasy” film, +so I. +BARTlarge +Yes, the ride is based on the characters and settings +of the 2003 Disney · Pixar film +GPT-2small +No, it was originally a way to show that you love +Finding Nemo. +GPT-2medium +Yes, it has operated from 1959 to 1998. +INFO (Ours) +No, this attraction is actually a re-theme of the +classic submarine voyage attraction that operated +from 1959 to 1998. The attraction is based on the +characters and settings of the 2003 Disney Pixar +film Finding Nemo, which is your favorite cartoon. +Ground Truth Response +No, your favorite cartoon is a new addition to this ride. The current +Finding Nemo ride is a re-theming of the classic “Submarine Voyage” +attraction that operated here from 1959 to 1998. +Table 7: Qualitative result. All the predicted results +in grounding task are from our model, INFO and it +predicts the correct answers in both tasks. We add other +baselines’ responses for comparative analysis. +truth knowledge resulting in less engaged answers +without any persona-related phrases. Although +BARTbase seems to employ a persona sentence in +the form of the phrase “You’re fan of the Fantasy +film”, its used sentence does not appear in human’s +personal profiles. This result also indicates that +the utterance is hard to identify its provenance +on the knowledge source. Moreover, GPT-2small +generates the utterance that contradicts the ground +truth knowledge. From the result, we can find that +the generated responses from the baselines show +hallucinations on both persona and knowledge. +Unlike other baselines, our model blends ground +truth knowledge and persona sentence into the + +response with less hallucination and engagingness. +In addition, the retrieved knowledge source that +our model refers to provides interpretability and +provenance of the responses to the users. More +examples are also depicted in Appendix C. +6 +Conclusions +In this paper, we presented a conversational +agent that generates responses grounding the +user’s persona and external knowledge. We +utilized poly-encoder-based candidate scoring +for +each +grounding +task. +We +additionally +implement persona level indicator to consider +multiple persona selections for delicate persona +grounding. With predicted sources, we construct +a +knowledge-persona +enhanced +query +to +retrieve latent paragraphs, and they are used +to generate informative and engaging responses by +marginalizing loss for each token. We show that +our method achieves the state-of-the-art (SoTA) +score in both grounding and generation tasks in the +persona-knowledge conversation dataset. We also +demonstrate that the responses from INFO show +less hallucination and more engagingness through +human evaluation and qualitative analysis. We also +compare the grounding modules and retrievers to +show INFO’s effectiveness. +7 +Limitations +The proposed model INFO has limitations. Given +the INFO’s settings, the model cannot deal with +real-world application, which means the absence +of ground truth knowledge or persona candidates +in the grounding task. We also conducted the +human evaluation to evaluate the capability of +the proposed model’s mitigating hallucination +in dialogue generation. However, the number +of cases is relatively small for evaluating the +capability of mitigating hallucination. Finally, +INFO demands high GPU computation resources, +since it marginalizes loss at the token level. +We plan to improve the INFO for future work. +We will train and evaluate the INFO in open- +domain settings as well as real-world settings for +the applicable conversational agents. Moreover, we +will conduct human evaluations with more cases. +Especially, we will enhance the way of quantitative +measurement for the model’s hallucinated answers. +Last but not least, we will improve the generator +of INFO with more computationally efficient +components. +8 +Acknowledgement +This +work +was +supported +by +Institute +of +Information +& +communications +Technology +Planning & Evaluation(IITP) grant funded by the +Korea government(MSIT) (No. 2020-0-00368, +A +Neural-Symbolic +Model +for +Knowledge +Acquisition and Inference Techniques), This +research was supported by the MSIT(Ministry +of +Science +and +ICT), +Korea, +under +the +ITRC(Information Technology Research Center) +support +program(IITP-2022-2018-0-01405) +supervised by the IITP(Institute for Information +& Communications Technology Planning & +Evaluation), This work was supported by Institute +for Information & communications Technology +Planning & Evaluation(IITP) grant funded by the +Korea government(MSIT) (No. 2022-0-00369, +(Part 4) Development of AI Technology to support +Expert Decision-making that can Explain the +Reasons/Grounds for Judgment Results based on +Expert Knowledge) +References +Glenn W Brier et al. 1950. 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Modeling multi- +turn conversation with deep utterance aggregation. +In Proceedings of the 27th International Conference +on Computational Linguistics, pages 3740–3752. + +A +Automatic Evaluation on Official Test Set +Models +Generation +Grounding (Acc.) +chrF++ +BLEU +R-1 +R-2 +R-L +Persona +Knowledge +GPT2small +28.83 +11.60 +36.28 +19.56 +32.42 +67.83 +70.95 +GPT2medium +30.34 +12.58 +38.35 +21.16 +34.34 +67.64 +72.46 +BARTbase +29.80 +12.15 +36.26 +19.73 +32.06 +67.66 +72.02 +BARTlarge +30.63 +11.86 +36.36 +19.42 +31.73 +67.62 +70.53 +INFO (RS) +52.81 +29.41 +56.37 +40.41 +51.16 +82.74 +98.88 +INFO (RT) +54.61 +32.33 +58.27 +42.39 +53.09 +80.83 +99.10 +Table 8: Main results on the official test set. RT indicates the model with RAG-Token model as generator. The +models are evaluated by generation metrics, including chrF++, BLEU, ROUGE-1 (R-1), ROUGE-2 (R-2) and +ROUGE-L (R-L). The accuracy for persona grounding task and knowledge grounding task are also noted. Since +BERTscore is not the official generation metric, we cannot evaluate the result on the metric as the ground truth of +the test is not yet disclosed. +B +Human Evaluation Distribution on Each Criteria +(a) Adequacy +(b) Fluency +Figure 2: The distribution of the rank on the adequacy and fluency criteria. Guide A to E indicates INFO, BARTbase, +BARTlarge, GPT-2small, and GPT-2medium, in the order. + +Guide A +Guide B +100 +Guide C +Guide D +Guide E +80 +f evaluation +60 +of +40 +# +20 +0 +1 +2 +3 +4 +5 +RankGuide A +Guide B +100 +Guide C +Guide D +Guide E +80 +f evaluation +60 +JO +40 +# +20 +0 +1 +2 +3 +4 +5 +Rank(a) Provenance +(b) Engagingness +Figure 3: The distribution of the rank on the provenance and engagingness criteria. Guide A to E indicates INFO, +BARTbase, BARTlarge, GPT-2small, and GPT-2medium, in the order. +Figure 4: The distribution of the rank on the less hallucination criterion. Note that the highest rank (1) means the +most hallucinated. Guide A to E indicates INFO, BARTbase, BARTlarge, GPT-2small, and GPT-2medium, in the +order. + +Guide A +Guide B +100 +Guide C +Guide D +Guide E +80 +f evaluation +60 +of +40 +# +20 +0 +1 +2 +3 +4 +5 +RankGuide A +100 +Guide B +Guide C +Guide D +Guide E +80 +f evaluation +60 +of +40 +# +20 +0 +1 +2 +3 +4 +5 +RankGuide A +Guide B +100 +Guide C +Guide D +Guide E +80 +f evaluation +60 +of +40 +# +20 +0 +2 +3 +5 +1 +4 +RankC +Qualitative Results +Given Landmark +Nocton Hall +Dialogue +Human: I know this place, but I don’t remember the name of this place. +Human’s Persona +I have a hall in my house. +I have worked with military hospital. +I would like to go to England. +I have heard about Frederick John Robinson. +I have respect to American Officers. +Ground Truth Knowledge (Grounding) +Today the site is maintained by English Heritage. +Retrieved Knowledge (Generation) +“Hogesdon” is first recorded in the Domesday Book, meaning an Anglo- +Saxon farm (or “fortified enclosure”) belonging to Hoch, or Hocq.... +By 1677, the church was in such poor structural condition that the Dean +of Chichester convened a consistory court with three parishioners and +asked them why repairs had not been carried out... +he south transept had historically served as the mausoleum for the +Cavendish family, who resided at nearby Latimer... +Following the diminished influence of the ancient Priory, predecessor of +the two parishes of St Bartholomew, disputes began to arise over rights +to tithes and taxes payable by lay residents who claimed allegiance with +the nearby and anciently associated parish of St Botolph Aldersgate - +an unintended consequence and legacy of King Kenry VIII’s religious +reforms. +a wall-mounted stone pulpit, a stone altar, a series of tall, pointed +windows high in the walls, an unadorned stone font and a short wooden +steeple-little more than an extended belfry-extending from the nave +roof...a wall-mounted stone pulpit, a stone altar, a series of tall, pointed +windows high in the walls, an unadorned stone font and a short wooden +steeple-little more than an extended belfry-extending from the nave roof... +Predicted Answers +BARTbase +This is the Nocton Hall, a historic Grade II listed +building in the village of Nocton, +BARTlarge +Nocton Hall is a historic Grade II listed building in +the village of Nocton, in Lincoln +GPT-2small +This is Nocton Hall, a historic Grade II listed +building in the village of Nocton, in +GPT-2medium +This is the Nocton Hall, a historic Grade II listed +building in the village of Nocton, +INFO (Ours) +This will be a nice vision for you. Nocton Hall is +a historic Grade II listed building in England that +you would like to visit. +Ground Truth Response +You can see Nocton Hall in the village of Nocton, in Lincolnshire of +England, the country you want to visit. +Given Landmark +Maiden Castle, Dorset +Dialogue +Human: Wow, this is amazing! What is this? +Machine: It is Maiden Castle in Dorset. I thought you would like it since +you are interested in historic forts. +Human: Who owns the site today? +Human’s Persona +I like Britain. +I have been to Dorset. +I am interested in historic forts. +I hope to work for English Heritage. +I would like to visit an old fort. +Ground Truth Knowledge (Grounding) +Today the site is protected as a Scheduled Ancient Monument and is +maintained by English Heritage. +Retrieved Knowledge (Generation) +Portland Castle is an artillery fort constructed by Henry VIII on the Isle +of Portland, Dorset, between 1539 and 1541... +this version of events, or even that the hill fort was attacked by the +Romans... +Between 1985 and 1986 further excavations under Niall Sharples were +prompted by the hill fort’s deteriorating condition, partly caused by the +large number of visitors to the site... +a Tudor rose and the initials E.R. (Elizabeth Regina), has been preserved +and can be seen in the inner bailey of the castle mounted on a replica +carriage... +Constructed on a territorial boundary in about 600 BC, the first hill fort +at Maiden Castle was a 6.4-hectare (16-acre) area surrounded by a single +ditch... +Predicted Answers +BARTbase +The site is maintained by English Heritage, the +country you are from. +BARTlarge +Today the site is owned by English Heritage..... +GPT-2small +Today the site is protected as a Scheduled Ancient +Monument and is maintained by English Heritage. +GPT-2medium +Today the site is maintained by English Heritage. +INFO (Ours) +Today the site is owned by English Heritage. You +may wish to research this further since you hope to +work for English Heritage. +Ground Truth Response +It is owned by English Heritage; a company you hope to work for. +Table 9: Qualitative results. All the predicted results in grounding task are from our model, INFO and it predicts +the correct answers in both tasks. We add other baselines’ responses for comparative analysis. + diff --git a/7dE0T4oBgHgl3EQffQBI/content/tmp_files/load_file.txt b/7dE0T4oBgHgl3EQffQBI/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9b0255946ac0026e80ec2860650c26872b4bdbbb --- /dev/null +++ b/7dE0T4oBgHgl3EQffQBI/content/tmp_files/load_file.txt @@ -0,0 +1,903 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf,len=902 +page_content='You Truly Understand What I Need : Intellectual and Friendly Dialogue Agents grounding Knowledge and Persona Jungwoo Lim1, Myunghoon Kang1∗, Yuna Hur1∗, Seungwon Jung1∗, Jinsung Kim1∗, Yoonna Jang1, Dongyub Lee3, Hyesung Ji2, Donghoon Shin2, Seungryong Kim1§ and Heuiseok Lim1§ 1Korea University, 2Dialogue Tech Division, NCSOFT, 3Naver Corporation {wjddn803,chaos8527,yj72722,redlion0929,jin62304,seungryong_kim,limhseok}@korea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='kr, {hyesung84,dhshin}@ncsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='com, dongyub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='lee@navercorp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='com Abstract To build a conversational agent that interacts fluently with humans, previous studies blend knowledge or personal profile into the pre-trained language model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' However, the model that considers knowledge and persona at the same time is still limited, leading to hallucination and a passive way of using personas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' We propose an effective dialogue agent that grounds external knowledge and persona simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The agent selects the proper knowledge and persona to use for generating the answers with our candidate scoring implemented with a poly-encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Then, our model generates the utterance with lesser hallucination and more engagingness utilizing retrieval augmented generation with knowledge-persona enhanced query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' We conduct experiments on the persona- knowledge chat and achieve state-of-the-art performance in grounding and generation tasks on the automatic metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Moreover, we validate the answers from the models regarding hallucination and engagingness through human evaluation and qualitative results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' We show our retriever’s effectiveness in extracting relevant documents compared to the other previous retrievers, along with the comparison of multiple candidate scoring methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Code is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='com/dlawjddn803/INFO 1 Introduction To build an ultimate conversational agent that interacts with humans fluently, previous studies provide generative neural network-based models (Sordoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Vinyals and Le, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Although the answers generated from those models are plausible, they lack informativeness and engagingness resulting in bland responses compared to humans (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', Equal Contributors § Corresponding author Dialogue Human: Is it in England?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Machine: No, it is actually in Scotland where you are going.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Human: Where in Scotland?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Human’s Persona I will travel through North Ayrshire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' I am going to Scotland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' I like history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' I am interested in architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' I love to garden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Ground Truth Knowledge Eglinton Castle was a large Gothic castellated mansion in Kilwinning, North Ayrshire, Scotland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Predicted Answers BARTbase It is in Scotland, which is a place you love.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' BARTlarge It is in Scotland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' in Scotland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' in Scotland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' in Ground Truth Response It is in North Ayrshire so you could visit when you travel through.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Table 1: Example of the generated answers from a typical generative model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', BART.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' We can find that BARTbase uses different persona sentence which has not appeared human’s personal profiles resulting in hallucinated answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Also, BARTlarge generates less engaging answers by making use of the knowledge only to answer the question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Both generated responses are in the situation of hallucination and are less engaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' However, for knowledgeable and attractive conversation, people usually provide informative replies by considering the background of the person whom they are talking to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Towards a human-like manner of dialogue, Ghazvininejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' (2018) and Dinan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' (2018) introduce the knowledge- grounded conversation for the knowledgeable and informative responses, whereas Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' (2018a) suggest the persona-grounded dialogue for the personalized responses to the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' To improve the machine’s answer with the external knowledge base, one injects the factual knowledge into the parameters of the language model (Raffel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Roberts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Despite the models’ capability of utilizing external knowledge implicitly, they produce “hallucinations” in the responses (Marcus, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The hallucination arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='02401v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='CL] 6 Jan 2023 in the dialogue involves the situation where the generated output contradicts the reference knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Also, it includes the situation when the generated output cannot be confirmed from the knowledge source (Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' To mitigate these hallucinated answers, hybrid models employing parametric memory with non-parametric (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', retrieval-based) memory are introduced to directly access external memories, leading the source to be inspected and interpreted (Karpukhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Petroni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' On the other hand, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' (2018a) suggest persona-chat dialogues with the corresponding personal profiles of each interlocutor to avoid general and monotonous answers from the machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Though See et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' (2020) show comparable quality in generating personalized conversation, the generated utterances merely confirm each interlocutor’s persona resulting in a passive manner of speaking such as “I have four children”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' In addition, the incoherent topics of the dialogues lead to shallow levels of conversation between the interlocutors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' To elaborate on this chit-chat conversation supported by external knowledge, Jang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' (2022) presents a novel persona-knowledge chat with a generative model that considers persona information and world knowledge altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Despite obtaining the knowledge and persona when generating the answers, the generative models’ responses still exhibit both hallucination and lesser engagingness as in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' In this paper, we propose INFO (Intellectual and Friendly dialOg agents) that responds with external knowledge and persona simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Owing to the enhanced capturing relevancy between the context and each candidate set, the knowledge selector and persona selector for the grounding task are implemented with the poly-encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' To alleviate hallucinated responses from the model, we adopt retrieval-augmented generation (RAG) (Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', 2020b) by utilizing non-parametric memory and parametric generator in addition to the enhanced input query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' By injecting predicted sources as input to the retrieved-augmented generator, our model maintains consistency between grounding and generation while training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Therefore, our model generates more knowledgeable and engaging answers in an active manner with less hallucination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' We show that INFO achieves the highest scores on both grounding and generation tasks in empirical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Also, we compare diverse candidate scoring modules including bi-encoder, cross-encoder, and poly-encoder and demonstrate their effect on generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' We additionally conduct experiments to show the effectiveness of the retriever module compared to sparse and dense retrievers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The qualitative results and human evaluation are also presented to validate our model’s capability to generate human-like answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Our contributions are as follows: We propose the model that grounds persona information and external knowledge with lesser hallucination and adequate utilization of persona in an active manner simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Our approach suggests that the generated responses from the model are interpretable regarding what the model refers to while generating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' We show that INFO achieves the SoTA performance in all of the automatic metrics and demonstrate its comparable quality with human evaluation and qualitative analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' 2 Related Works 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='1 Knowledge Grounded Conversation To let the neural network models ground external knowledge and generate informative answers, Ghazvininejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' (2018) suggests a data- driven neural conversational agent that provides knowledgeable answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Also, Dinan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' (2018) introduces open-domain dialogue where the two speakers are talking with Wikipedia knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' To inject the external knowledge into the pre-trained language model efficiently, Raffel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Roberts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' (2020) success in equipping the knowledge into the parameters and show comparable performance in open-domain question and answering tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' However, the approach is not capable of expand or revise their inherent knowledge and provides hallucination (Marcus, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' To overcome the limitations, Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' (2020b) combines a pre-trained parametric model and non-parametric memory for the open-domain question and answering to reduce hallucination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Since their non- parametric memory can be updated without extra pre-training, revising knowledge is more efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Furthermore, it is found that a retrieval-augmented Figure 1: Overview of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' U is the input comprises dialogue history and knowledge snippet, and cand denotes each candidate from the grounding tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The grounding score is obtained through the dot product operation with the representation of input context Udial and candidate at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The predicted sources convert into the knowledge-persona enhanced query (KPEQ) with dialogue history and KPEQ is fed into the retrieval-augmented generator to generate the responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' generator also reduces hallucination in knowledge- grounded conversation as well (Shuster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', 2021), and a similar approach recently achieves outstanding performance in knowledge-grounded conversation (Paranjape et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='2 Persona Grounded Conversation In order to alleviate bland and general answers with consistent personality, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' (2018a) constructs a persona-chat dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' In the dataset, the two interlocutors chat with the persona profile sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Along with this dataset, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' (2018a) introduces the model with a profile memory network by considering the dialogue history to perform attention over the persona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' They enlarge the persona-chat dataset with Reddit corpus, and pre-trained the model with these dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' After that, they fine-tune pre- trained model on the persona-chat (Mazare et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Also, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' (2020) trains a receiver to reinforce the mutual persona understanding between interlocutors, and Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' (2019) utilize pre-trained models (Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', 2019) to build personalized dialogue agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='3 Encoders for Sentence Scoring There exist diverse encoder structures for sentence scoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Bi-encoder scores the relevance between sentences by feeding context and candidates into separate encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' An example of bi-encoders are memory networks (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', 2018a), transformer memory networks (Dinan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', 2018), LSTM (Lowe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Since bi- encoder calculates with cached encoded sentence representations, it is relatively fast in computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' However, the bi-encoder has a limitation of capturing mutual information between context and candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Cross-encoder, on the other hand, scores by aligning context and candidates in one sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' A type of cross-encoders is a sequential matching network that is based on deep matching networks (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', 2018) and gated self-attention (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', 2018b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Although using a cross-encoder can achieve rich interaction between the sentences within the encoder, the problem of slow processing still remains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' To exploit both benefits of each model, poly-encoder adopts attention mechanism into the bi-encoder architecture and shows satisfactory performances as cross-encoder with fast inference time (Humeau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' For the enhanced representation of grounding knowledge and persona, we employ a poly-encoder as a selector for each grounding task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' 3 Method To generate more knowledgeable and engaging dialogue, we introduce our conversational model that grounds external knowledge and persona information as in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' We first encode the input with the pre-trained language model, and then choose the proper knowledge and persona from the given candidates for each selector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' We employ poly-encoder (Humeau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', 2019) as knowledge selector and persona selector to exploit its enhanced capability of capturing relevance between candidate set and context (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', dialogue history).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' the predicted persona ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='and knowledge are aligned into one sequence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='KPEQ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='Retriever(Non-Parametric) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='Poly-encoder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='Knowledge Selector ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='Document Index ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='Uaial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='O- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='Score ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='Poly- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='U ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='Z1-08 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='encoder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='Attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='Z7777 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='Z1-03 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='Z2-02 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='Uaial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='Uaal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='acand ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='Knowledge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='Z1-01 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='Candidate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='Z2-09 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='CM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='Dialogue ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='Z2-05 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='Z2-07 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='Attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='Attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='Candidate Aggregator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='Persona ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='Candidate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='个 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='↑ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='h1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='hn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='h2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='a1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='aT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='Persona Selector ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='↑ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='↑ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='Persona ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='Marginalize ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='Generator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='Generated ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='Context Encoder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='Candidate Encoder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='Poly- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='Level ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='(Parametric) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='Answer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='↑ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='↑ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='encoder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='Indicator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='U ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='candi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='cand2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='candTto the dialogue history for consistency between ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='grounding and generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The sequence is defined as a knowledge-persona enhanced query (KPEQ), then it feeds into the retriever-augmented generator (RAG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The generator then extracts the relevant paragraphs to refer from the knowledge index to reduce hallucination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='1 Input Construction The given dialogue is notated as {(uhm 1 , umc 1 ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='(uhm o , umc o )}, where o is the number of rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' uhm and umc indicate the utterances of human and machines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' We first take o-th round dialogue history, except for the final machine’s reply umc o , for the initial input for the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' We define the clue of the dialogue as knowledge snippet clk to inform the machine of which topic the user is interested in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The knowledge snippet is the name of the landmark that the user encounters, which is given topic from the dialogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' We then align the dialogue history and knowledge snippet into the one sequence for the model input as U = {uhm 1 , umc 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='uhm o , clk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='2 Model Components 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='1 Poly-Encoder Based Candidate Scoring For knowledge and persona grounding tasks, we suggest poly-encoder-based candidate scoring to leverage the capability of capturing the semantic similarities between the context input and the candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' It is employed to select proper sources to be used when generating the utterance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' When the context input U comes in, we compute the grounding scores of each candidate utilizing the embeddings of context input and encoded candidates in the poly-encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The grounding score is used to select the most suitable source(s) in the knowledge selector and persona selector, which will be introduced in the following Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' In poly-encoder architecture (Humeau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', 2019), candidates are fed into the candidate encoder and denoted as {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', aT } where T is the number of candidates in the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Each candidate embedding at is the first output of the candidate encoder, which is represented by the transformer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' After encoding candidates, the context input (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', dialogue history) is embedded with a separate context encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Unlike the candidate encoder, the context encoder embeds the dialogue into multiple vectors through M context codes {c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='cM}, which are learned for capturing diverse aspects of a given context rather than using one embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Each context code is used to extract U m dial by attending over all the previous layer’s output as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' U m dial = � j wcm j hj (1) Note that the h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', hn is the output of the pre- trained language model and n is the number of tokens in the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The weights are computed as (wcm 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', wcm n ) = softmax(cm · h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', cm · hn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Then, the final attention proceeds between the global features of the input and a given candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' In other words, the final dialogue feature Udial is obtained by aggregating each dialogue feature U m dial, while gaining richer interactions with context codes as in Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Udial = � m wmU m dial, (2) where w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', wM can be obtained from softmax(at · U 1 dial, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', at · U M dial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The final predicted candidate is chosen based on the highest score that is acquired from the dot product operation as (Udial · at).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='2 Knowledge Selector (KS) We build a knowledge selector for the knowledge grounding task, employing poly-encoder-based candidate scoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' When the grounding scores are produced from the candidate scoring module, the label with the highest score is selected as the predicted knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The knowledge loss LKG for the knowledge grounding task is computed with cross-entropy loss (Brier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', 1950) as in Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' LKG = − � j klj · log ˆ klj, (3) klj is the ground-truth label from the knowledge candidates of the j-th example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='3 Persona Selector (PS) We also implement a persona selector for the persona grounding task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Since multiple personas can be chosen to generate the responses, consideration of one or more persona sentences are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Similar to the knowledge selector, we assign the grounding score to each persona candidate with the candidate scoring module as in Equation 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' When the scores of each candidate are computed from the candidate scoring module, then the persona level indicator classifies which the number of the persona should be selected with the [CLS] token of the model input U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' After predicting the level of persona-engagingness, we pick persona sentences to be grounded according to the number predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' For example, if the persona level indicator predicts 2, then top-2 persona sentences are chosen in the persona grounding task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The selected persona sentence(s) are marked as 1 otherwise, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' We use binary cross-entropy loss for persona grounding as in Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' LPG = − � j plj · log ˆ plj + (1 − plj) · log(1 − ˆ plj) (4) Note that plj is the ground-truth label from the knowledge candidates of the j-th example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='4 Query-Enhanced Generator Following the works of Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' (2020b), we exploit the retrieval augmented generation’s capability to reduce hallucination and access the memory directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' For a consistent way of training while solving grounding and generation tasks, we reconstruct the query that feeds into the retriever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' When the knowledge and persona are predicted from each selector, we aggregate them with dialogue history into one sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Then, the final query is denoted as KPEQ = {U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' ˆP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' ˆK} and defined as a knowledge-persona enhanced query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' ˆP and ˆK are predicted persona and knowledge from each candidate set, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The retriever rη aims to search top-K latent paragraphs with the KPEQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' We utilize a pre- trained dense passage retriever (DPR) (Karpukhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', 2020) trained on natural question dataset (Kwiatkowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', 2019) which has parametric memory and bi-encoder architecture to retrieve a latent document embedding following Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' (2020b) : rη(z|KPEQ) ∝ exp(d(z)⊤q(KPEQ)), (5) where d(·) is an embedding from a document encoder and q(·) is a representation from query encoder, both implemented with BERTbase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' z denotes the list of document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' With the relevant paragraphs from the retriever, we employ RAG-Token architecture as the generator to borrow its strength of predicting each target token based on top-K different paragraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Since RAG-Sequence, which has a different architecture to RAG-Token, uses the same document from the retriever to predict each token as depicted in Equation 6, the result may opt to depend on the retrieved document (Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The two different versions of RAGs (Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', 2020b) are as follows: SRS(y|x) ≈ � z∈top-k(p(·|x)) rη(z|x) N � i gθ(yi|x, z, y1:i−1) (6) SRT(y|x) ≈ N � i � z∈top-k(p(·|x)) rη(z|x)gθ(yi|x, z, y1:i−1), (7) where SRS indicates our method with RAG- Sequence architecture and SRT denotes ours with the RAG-Token model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' x is a token of KPEQ and yi is a single token from the ground truth responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Also, z is a retrieved paragraph from the retriever and N is the maximum sequence length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The SRT generator g(·) marginalizes the loss from different paragraphs when generating answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' In detail, the generator outputs a distribution for the next token for each document before marginalizing as in Equation 7 where η denotes the parameter of the retriever, and θ indicates the parameter of the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' After that, the generator repeats the process with the following output token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Finally, the SRT aims to generate the next token following an auto-regressive manner with a standard beam search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' In other words, the model minimizes the negative marginal log-likelihood for each input/output pair (KPEQj, yj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The language model loss is formulated as : LS = − � j logp(yj|KPEQj) (8) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='3 Final Objectives We then train the full model in the multi-tasking manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The full objectives of the model is indicated as Equation 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' L = λKGLKG + λPGLPG + λSLS (9) Models Generation Grounding (Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=') chrF++ BLEU R-1 R-2 R-L BERTScore Persona Knowledge GPT2small 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='73 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='43 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='58 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='44 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='62 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='56 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='44 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='59 GPT2medium 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='12 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='31 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='29 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='17 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='12 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='92 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='44 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='42 BARTbase 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='77 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='99 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='24 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='73 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='13 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='35 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='45 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='18 BARTlarge 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='69 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='91 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='57 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='83 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='05 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='10 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='44 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='01 INFO (SRS) 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='33 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='36 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='36 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='36 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='16 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='00 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='70 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='24 INFO (SRT ) 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='29 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='46 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='26 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='35 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='06 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='29 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='87 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='22 Table 2: Main results on the official validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' SRS denotes our method with RAG-Sequence architecture and SRT indicates the model with RAG-Token model as generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The models are evaluated by generation metrics, including chrF++, BLEU, ROUGE-1 (R-1), ROUGE-2 (R-2), ROUGE-L (R-L), and BERTScore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' We control the proportion of each task and we set λKG, λPG, and λS as 1:1:5 for the experiments, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' We find the value of each λ with manual search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' 4 Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='1 Experiment Details Dataset FoCus (Jang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', 2022) is the dataset for customized dialogue benchmark, where each conversation is directly grounded with knowledge and persona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The dataset includes knowledge- aware dialogue with personal profiles between humans and machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' There are 12,484 dialogues about 5,152 knowledge sources from Wikipedia and 32,855 persona sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' To validate the knowledge grounding capability and customized dialogue generation, we evaluate our method with the official FoCus validation set for the effectiveness of experiments since the result from the official test set can be tested only through the leaderboard*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Experimental Setup For each candidate scoring module, we implement poly-encoder (Humeau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', 2019) with BERTlarge, and the number of context codes is 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' For the dialogue generation, we implement our method with Hugging Face (Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', 2020) and use facebook/rag-token-nq as the backbone model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' We use the same architecture of retriever and generator from RAG along with the decoding and leverage our knowledge index for non-parametric query-document ranking with FAISS library (Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The knowledge index consists of the paragraphs from the given Wikipedia knowledge entitled with the name of the given landmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' We set learning rate as 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='25e-6 with AdamW (Kingma and Ba, 2014) https://codalab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='lisn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='upsaclay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='fr/competitions/3754 for the optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The batch size is set as 32, and the number of dialogue history is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The whole model was trained for three epochs on RTX A6000 GPU and took 8 hours per one epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Baselines We implement the baselines from previous study (Jang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', 2022) and we conduct experiments with GPT-2 (Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', 2019) and BART (Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', 2020a) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' For a fair comparison, we demonstrate the results on GPT- 2small, which has 12 layers, and BARTbase, which has 6 encoders and 6 decoder layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Also, GPT- 2medium contains 24 layers of the decoder, and BARTlarge possesses 12 layers for each encoder and decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='2 Automatic Evaluation We show the main results on the FoCus dataset with automatic metrics in grounding and generation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The official metrics for the benchmark are chrF++ (Popovi´c, 2017), BLEU (Papineni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', 2002), ROUGE-1, ROUGE-2, and ROUGE-L (Lin, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' To consider the semantic similarity score for each token between candidate and reference sentences using contextual representation, we additionally adopt BERTscore (Zhang* et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' For grounding task, we used accuracy for both knowledge and persona grounding, and F1 score for the persona grounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' In Table 2, it is found that our method shows substantial improvements in all the metrics from generation to grounding compared to the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Especially, the performances of INFO increase over 18% at least regarding the generation metrics except for BERTScore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Furthermore, our model achieves remarkable success in persona and knowledge accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Unlike the performance in other generation metrics, SRS demonstrates better persona accuracy than SRT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' This result might be Model Generation Grounding chrF++ BLEU R-1 R-2 R-L BERTScore Persona (Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=') Persona (F1) Knowledge (Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=') SRT Bi-encoder 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='83 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='51 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='35 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='80 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='37 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='86 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='10 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='20 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='18 Cross-encoder 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='90 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='18 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='57 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='25 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='29 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='52 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='09 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='32 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='49 Poly-encoder 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='29 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='46 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='26 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='35 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='06 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='29 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='87 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='56 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='22 Table 3: Performances comparison between the encoding modules for grounding tasks attributed to the architecture of the generator, which is more applicable to sentence classification tasks such as persona grounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The official test result is also demonstrated in Appendix A, but BERTscore is missing due to the unreleased ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='3 Human Evaluation We conduct a human evaluation to validate the responses from our model through Amazon Mturk services†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The assessment criteria are fluency, adequacy, provenance, engagingness, and hallucination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' In specific, provenance is the level of utilization of the ground truth knowledge into the responses, whereas engagingness means how much the answers are persona-related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Also, hallucination indicates whether the answer contradicts the persona and knowledge or cannot be verified from the source content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' We randomly chose 50 dialogues from the official test set, and three workers were allocated to evaluate each dialogue generated by our model and baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' We asked the workers to rank the answers according to each criterion following Cho and May (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Rank is scaled from 1 to 5, and the lower number is mapped to the better quality except for hallucination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The agreement between the annotators is calculated with Fleiss’ Kappa coefficient and is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='4185 indicating fair agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The relations between the annotators hardly exist since we collect the results from the Amazon Mturk workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' As in Table 4, INFO surpasses BARTbase, BARTlarge, GPT-2small and GPT-2medium in all of the criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' INFO achieves the highest rank in adequacy, fluency, and provenance and generates a more human-like response than other generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Also, the workers ranked our model the lowest when they were asked to rank the responses in the most hallucinated order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Thus, it can be found that INFO generates more engaging and fewer hallucination utterances with respect to the human.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The distribution of the rank per each criterion is illustrated in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' †https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='mturk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='com/ Models Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Rank Ad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' ↓ Fl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' ↓ Prov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' ↓ Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' ↓ Hall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' ↑ GPT-2small 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='57 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='41 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='58 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='46 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='49 GPT-2medium 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='04 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='02 BARTbase 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='43 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='29 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='47 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='22 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='45 BARTlarge 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='31 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='63 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='29 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='44 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='69 INFO (Ours) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='57 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='57 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='62 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='63 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='35 Table 4: Human evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The value in the table is the average rank of the each model’s response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The abbreviation Ad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Fl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Prov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' and Hall denote adequacy, fluency, provenance, engaginess, and hallucination, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' 5 Results and Analysis 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='1 Variants on Candidate Scoring Module To validate the poly-encoder as a candidate scoring module, we apply diverse candidate scoring modules, including the bi-encoder and cross- encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' From the results in Table 3, we can find that the poly-encoder outperforms in the generation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' In the grounding task, SRT with cross-encoder scoring shows improved accuracy on grounding persona and knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The result seems to be SRT with bi-encoder and cross-encoder are better than that with poly-encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' However, the F1 score of INFO is higher than the two candidate scoring modules implying that low accuracy in persona is due to the tendency of active use on the persona in poly-encoder while the other two models opt to predict not to use persona sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The results suggest that the high accuracy of persona not always guarantees the engagingness in the dialogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='2 Comparison on other Retrievers We show that INFO is effective in retrieving knowledge compared to other sparse and dense retrievers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' We retrieve the knowledge from our knowledge index built with Wikipedia paragraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' We utilize TF-IDF (Joachims, 1996), and deep passage retrieval (DPR) (Karpukhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' In the case of TF-IDF, we set the sum of query and knowledge tokens less than or equal to 512, which is the maximum sequence length of DPR and INFO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' We use bert-base-uncased as the tokenizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' For DPR, we extract less than 40 knowledge using TF-IDF due to memory limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' We first retrieve the five paragraphs related to the query that comprises knowledge snippet, dialogue history, predicted knowledge candidate, and selected persona sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' In Table 5, we find that the retriever we used outperforms compared to the TF-IDF and DPR in all the metrics, including BERTscore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The results imply that INFO’s retriever is suitable for extracting similar paragraphs rather than other retrievers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Model chrF++ BLEU R-1 R-2 R-L BERTScore TF-IDF 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='91 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='52 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='91 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='96 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='43 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='54 DPR 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='57 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='86 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='44 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='55 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='20 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='48 INFO 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='36 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='40 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='48 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='18 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='32 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='14 Table 5: Comparison with other retrievers 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='3 Effect of Selectors on Generation We measure each selector module’s effect on the generation task by changing the query which feds into the retriever on a validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The experimental results are shown in Table 6, where GTK, GTP represents ground truth knowledge and persona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Although the query that comprises the ground truth source shows the highest scores, INFO demonstrates comparable results on the generation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' From the result where the performance increase of INFO + GTP is larger than that of INFO + GTK about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='8%p, we can identify that our persona selector still has more space to achieve its maximum level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Query chrF++ BLEU R-1 R-2 R-L BERTScore INFO (RT) 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='29 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='46 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='26 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='35 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='06 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='29 +GTK 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='35 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='56 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='31 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='55 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='18 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='29 +GTP 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='19 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='39 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='61 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='46 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='01 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='79 +GTK+GTP 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='40 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='60 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='88 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='64 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='16 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='84 Table 6: Comparison between the generation performances based on the variants of query with ground truth knowledge and persona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Note that all the performance is evaluated with the official validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='4 Qualitative Analysis In Table 7, an example from the predicted results is illustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' In the case of BARTlarge, and GPT- 2medium, the responses only reflect the ground Given Landmark Finding Nemo Submarine Voyage Dialogue Human: What area of the park is this ride in?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Machine: This ride is located in the Tomorrowland area of Disneyland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Human: Has this ride always been about Finding Nemo?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Human’s Persona I’ve never been to California.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' My favorite cartoon is Finding Nemo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' I would like to visit Disneyland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' My favorite color is yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' I enjoy swimming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Ground Truth Knowledge (Grounding) Based on the characters and settings of the 2003 Disney · Pixar, Finding Nemo, it is a re-theming of the classic Submarine Voyage attraction that operated from 1959 to 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Retrieved Knowledge (Generation) The original Submarine Voyage was built in 1959 as part of the then new Tomorrowland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' In 2008, Finding Nemo Submarine Voyage received an award for outstanding achievement from the Themed Entertainment Association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Finding Nemo Submarine Voyage is .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' which opened on June 11, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' it is a re-theming of the classic Submarine Voyage attraction that operated from 1959 to 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' On July 15, 2005, two days before the 50th Anniversary of Disneyland, the Finding Nemo Submarine Voyage was officially announced at the new Turtle Talk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The attraction reuses the eight original 1959 Submarine Voyage through Liquid Space attraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Predicted Answers BARTbase Yes, that’s right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' You’re a fan of the “Fantasy” film, so I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' BARTlarge Yes, the ride is based on the characters and settings of the 2003 Disney · Pixar film GPT-2small No, it was originally a way to show that you love Finding Nemo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' GPT-2medium Yes, it has operated from 1959 to 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' INFO (Ours) No, this attraction is actually a re-theme of the classic submarine voyage attraction that operated from 1959 to 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The attraction is based on the characters and settings of the 2003 Disney Pixar film Finding Nemo, which is your favorite cartoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Ground Truth Response No, your favorite cartoon is a new addition to this ride.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The current Finding Nemo ride is a re-theming of the classic “Submarine Voyage” attraction that operated here from 1959 to 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Table 7: Qualitative result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' All the predicted results in grounding task are from our model, INFO and it predicts the correct answers in both tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' We add other baselines’ responses for comparative analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' truth knowledge resulting in less engaged answers without any persona-related phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Although BARTbase seems to employ a persona sentence in the form of the phrase “You’re fan of the Fantasy film”, its used sentence does not appear in human’s personal profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' This result also indicates that the utterance is hard to identify its provenance on the knowledge source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Moreover, GPT-2small generates the utterance that contradicts the ground truth knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' From the result, we can find that the generated responses from the baselines show hallucinations on both persona and knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Unlike other baselines, our model blends ground truth knowledge and persona sentence into the response with less hallucination and engagingness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' In addition, the retrieved knowledge source that our model refers to provides interpretability and provenance of the responses to the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' More examples are also depicted in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' 6 Conclusions In this paper, we presented a conversational agent that generates responses grounding the user’s persona and external knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' We utilized poly-encoder-based candidate scoring for each grounding task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' We additionally implement persona level indicator to consider multiple persona selections for delicate persona grounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' With predicted sources, we construct a knowledge-persona enhanced query to retrieve latent paragraphs, and they are used to generate informative and engaging responses by marginalizing loss for each token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' We show that our method achieves the state-of-the-art (SoTA) score in both grounding and generation tasks in the persona-knowledge conversation dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' We also demonstrate that the responses from INFO show less hallucination and more engagingness through human evaluation and qualitative analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' We also compare the grounding modules and retrievers to show INFO’s effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' 7 Limitations The proposed model INFO has limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Given the INFO’s settings, the model cannot deal with real-world application, which means the absence of ground truth knowledge or persona candidates in the grounding task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' We also conducted the human evaluation to evaluate the capability of the proposed model’s mitigating hallucination in dialogue generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' However, the number of cases is relatively small for evaluating the capability of mitigating hallucination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Finally, INFO demands high GPU computation resources, since it marginalizes loss at the token level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' We plan to improve the INFO for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' We will train and evaluate the INFO in open- domain settings as well as real-world settings for the applicable conversational agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Moreover, we will conduct human evaluations with more cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Especially, we will enhance the way of quantitative measurement for the model’s hallucinated answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Last but not least, we will improve the generator of INFO with more computationally efficient components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' 8 Acknowledgement This work was supported by Institute of Information & communications Technology Planning & Evaluation(IITP) grant funded by the Korea government(MSIT) (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' 2020-0-00368,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' A Neural-Symbolic Model for Knowledge Acquisition and Inference Techniques),' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' In International Conference on Learning Representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Fabio Petroni, Patrick Lewis, Aleksandra Piktus, Tim Rocktäschel, Yuxiang Wu, Alexander H Miller, and Sebastian Riedel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' How context affects language models’ factual predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' In Automated Knowledge Base Construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Maja Popovi´c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' chrF++: words helping character n-grams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' In Proceedings of the Second Conference on Machine Translation, pages 612–618, Copenhagen, Denmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Alec Radford, Jeffrey Wu, Rewon Child, David Luan, Dario Amodei, Ilya Sutskever, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Language models are unsupervised multitask learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' OpenAI blog, 1(8):9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Exploring the limits of transfer learning with a unified text-to-text transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Journal of Machine Learning Research, 21:1–67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Adam Roberts, Colin Raffel, and Noam Shazeer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' How much knowledge can you pack into the parameters of a language model?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' In Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP), pages 5418–5426.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Abigail See, Stephen Roller, Douwe Kiela, and Jason Weston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' What makes a good conversation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' how controllable attributes affect human judgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' In Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers), pages 1702–1723.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Kurt Shuster, Spencer Poff, Moya Chen, Douwe Kiela, and Jason Weston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Retrieval augmentation reduces hallucination in conversation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' In Findings of 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pages 196–205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Oriol Vinyals and Quoc V Le.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' A neural conversational model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' arXiv preprint arXiv:1506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='05869.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Thomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi, Pierric Cistac, Tim Rault, Rémi Louf, Morgan Funtowicz, Joe Davison, Sam Shleifer, Patrick von Platen, Clara Ma, Yacine Jernite, Julien Plu, Canwen Xu, Teven Le Scao, Sylvain Gugger, Mariama Drame, Quentin Lhoest, and Alexander M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Rush.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Transformers: State-of-the-art natural language processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' In Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing: System Demonstrations, pages 38–45, Online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Thomas Wolf, Victor Sanh, Julien Chaumond, and Clement Delangue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Transfertransfo: A transfer learning approach for neural network based conversational agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' arXiv preprint arXiv:1901.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Saizheng Zhang, Emily Dinan, Jack Urbanek, Arthur Szlam, Douwe Kiela, and Jason Weston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' 2018a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Personalizing dialogue agents: I have a dog, do you have pets too?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' In Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 2204– 2213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Tianyi Zhang*, Varsha Kishore*, Felix Wu*, Kilian Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Weinberger, and Yoav Artzi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Bertscore: Evaluating text generation with bert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' In International Conference on Learning Representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Zhuosheng Zhang, Jiangtong Li, Pengfei Zhu, Hai Zhao, and Gongshen Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' 2018b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Modeling multi- turn conversation with deep utterance aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' In Proceedings of the 27th International Conference on Computational Linguistics, pages 3740–3752.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' A Automatic Evaluation on Official Test Set Models Generation Grounding (Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=') chrF++ BLEU R-1 R-2 R-L Persona Knowledge GPT2small 28.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='61 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='33 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='27 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='39 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='09 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='83 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='10 Table 8: Main results on the official test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' RT indicates the model with RAG-Token model as generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The models are evaluated by generation metrics, including chrF++, BLEU, ROUGE-1 (R-1), ROUGE-2 (R-2) and ROUGE-L (R-L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' The accuracy for persona grounding task and knowledge grounding task are also noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Since BERTscore is not the official generation metric, we cannot evaluate the result on the metric as the ground truth of the test is not yet disclosed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' B Human Evaluation Distribution on Each Criteria (a) Adequacy (b) Fluency Figure 2: The distribution of the rank on the adequacy and fluency criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Guide A to E indicates INFO, BARTbase, BARTlarge, GPT-2small, and GPT-2medium, in the order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Guide A Guide B 100 Guide C Guide D Guide E 80 f evaluation 60 of 40 # 20 0 1 2 3 4 5 RankGuide A Guide B 100 Guide C Guide D Guide E 80 f evaluation 60 JO 40 # 20 0 1 2 3 4 5 Rank(a) Provenance (b) Engagingness Figure 3: The distribution of the rank on the provenance and engagingness criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Guide A to E indicates INFO, BARTbase, BARTlarge, GPT-2small, and GPT-2medium, in the order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Figure 4: The distribution of the rank on the less hallucination criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Note that the highest rank (1) means the most hallucinated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Guide A to E indicates INFO, BARTbase, BARTlarge, GPT-2small, and GPT-2medium, in the order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Guide A Guide B 100 Guide C Guide D Guide E 80 f evaluation 60 of 40 # 20 0 1 2 3 4 5 RankGuide A 100 Guide B Guide C Guide D Guide E 80 f evaluation 60 of 40 # 20 0 1 2 3 4 5 RankGuide A Guide B 100 Guide C Guide D Guide E 80 f evaluation 60 of 40 # 20 0 2 3 5 1 4 RankC Qualitative Results Given Landmark Nocton Hall Dialogue Human: I know this place, but I don’t remember the name of this place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Human’s Persona I have a hall in my house.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' I have worked with military hospital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' I would like to go to England.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' I have heard about Frederick John Robinson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' I have respect to American Officers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Ground Truth Knowledge (Grounding) Today the site is maintained by English Heritage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Retrieved Knowledge (Generation) “Hogesdon” is first recorded in the Domesday Book, meaning an Anglo- Saxon farm (or “fortified enclosure”) belonging to Hoch, or Hocq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='. By 1677, the church was in such poor structural condition that the Dean of Chichester convened a consistory court with three parishioners and asked them why repairs had not been carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' he south transept had historically served as the mausoleum for the Cavendish family, who resided at nearby Latimer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Following the diminished influence of the ancient Priory, predecessor of the two parishes of St Bartholomew, disputes began to arise over rights to tithes and taxes payable by lay residents who claimed allegiance with the nearby and anciently associated parish of St Botolph Aldersgate - an unintended consequence and legacy of King Kenry VIII’s religious reforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' a wall-mounted stone pulpit, a stone altar, a series of tall, pointed windows high in the walls, an unadorned stone font and a short wooden steeple-little more than an extended belfry-extending from the nave roof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='a wall-mounted stone pulpit, a stone altar, a series of tall, pointed windows high in the walls, an unadorned stone font and a short wooden steeple-little more than an extended belfry-extending from the nave roof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Predicted Answers BARTbase This is the Nocton Hall, a historic Grade II listed building in the village of Nocton, BARTlarge Nocton Hall is a historic Grade II listed building in the village of Nocton, in Lincoln GPT-2small This is Nocton Hall, a historic Grade II listed building in the village of Nocton, in GPT-2medium This is the Nocton Hall, a historic Grade II listed building in the village of Nocton, INFO (Ours) This will be a nice vision for you.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Nocton Hall is a historic Grade II listed building in England that you would like to visit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Ground Truth Response You can see Nocton Hall in the village of Nocton, in Lincolnshire of England, the country you want to visit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Given Landmark Maiden Castle, Dorset Dialogue Human: Wow, this is amazing!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' What is this?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Machine: It is Maiden Castle in Dorset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' I thought you would like it since you are interested in historic forts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Human: Who owns the site today?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Human’s Persona I like Britain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' I have been to Dorset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' I am interested in historic forts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' I hope to work for English Heritage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' I would like to visit an old fort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Ground Truth Knowledge (Grounding) Today the site is protected as a Scheduled Ancient Monument and is maintained by English Heritage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Retrieved Knowledge (Generation) Portland Castle is an artillery fort constructed by Henry VIII on the Isle of Portland, Dorset, between 1539 and 1541.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' this version of events, or even that the hill fort was attacked by the Romans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Between 1985 and 1986 further excavations under Niall Sharples were prompted by the hill fort’s deteriorating condition, partly caused by the large number of visitors to the site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' a Tudor rose and the initials E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' (Elizabeth Regina), has been preserved and can be seen in the inner bailey of the castle mounted on a replica carriage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Constructed on a territorial boundary in about 600 BC, the first hill fort at Maiden Castle was a 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='4-hectare (16-acre) area surrounded by a single ditch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Predicted Answers BARTbase The site is maintained by English Heritage, the country you are from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' BARTlarge Today the site is owned by English Heritage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' GPT-2small Today the site is protected as a Scheduled Ancient Monument and is maintained by English Heritage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' GPT-2medium Today the site is maintained by English Heritage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' INFO (Ours) Today the site is owned by English Heritage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' You may wish to research this further since you hope to work for English Heritage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Ground Truth Response It is owned by English Heritage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' a company you hope to work for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' Table 9: Qualitative results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' All the predicted results in grounding task are from our model, INFO and it predicts the correct answers in both tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} +page_content=' We add other baselines’ responses for comparative analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7dE0T4oBgHgl3EQffQBI/content/2301.02401v1.pdf'} diff --git a/9NFQT4oBgHgl3EQf5jYf/content/2301.13435v1.pdf b/9NFQT4oBgHgl3EQf5jYf/content/2301.13435v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..13824012394dc817c6afa40e416bb431f683e434 --- /dev/null +++ b/9NFQT4oBgHgl3EQf5jYf/content/2301.13435v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:bda137ef3a8f36545d66dcde9d276f814317fe7b42111521722b4ff4759ff344 +size 1641344 diff --git 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100644 index 0000000000000000000000000000000000000000..0ebed41a23923f1823a7f8673cda02fe95b167c8 --- /dev/null +++ b/9dE4T4oBgHgl3EQf3Q3I/content/tmp_files/2301.05305v1.pdf.txt @@ -0,0 +1,787 @@ +Reinforcement Learning-based Joint Handover and +Beam Tracking in Millimeter-wave Networks +Sara Khosravi∗, Hossein S. Ghadikolaei‡, Jens Zander∗, and Marina Petrova ∗† +∗School of EECS, KTH Royal Institute of the Technology, Stockholm, Sweden, +† Mobile Communications and Computing, RWTH Aachen University, Germany, ‡ Ericsson Research, Sweden +Email: {sarakhos, jenz, petrovam} @kth.se, hossein.shokri.ghadikolaei@ericsson.com +Abstract—In this paper, we develop an algorithm for joint +handover and beam tracking in millimeter-wave (mmWave) +networks. The aim is to provide a reliable connection in terms of +the achieved throughput along the trajectory of the mobile user +while preventing frequent handovers. We model the association +problem as an optimization problem and propose a reinforcement +learning-based solution. Our approach learns whether and when +beam tracking and handover should be performed and chooses +the target base stations. In the case of beam tracking, we +propose a tracking algorithm based on measuring a small spatial +neighbourhood of the optimal beams in the previous time slot. +Simulation results in an outdoor environment show the superior +performance of our proposed solution in achievable throughput +and the number of handovers needed in comparison to a multi- +connectivity baseline and a learning-based handover baseline. +Index Terms—Millimeter-wave, user association, beam track- +ing, handover, reinforcement learning. +I. INTRODUCTION +Millimeter-wave (mmWave) is a key radio access technol- +ogy for beyond 5G communication systems, offering ultra- +high data rates due to a large amount of free spectrum [1]. +However, due to the fewer scattering paths and significant +penetration loss, mmWave links are vulnerable to static or +dynamic obstacles. To overcome such severe loss, both base +station (BS) and user equipment (UE) may need directional +communication using a large number of antennas, which may +result in frequent misalignment of beams due to mobility and +blockage. Hence, finding and maintaining the optimal beam +directions (beam alignment) is necessary. The lengthy period +to achieve the beam alignment (hundreds of milliseconds +to seconds [2]) results in a high cell search time or BS +discovery time in mmWave systems. As reported in [3], the +BS discovery time which is the time required to search the +target BS when the handover command is received by the +UE is about 200 ms. Moreover, to improve the capacity and +coverage the density of the BSs is usually high in mmWave +systems [1]. Hence, conventional handover methods based on +instantaneous received signal power can cause unnecessarily +frequent handovers and a ping-pong effect. This leads to a +severe drop in service reliability. Therefore, fast BS discovery +(finding target BS in the handover process), and efficient +handover execution techniques, will be required to use the +full promise of mmWave cellular networks. +The spatial mmWave channel can be approximated by a +few dominant paths, where each path can be defined with +its angle of departure (AoD), angle of arrival (AoA) and +gain [4]. Hence, one can only estimate these path parameters +instead of a large dimensional channel matrix [5], [6]. The +process of identifying the dominant paths is called beam +training. However, due to the dynamic environment, frequent +beam training may cause high overhead1. Temporal correlation +of spatial mmWave channel can be employed to accelerate +the beam training process by tracking the variation of the +dominant path directions [6]. +A. Related Work +To address the link failure and throughput degradation in +a dynamic environment, the multi-connectivity technique has +been vastly analyzed in literature [7], [8]. In this technique, the +UE keeps its connection to multiple BSs (either at mmWave +band or sub-6 GHz band). However, power consumption, +synchronization and the need for frequent tracking are the +main challenges. In the 3GPP standard (release 16) two +handover techniques are introduced to improve the link robust- +ness during mobility: dual active protocol stack (DAPS), and +conditional handover (CHO) [9]. In the DAPS, the connection +to the current serving BS is maintained until the connection +to the target BS is fully established. In the CHO, the UE is +configured with multiple target BSs. During the handover, the +UE can select one of the configured BSs as the target BS +during the RRC reconfiguration message. Although CHO can +decrease the handover failure probability, it may increase the +handover latency if the UE asks for multiple handovers during +a single RRC reconfiguration [7]. +Applying machine learning as the main decision-maker tool +to make the optimal handover decision and choose the target +BS has been also studied in the literature [10], [11]. The +authors in [10] proposed a reinforcement learning (RL) based +handover policy to reduce the number of handovers while +keeping the quality of service in heterogeneous networks. +In [11] an intelligent handover method based on choosing +the backup solution for each serving link to maximize the +aggregate rate along a trajectory has been proposed. +1Overhead depends on the training time compared with the changes in the +environment. +arXiv:2301.05305v1 [eess.SY] 12 Jan 2023 + +In terms of beam tracking, authors in [12] applied the +correlation of spatial mmWave channel in adjacent locations +and proposed the beam steering method based on searching +over a small angular space in the vicinity of the previously +known valid beams. The authors in [6] applied machine +learning to the tracking procedure to extract useful information +from the history of AoD tracking. +All the aforementioned works only take handover or beam +tracking issues into account. Additionally, they do not study +the impact of selecting beam tracking and handover on the +achieved throughput of the UE along its trajectory and instead +focus on the achieved rate as the primary performance metric. +B. Our Contributions +In this paper, we develop a novel joint handover and beam +tracking algorithm in a mmWave network under mobility. The +algorithm aims to associate the UEs to BSs that maximize +the sum achieved throughput along the trajectory and ensure +the achieved throughput in each location of the trajectory +is higher than a pre-defined threshold. The user association +process is defined as the process of determining whether a user +is associated with a particular BS before data transmissions +commence. In the case of handover, the UE is associated with a +new BS, whereas in the case of beam tracking, the UE remains +associated with the serving BS from the previous time slot. The +main contributions of our paper are summarized as below: +• System Modeling: We model the user association prob- +lem as a non-convex optimization problem. Unlike the +existing works in the literature, we consider achieved +throughput as the main performance metric to measure +the effect of handover or beam tracking on the UEs’ +quality of service. +• Learning-based Solution: The objective function in our +proposed user association problem highly depends on the +user association mechanism. We utilize the reinforcement +learning (RL) algorithm to approximate the solution to +this problem. The aim is to decide whether to run a beam +tracking algorithm or a handover algorithm. +• Joint Handover and Beam Tracking Algorithm: In the +case of a handover decision, the target BS will be +recognized as the output of the RL algorithm. In the +case of beam tracking, the search space will be defined +based on our proposed tracking algorithm by searching +the directions in the small spatial neighbourhood of the +previously selected optimal directions. +• Empirical Evaluation: We apply ray tracing with a +real building data map as the input. The results show +the effectiveness of our proposed method in achieving +throughput along trajectories and decreasing the number +of handovers. +The rest of the paper is organized as follows. We introduce +the system model and problem formulation in Section II. In +Section III, we propose our method. We present the numerical +results in Section IV and, conclude our work in Section V. +Notations: Throughout the paper, vectors and scalars are +shown by bold lower-case (x) and non-bold (x) letters, respec- +tively. The conjugate transpose of a vector x is represented by +xH. We define set [M] := {1, 2, .., M} for any integer M. The +indicator function 1{·} equals to one if the constraint inside +{·} is satisfied. +II. SYSTEM MODEL AND PROBLEM FORMULATION +In this section, first, we introduce the mmWave channel +model. Then, we present the user association problem formu- +lation. +We consider a downlink communication with |B| mmWave +BSs, where each is equipped with NBS antennas, communi- +cating with a single antenna mobile UE. We consider analog +beamforming with a single RF chain. We assume all BSs +allocate equal resources to their serving UEs. The channel +between BS j ∈ B and its serving UE during time slot i is +[13]: +hj = +L +� +ℓ=1 +hℓaH(φℓ, θℓ), +(1) +where L is the number of available paths. Each path ℓ has +complex gain hℓ (include path-loss) and horizontal φℓ and +vertical θℓ, AoD. Due to the notation simplicity, we drop the +index j and i from the channel parameters. The array response +vector is a(.) where its exact expression depends on the array +geometry and possible hardware impairments. The signal-to- +noise ratio (SNR) in time slot i is +SNR(i) +j += p|hH +j fj|2 +σ2 +, +(2) +where σ2 is the noise power, p is the transmit power, fj ∈ CNBS +is the beamforming vector of BS j. +We define variable x(i) +j +∈ {0, 1} for j ∈ B as an association +indicator in time slot i, where is equal 1 if UE is associated to +the BS j and 0 otherwise. Hence, the achieved rate per second +per hertz in time slot i is +R(i) = x(i) +jS log2(1 + SNR(i) +jS ) = +� +j∈B +x(i) +j log2(1 + SNR(i) +j ), +where jS is the index of the serving BS of the UE during time +slot i. Here, we assume each UE is served by only one BS. +We define the achievable throughput per hertz of the UE by +multiplying its rate by the data transmission time as +Γ(i) = (1 − τ (i) +b +τc +)R(i), +(3) +where, τ (i) +b +is the beam training duration which may have a +different value in each time slot i, and τc is the duration of +the time slot that is a fixed value for all time slots, see Fig. 1. +A. Beam Training and Beam Tracking +As depicted in Fig. 1a, when the UE is connected to a +BS j ∈ B, initial beam training is performed by sending +pilots over all combination of the beam directions in the +codebook during τb. Based. on the UE’s feedback of the +received signal strength (or estimated SNR), the best beam pair +directions are selected. Then, the BS and the UE would use this + +Initial beam training +Data Transmission +τb +τc +(a) +Tracking +Data Transmission +τb +(b) +Fig. 1: τc is the time slot duration. τb is (a) the initial beam +training duration when the UE is associated with the new +BS (handover case), (b) the beam tacking duration when the +serving BS is the same for the consecutive slots. +direction (φℓ⋆, θℓ⋆) during the data transmission phase. The +beamforming vector, f is chosen to maximize the achievable +rate of the UE. Due to the monotonicity of the logarithm +function, this is equivalent to maximising the SNR term in +(2). Hence +f ∗ +j = arg max +fj∈F +|hH +j fj|2 +(4) +where F is the beamforming codebook that contains all +the feasible beamforming vectors. The n-th element of the +codebook F is defined as f(n) = a(φn, θn), where (φn, θn) +are steering angles and a(.) is the array response vector. +When the BS continues serving the same UE in a consecu- +tive time slot, only searching the neighbouring beam directions +of the main directions can be sufficient to maintain the link +quality. This process is called beam tracking. As shown in +Fig. 1b, the duration of τb is much smaller than the initial +beam training duration. +B. Problem Formulation +The UE association depends on the channel quality between +the BS and the UE. Due to UE mobility or temporary +blockage, the channel quality changes and consequently the +UE association. Based on the UEs’ velocity, we determine how +quickly the channel quality can change and predict the time +at which the current UE association needs to be updated. We +define TA seconds as the frequency of updating the association. +Hence, we need to make the decision every TA whether to run +the handover execution or beam tracking procedure if SNR is +lower than the pre-defined SNR threshold (SNRthr). Note that +we can have an on-demand reactive handover at any time slot +if the link toward the serving BS fails abruptly. However, with +a proper choice of TA, the frequency of those reactive events +could be very small. We define the duration of the trajectory +as M and consider the discrete time index i to describe the +association update at each interval. +The goal is to maximize the aggregate throughput of the UE +along the trajectory while ensuring the achieved throughput in +each time slot i is higher than a predefined threshold. To this +end, we define functions F1 and F2 as +• F1 is the averaged throughput along the trajectory as +F1 = +M +� +i=1 +E +� +Γ(i)� +, +where the expectation is with respect to the randomness +of channel fading and the blockage, M is the duration of +the trajectory, and Γ(i) is defined in (3). +• F2 is the expected number of time slots whose throughput +is lower than the threshold (Γthr). +F2 = E +� M +� +i=1 +1 +� +Γ(i) ≤ Γthr +�� += +M +� +i=1 +Pr +� +Γ(i) ≤ Γthr +� +. +We formulate the user association at time slot i ∈ [M] +as an optimization problem which involves finding the x(i) +j +corresponding to the association indicator as +max +{x(i) +j +}i,j +F1 − λF2 +(5a) +s.t. +� +j∈B +x(i) +j += 1, ∀, i ∈ [M] +(5b) +x(i) +j +∈ {0, 1}, +∀j ∈ B, i ∈ [M] +(5c) +where λ is a large constant controlling the importance of F2. +Constraint (5b) guarantees that each UE is served by one BS. +The optimization problem (5) is nonlinear. Solving this +optimization problem requires estimating the expectation value +in F1 and F2 which requires running many realizations. +Moreover, the impact of choosing the x(i) +j +(the target BSs +in the handover case or choosing beam tracking procedure) +propagates in time and can affect the UEs’ performance in +the next time slots. Therefore, we need to consider the long- +term benefits of selecting association indicators besides their +immediate effects on the UEs’ performance. Furthermore, In +order to select the target BSs, we need to model or predict the +UEs’ performance in the next time slots, which can add more +complexity to the network due to the mobility of the UE and +obstacles in mmWave networks. These motivate us to utilize +the RL to approximate the solution of (5). +III. +PROPOSED METHOD +We transform the problem (5) to an RL problem in which +the objective function is turned into a reward function, and +the constraints are transformed into the feasible state and +action spaces. In the following, first, we start with defining +the Markov decision process, and then we will describe our +joint handover and beam tracking algorithm. +A. Markov Decision Process Formulation +RL problems are formulated based on the idea of the +Markov decision process (MDP), which is the agent’s interac- +tion with different states of the environment to maximize the +expected long-term reward. The agent is the main decision- +maker who can sit on the edge cloud. All BSs are connected +to the agent. Now, we define different elements of an MDP. + +1) State Space: The state space describes the environ- +ment by which the agent is interacting through different +actions. We define the state at time slot i as s(i) += +(ℓ(i)), j(i) +S , SNR(i), I(i)) ∈ S, where ℓ(i) is the location index +of the UE along the trajectory 2, j(i) +S is the index of the serving +BS, SNR(i) is the SNR value of the UE with serving BS j(i) +S +in time slot i. I(i) ∈ {0, 1} is the beam tracking activation +indicator. I(i) = 1 means the i-th time slot is the tracking slot +for the UE. +2) Action Space: The action space includes all possible +actions that can be taken by the agent. The action can change +the state of the environment from the current state to the target +state. In our problem, a(i) ∈ A = {0, 1, 2, ..., [|B|]} is the +decision regarding beam tracking (a(i) = 0) or choosing the +index of new serving BS in the case of handover decision +(a(i) ∈ [|B|]). In other words, if a(i) ̸= 0 means the handover +decision is made and the value of a(i) shows the target BS. +Hence, the action is to specify a serving BS for the UE along +its trajectory. +3) Policy: A policy π(.) maps the state of the environment +to the action of the agent. In our case, π is a function from S +to A, i.e., π : S → {0, 1, ..., [|B|]} +4) Rewards: The agent obtains the reward after taking an +action a(i) when current state is s(i) and moves to next state +s(i+1). Here we define reward r(s(i), a(i), s(i+1)) as +r(s(i), a(i), s(i+1)) = Γ(i) − λ1 +� +Γ(i) ≤ Γthr +� +, +(6) +where Γ(i) is defined in (3). +5) State-action value: The function Qπ(s, a) is the long- +term reward and is defined as the expected summation of +discounted reward in the future for the action a ∈ A that +agent takes in state s under policy π. The RL algorithm aims +to choose the optimal policy π⋆ in each state s that maximizes +the Qπ(s, a). With discount factor η ∈ [0, 1], we have +Qπ(s, a) = E +�� +i +ηir(s(i), s(i), s(i+1)) +� +, +where the expectation is over the transition probabilities. In +our problem, transition probabilities model the SNR variations +due to the randomness of the channel fading and blockage. +We assume mobility information including the UEs’ current +location and its trajectory is known3. Therefore, the transition +to the next location is deterministic. +The optimal policy in state s ∈ S is found by +π⋆(s) = arg max +a∈A +Qπ(s, a). +(7) +Due to the continuous and large number of state spaces, we +apply deep Q-learning (DQL) [14] to solve (7). In DQL, the +state-action value function is estimated by the deep neural +network function approximators. +2Note that, we discretize the location of the UE along the trajectory. Hence, +every location dimension (x, y) a trajectory with length M is mapped to a +location index ℓ(i) ∈ [M]. +3Note that the location information can be easily fed back through lower- +frequency links. +B. Joint Handover and Beam Tracking Algorithm +Algorithm 1 describes our proposed joint handover and +beam tracking algorithm along a trajectory with duration M. +If the current association cannot offer the required SNR level, +the decision regarding handover or beam track is made based +on a(i) as the output of the RL algorithm. In the case of the +handover decision, the value of a(i) represents the target BS. +The beam tracking algorithm based on small spatial mea- +surement in time slot i is shown in Algorithm 2. In slot i, the +algorithm starts by using the main beam of the same serving +BS in the previous time slot i − 1. If the SNR value is lower +than the threshold, then starts a small spatial measurement over +the AoD direction of the main beam. To quantify the size of the +spatial neighbourhood, we define ∆φ and ∆θ as the maximum +absolute horizontal and vertical deviation from the main AoD +direction. We define δφ and δθ as the measurement resolution +in horizontal and vertical, respectively. Inspired by [15], the +spatial neighbourhood N surrounding the main AoD direction +can be expressed using the horizontal neighbourhood Nφ and +vertical neighbourhood Nθ as +Nφ(∆φ, δφ) = +� +i.δφ : i ∈ +� +− +�∆φ +δφ +� +, +�∆φ +δφ +��� +(8) +Nθ(∆θ, δθ) = +� +j.δθ : j ∈ +� +− +�∆θ +δθ +� +, +�∆θ +δθ +��� +(9) +where ⌊.⌋ is the floor operation. The complete neighbourhood +is the Cartesian product of the horizontal and vertical neigh- +bourhoods as +N(∆φ, ∆θ, δφ, δθ) = Nφ(∆φ, δφ) × Nθ(∆θ, δθ) += {(φ, θ) : φ ∈ Nφ(∆φ, δφ), θ ∈ Nθ(∆θ, δθ)}(10) +The spatial neighborhoods T (i) in time slot i surrounding the +main AoD directions (φ(i−1) +ℓ⋆ +, θ(i−1) +ℓ⋆ +) in previous time slot is +T (i) = (φ(i−1) +ℓ⋆ +, θ(i−1) +ℓ⋆ +, ) + N(∆φ, ∆θ, δφ, δθ). +(11) +Now given the main AoD direction, we need to find the +transmit direction from neighbourhoods T (i) that offers the +SNR threshold. We represent the sorted direction pairs as +[T (i)]I, where I is the sorted indices. It means the directions +in [T (i)]I increase in distance from the main AoD direction. +Starting from the main AoD direction, the SNR of each trans- +mit direction in [T (i)]I is measured until a beam pair meets +the required SNR level. Afterwards, no further measurements +are required. If no direction meets the threshold, the entire +(∆φ, ∆θ)-neighbourhood is measured to find the beam pairs +that offer the SNR threshold. +Note that in the worse scenario, if the selected target BS +based on our proposed algorithm cannot offer the required +SNR level due to very sudden blockage, the conventional +handover methods based on searching over the candidate BSs +in UEs vicinity can be applied. However, as shown in the +numerical results, such extreme case is rare. + +Algorithm 1 Joint handover and beam tracking +Input: Trajectory with duration M +1: Initialization: for i = 1 set j(1) +S =1 +2: for i ∈ 1, ..., M do +3: +if SNR(i) +jS < SNRthr then +4: +Choose the optimal action a(i) based on current +s(i). +5: +if a(i) ̸= 0 then. +▷ handover execution +6: +Set j(i) +S += a(i) and run the initial beam training +process and compute the achieved throughput Γ(i) as (3). +7: +else +8: +Run Algorithm 2 and compute Γ(i). +9: +end if +10: +end if +11: end for +Output: Γ(i) +Algorithm 2 Beam tracking in time slot i at the BS j +Input: [T (i)]I, SNRthr, duration of each beam pair testing (β), +cnt(i) = 0. +1: for (φ, θ) ∈ [T ]I do +2: +Set f (i) +j += a(φ, θ). +3: +Measure SNR(i) +j +as (2). +4: +Set cnt(i) = cnt(i) + 1. +▷ number of beam pair +testing +5: +if SNR(i) +j +>= SNRthr then +6: +(φ(i) +ℓ⋆ , θ(i) +ℓ⋆ ) = (φBS, θBS) +7: +τ (i) +b += β.cnt(i) +8: +break; +9: +end if +10: end for +compute the achieved throughput Γ(i) as (3) +IV. NUMERICAL RESULTS +We evaluate the performance of the proposed method in an +urban environment using the ray tracing tool in the MATLAB +toolbox. The output of the ray tracing tool is the L available +paths between a BS and a UE in a specific location. The ray +tracing maintains the spatial consistency of mmWave channels. +As depicted in Fig. 2, we extracted the building map of Kista +in Stockholm city, Sweden and used it as the input data for +the ray tracing simulation. In our scenario, we assumed the +building material is brick and the terrain material is concrete. +We also add some random obstacles in the street with different +heights (1 m and 3 m) and widths (2 m and 4 m) as the human +bodies and various vehicles. These temporary obstacles are +distributed randomly in the street with density 10−2 per m2. +The material loss and the location of the temporary obstacles +are chosen randomly in each realization of the channel. The +BSs are located on the wall of buildings. The location of the +BSs is chosen randomly while covering the entire trajectory. +The BSs’ height is 6 m. We consider a pedestrian mobility +Fig. 2: Simulation area in Kista, Stockholm. The yellow line +shows the trajectory. Stars show the location of the BSs. +model with a speed of 1 m/s. We consider the different lengths +of the trajectories as 100TA, 200TA, 300TA, 400TA, 500TA. +The main simulation parameters are listed in Table I. +In the simulation, we consider the SNRthr = 2 dB and the +throughput threshold Γthr = 1 bit/Hz. The value of τc is 10 ms. +In the case of handover, we fix the initial beam training dura- +tion as τb = 1 +3τc. In the case of beam tracking, τb is not fixed +and equals the size of measuring neighbourhood multiplied +by the duration of each beam pair testing (β = 10 µs). We +compare the performance of our proposed method with two +baselines. To have a fair comparison, we choose two baselines +in which the target BS for the handover is pre-determined. +Hence, we do not take into account the discovery time of +finding the target BS in the baselines. Just like in our method, +the handover is triggered if SNR < SNRthr. +As Baseline 1 we consider the multi-connectivity method +[8]. We implement a scenario where the UE maintains its +connection with a nearby BS as a backup solution while +being connected to the serving BS and once it experiences the +blockage of the serving link, starts connecting to the backup +solution. As Baseline 2 we select the learning-based handover +in [11]. The method shows very good performance in maxi- +mizing the achieved rate along the trajectory. In this baseline, +the target BS during the handover process is determined by a +learning algorithm. Although the target BSs are selected based +on the long-term effect on the achieved rate, still can cause +frequent handovers and throughput degradation. +First, we fix the number of BSs to 10 (see Fig. 2). We +consider 104 different channel realization as the input of the +RL algorithm. After getting the optimal policy, we test it +over real-time measurements and report the average of the +performance over 500 channel realizations. Fig. 3 shows the +average number of locations with unmet throughput thresholds +along the trajectory with different lengths and Fig. 4 shows the +average number of handovers needed. In comparison to the +other two baselines, our method provides better throughput +results by selecting to perform either beam tracking or a +handover. Furthermore, we note that the two baselines have +a higher number of handovers than our method due to only +considering the handover solution. Hence, by considering the +joint handover and beam tracking problem our method pro- +vides better-achieved throughput while decreasing the number +of handovers. +Fig. 5 shows the average aggregate achieved + +Table I: Simulation parameters. +Parameters +Values in Simulations +BS transmit power +10 dBm +Noise power level +σ2=-174 dBm/Hz +Signal bandwidth +100 MHz +BS antenna +8 × 8 uniform planar array [11] +Time interval duration +TA = 1s +Neighborhood size +(∆φ, ∆θ) = (10◦, 10◦) +Measurement resolution +(δφ, δθ) = (5◦, 5◦) +Discount factor +η = 0.99 +λ +100 +100 +200 +300 +400 +500 +0 +200 +400 +Trajectory length (m) +Number of locations satisfying Γthr +Our method +Baseline 1 +Baseline 2 +Fig. 3: The average number of locations with unmet through- +put threshold for different lengths of the trajectory. +throughput along the trajectory with length 300 m for different +numbers of BSs. By increasing the number of BSs the number +of the locations satisfying the Γthr also increases hence the +aggregate throughput along the trajectory increases. Even with +a small number of BSs, our method outperforms baselines +in aggregate throughput along the trajectory by determining +whether to use a handover or beam tracking solution. +We consider 10000 iterations during the training in our +method and Baseline 2. With the training machine MacBook +Pro 2020 M1 with a memory of 16 GB, each iteration takes +about 15 seconds. Note that the absolute value of the training +time per iteration depends on the running machine. +V. CONCLUSIONS +In this work, we proposed and studied a learning-based joint +handover and beam tracking method in a mobile mmWave +network. The aim of our algorithm is to maximize the aggre- +gate throughput of the UE along a trajectory and ensure the +achieved throughput in each location is higher than the thresh- +old. 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Yum, “The smart handoff +policy for millimeter wave heterogeneous cellular networks,” IEEE Trans +Mob Comput., vol. 17, no. 6, pp. 1456–1468, Jun. 2018. +[11] S. Khosravi, H. S. Ghadikolaei, and M. Petrova, “Learning-based +handover in mobile millimeter-wave networks,” IEEE TCCN, vol. 7, +no. 2, pp. 663–674, 2021. +[12] A. Patra, L. Simi´c, and P. M¨ah¨onen, “Smart mm-wave beam steering +algorithm for fast link re-establishment under node mobility in 60 ghz +indoor wlans,” in Proceedings of the 13th ACM International Symposium +on Mobility Management and Wireless Access, 2015, pp. 53–62. +[13] M. R. Akdeniz, Y. Liu, M. K. Samimi, S. Sun, S. Rangan, T. S. +Rappaport, and E. Erkip, “Millimeter wave channel modeling and +cellular capacity evaluation,” IEEE J-SAC, vol. 32, no. 6, pp. 1164– +1179, Jun. 2014. +[14] D. Bertsekas, Reinforcement Learning and optimal control. +Athena +Scientific, 2019. +[15] I. P. Roberts, A. Chopra, T. Novlan, S. Vishwanath, and J. G. Andrews, +“Steer: Beam selection for full-duplex millimeter wave communication +systems,” IEEE Trans Commun., pp. 1–1, 2022. + diff --git a/9dE4T4oBgHgl3EQf3Q3I/content/tmp_files/load_file.txt b/9dE4T4oBgHgl3EQf3Q3I/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c0f282319bb80023f860ab0728092f1387bd268a --- /dev/null +++ b/9dE4T4oBgHgl3EQf3Q3I/content/tmp_files/load_file.txt @@ -0,0 +1,426 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf,len=425 +page_content='Reinforcement Learning-based Joint Handover and Beam Tracking in Millimeter-wave Networks Sara Khosravi∗, Hossein S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Ghadikolaei‡, Jens Zander∗, and Marina Petrova ∗† ∗School of EECS, KTH Royal Institute of the Technology, Stockholm, Sweden, † Mobile Communications and Computing, RWTH Aachen University, Germany, ‡ Ericsson Research, Sweden Email: {sarakhos, jenz, petrovam} @kth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content='se, hossein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content='shokri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content='ghadikolaei@ericsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content='com Abstract—In this paper, we develop an algorithm for joint handover and beam tracking in millimeter-wave (mmWave) networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' The aim is to provide a reliable connection in terms of the achieved throughput along the trajectory of the mobile user while preventing frequent handovers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' We model the association problem as an optimization problem and propose a reinforcement learning-based solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Our approach learns whether and when beam tracking and handover should be performed and chooses the target base stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' In the case of beam tracking, we propose a tracking algorithm based on measuring a small spatial neighbourhood of the optimal beams in the previous time slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Simulation results in an outdoor environment show the superior performance of our proposed solution in achievable throughput and the number of handovers needed in comparison to a multi- connectivity baseline and a learning-based handover baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Index Terms—Millimeter-wave, user association, beam track- ing, handover, reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' INTRODUCTION Millimeter-wave (mmWave) is a key radio access technol- ogy for beyond 5G communication systems, offering ultra- high data rates due to a large amount of free spectrum [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' However, due to the fewer scattering paths and significant penetration loss, mmWave links are vulnerable to static or dynamic obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' To overcome such severe loss, both base station (BS) and user equipment (UE) may need directional communication using a large number of antennas, which may result in frequent misalignment of beams due to mobility and blockage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Hence, finding and maintaining the optimal beam directions (beam alignment) is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' The lengthy period to achieve the beam alignment (hundreds of milliseconds to seconds [2]) results in a high cell search time or BS discovery time in mmWave systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' As reported in [3], the BS discovery time which is the time required to search the target BS when the handover command is received by the UE is about 200 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Moreover, to improve the capacity and coverage the density of the BSs is usually high in mmWave systems [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Hence, conventional handover methods based on instantaneous received signal power can cause unnecessarily frequent handovers and a ping-pong effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' This leads to a severe drop in service reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Therefore, fast BS discovery (finding target BS in the handover process), and efficient handover execution techniques, will be required to use the full promise of mmWave cellular networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' The spatial mmWave channel can be approximated by a few dominant paths, where each path can be defined with its angle of departure (AoD), angle of arrival (AoA) and gain [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Hence, one can only estimate these path parameters instead of a large dimensional channel matrix [5], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' The process of identifying the dominant paths is called beam training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' However, due to the dynamic environment, frequent beam training may cause high overhead1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Temporal correlation of spatial mmWave channel can be employed to accelerate the beam training process by tracking the variation of the dominant path directions [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Related Work To address the link failure and throughput degradation in a dynamic environment, the multi-connectivity technique has been vastly analyzed in literature [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' In this technique, the UE keeps its connection to multiple BSs (either at mmWave band or sub-6 GHz band).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' However, power consumption, synchronization and the need for frequent tracking are the main challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' In the 3GPP standard (release 16) two handover techniques are introduced to improve the link robust- ness during mobility: dual active protocol stack (DAPS), and conditional handover (CHO) [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' In the DAPS, the connection to the current serving BS is maintained until the connection to the target BS is fully established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' In the CHO, the UE is configured with multiple target BSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' During the handover, the UE can select one of the configured BSs as the target BS during the RRC reconfiguration message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Although CHO can decrease the handover failure probability, it may increase the handover latency if the UE asks for multiple handovers during a single RRC reconfiguration [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Applying machine learning as the main decision-maker tool to make the optimal handover decision and choose the target BS has been also studied in the literature [10], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' The authors in [10] proposed a reinforcement learning (RL) based handover policy to reduce the number of handovers while keeping the quality of service in heterogeneous networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' In [11] an intelligent handover method based on choosing the backup solution for each serving link to maximize the aggregate rate along a trajectory has been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' 1Overhead depends on the training time compared with the changes in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content='05305v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content='SY] 12 Jan 2023 In terms of beam tracking, authors in [12] applied the correlation of spatial mmWave channel in adjacent locations and proposed the beam steering method based on searching over a small angular space in the vicinity of the previously known valid beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' The authors in [6] applied machine learning to the tracking procedure to extract useful information from the history of AoD tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' All the aforementioned works only take handover or beam tracking issues into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Additionally, they do not study the impact of selecting beam tracking and handover on the achieved throughput of the UE along its trajectory and instead focus on the achieved rate as the primary performance metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Our Contributions In this paper, we develop a novel joint handover and beam tracking algorithm in a mmWave network under mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' The algorithm aims to associate the UEs to BSs that maximize the sum achieved throughput along the trajectory and ensure the achieved throughput in each location of the trajectory is higher than a pre-defined threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' The user association process is defined as the process of determining whether a user is associated with a particular BS before data transmissions commence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' In the case of handover, the UE is associated with a new BS, whereas in the case of beam tracking, the UE remains associated with the serving BS from the previous time slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' The main contributions of our paper are summarized as below: System Modeling: We model the user association prob- lem as a non-convex optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Unlike the existing works in the literature, we consider achieved throughput as the main performance metric to measure the effect of handover or beam tracking on the UEs’ quality of service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Learning-based Solution: The objective function in our proposed user association problem highly depends on the user association mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' We utilize the reinforcement learning (RL) algorithm to approximate the solution to this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' The aim is to decide whether to run a beam tracking algorithm or a handover algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Joint Handover and Beam Tracking Algorithm: In the case of a handover decision, the target BS will be recognized as the output of the RL algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' In the case of beam tracking, the search space will be defined based on our proposed tracking algorithm by searching the directions in the small spatial neighbourhood of the previously selected optimal directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Empirical Evaluation: We apply ray tracing with a real building data map as the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' The results show the effectiveness of our proposed method in achieving throughput along trajectories and decreasing the number of handovers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' We introduce the system model and problem formulation in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' In Section III, we propose our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' We present the numerical results in Section IV and, conclude our work in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Notations: Throughout the paper, vectors and scalars are shown by bold lower-case (x) and non-bold (x) letters, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' The conjugate transpose of a vector x is represented by xH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' We define set [M] := {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content='., M} for any integer M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' The indicator function 1{·} equals to one if the constraint inside {·} is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' SYSTEM MODEL AND PROBLEM FORMULATION In this section, first, we introduce the mmWave channel model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Then, we present the user association problem formu- lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' We consider a downlink communication with |B| mmWave BSs, where each is equipped with NBS antennas, communi- cating with a single antenna mobile UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' We consider analog beamforming with a single RF chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' We assume all BSs allocate equal resources to their serving UEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' The channel between BS j ∈ B and its serving UE during time slot i is [13]: hj = L � ℓ=1 hℓaH(φℓ, θℓ), (1) where L is the number of available paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Each path ℓ has complex gain hℓ (include path-loss) and horizontal φℓ and vertical θℓ, AoD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Due to the notation simplicity, we drop the index j and i from the channel parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' The array response vector is a(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=') where its exact expression depends on the array geometry and possible hardware impairments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' The signal-to- noise ratio (SNR) in time slot i is SNR(i) j = p|hH j fj|2 σ2 , (2) where σ2 is the noise power, p is the transmit power, fj ∈ CNBS is the beamforming vector of BS j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' We define variable x(i) j ∈ {0, 1} for j ∈ B as an association indicator in time slot i, where is equal 1 if UE is associated to the BS j and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Hence, the achieved rate per second per hertz in time slot i is R(i) = x(i) jS log2(1 + SNR(i) jS ) = � j∈B x(i) j log2(1 + SNR(i) j ), where jS is the index of the serving BS of the UE during time slot i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Here, we assume each UE is served by only one BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' We define the achievable throughput per hertz of the UE by multiplying its rate by the data transmission time as Γ(i) = (1 − τ (i) b τc )R(i), (3) where, τ (i) b is the beam training duration which may have a different value in each time slot i, and τc is the duration of the time slot that is a fixed value for all time slots, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Beam Training and Beam Tracking As depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' 1a, when the UE is connected to a BS j ∈ B, initial beam training is performed by sending pilots over all combination of the beam directions in the codebook during τb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' on the UE’s feedback of the received signal strength (or estimated SNR), the best beam pair directions are selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Then, the BS and the UE would use this Initial beam training Data Transmission τb τc (a) Tracking Data Transmission τb (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' 1: τc is the time slot duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' τb is (a) the initial beam training duration when the UE is associated with the new BS (handover case), (b) the beam tacking duration when the serving BS is the same for the consecutive slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' direction (φℓ⋆, θℓ⋆) during the data transmission phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' The beamforming vector, f is chosen to maximize the achievable rate of the UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Due to the monotonicity of the logarithm function, this is equivalent to maximising the SNR term in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Hence f ∗ j = arg max fj∈F |hH j fj|2 (4) where F is the beamforming codebook that contains all the feasible beamforming vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' The n-th element of the codebook F is defined as f(n) = a(φn, θn), where (φn, θn) are steering angles and a(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=') is the array response vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' When the BS continues serving the same UE in a consecu- tive time slot, only searching the neighbouring beam directions of the main directions can be sufficient to maintain the link quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' This process is called beam tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' 1b, the duration of τb is much smaller than the initial beam training duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Problem Formulation The UE association depends on the channel quality between the BS and the UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Due to UE mobility or temporary blockage, the channel quality changes and consequently the UE association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Based on the UEs’ velocity, we determine how quickly the channel quality can change and predict the time at which the current UE association needs to be updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' We define TA seconds as the frequency of updating the association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Hence, we need to make the decision every TA whether to run the handover execution or beam tracking procedure if SNR is lower than the pre-defined SNR threshold (SNRthr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Note that we can have an on-demand reactive handover at any time slot if the link toward the serving BS fails abruptly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' However, with a proper choice of TA, the frequency of those reactive events could be very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' We define the duration of the trajectory as M and consider the discrete time index i to describe the association update at each interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' The goal is to maximize the aggregate throughput of the UE along the trajectory while ensuring the achieved throughput in each time slot i is higher than a predefined threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' To this end, we define functions F1 and F2 as F1 is the averaged throughput along the trajectory as F1 = M � i=1 E � Γ(i)� , where the expectation is with respect to the randomness of channel fading and the blockage, M is the duration of the trajectory, and Γ(i) is defined in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' F2 is the expected number of time slots whose throughput is lower than the threshold (Γthr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' F2 = E � M � i=1 1 � Γ(i) ≤ Γthr �� = M � i=1 Pr � Γ(i) ≤ Γthr � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' We formulate the user association at time slot i ∈ [M] as an optimization problem which involves finding the x(i) j corresponding to the association indicator as max {x(i) j }i,j F1 − λF2 (5a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' � j∈B x(i) j = 1, ∀, i ∈ [M] (5b) x(i) j ∈ {0, 1}, ∀j ∈ B, i ∈ [M] (5c) where λ is a large constant controlling the importance of F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Constraint (5b) guarantees that each UE is served by one BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' The optimization problem (5) is nonlinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Solving this optimization problem requires estimating the expectation value in F1 and F2 which requires running many realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Moreover, the impact of choosing the x(i) j (the target BSs in the handover case or choosing beam tracking procedure) propagates in time and can affect the UEs’ performance in the next time slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Therefore, we need to consider the long- term benefits of selecting association indicators besides their immediate effects on the UEs’ performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Furthermore, In order to select the target BSs, we need to model or predict the UEs’ performance in the next time slots, which can add more complexity to the network due to the mobility of the UE and obstacles in mmWave networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' These motivate us to utilize the RL to approximate the solution of (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' PROPOSED METHOD We transform the problem (5) to an RL problem in which the objective function is turned into a reward function, and the constraints are transformed into the feasible state and action spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' In the following, first, we start with defining the Markov decision process, and then we will describe our joint handover and beam tracking algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Markov Decision Process Formulation RL problems are formulated based on the idea of the Markov decision process (MDP), which is the agent’s interac- tion with different states of the environment to maximize the expected long-term reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' The agent is the main decision- maker who can sit on the edge cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' All BSs are connected to the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Now, we define different elements of an MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' 1) State Space: The state space describes the environ- ment by which the agent is interacting through different actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' We define the state at time slot i as s(i) = (ℓ(i)), j(i) S , SNR(i), I(i)) ∈ S, where ℓ(i) is the location index of the UE along the trajectory 2, j(i) S is the index of the serving BS, SNR(i) is the SNR value of the UE with serving BS j(i) S in time slot i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' I(i) ∈ {0, 1} is the beam tracking activation indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' I(i) = 1 means the i-th time slot is the tracking slot for the UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' 2) Action Space: The action space includes all possible actions that can be taken by the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' The action can change the state of the environment from the current state to the target state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' In our problem, a(i) ∈ A = {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=', [|B|]} is the decision regarding beam tracking (a(i) = 0) or choosing the index of new serving BS in the case of handover decision (a(i) ∈ [|B|]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' In other words, if a(i) ̸= 0 means the handover decision is made and the value of a(i) shows the target BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Hence, the action is to specify a serving BS for the UE along its trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' 3) Policy: A policy π(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=') maps the state of the environment to the action of the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' In our case, π is a function from S to A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=', π : S → {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=', [|B|]} 4) Rewards: The agent obtains the reward after taking an action a(i) when current state is s(i) and moves to next state s(i+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Here we define reward r(s(i), a(i), s(i+1)) as r(s(i), a(i), s(i+1)) = Γ(i) − λ1 � Γ(i) ≤ Γthr � , (6) where Γ(i) is defined in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' 5) State-action value: The function Qπ(s, a) is the long- term reward and is defined as the expected summation of discounted reward in the future for the action a ∈ A that agent takes in state s under policy π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' The RL algorithm aims to choose the optimal policy π⋆ in each state s that maximizes the Qπ(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' With discount factor η ∈ [0, 1], we have Qπ(s, a) = E �� i ηir(s(i), s(i), s(i+1)) � , where the expectation is over the transition probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' In our problem, transition probabilities model the SNR variations due to the randomness of the channel fading and blockage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' We assume mobility information including the UEs’ current location and its trajectory is known3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Therefore, the transition to the next location is deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' The optimal policy in state s ∈ S is found by π⋆(s) = arg max a∈A Qπ(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' (7) Due to the continuous and large number of state spaces, we apply deep Q-learning (DQL) [14] to solve (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' In DQL, the state-action value function is estimated by the deep neural network function approximators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' 2Note that, we discretize the location of the UE along the trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Hence, every location dimension (x, y) a trajectory with length M is mapped to a location index ℓ(i) ∈ [M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' 3Note that the location information can be easily fed back through lower- frequency links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Joint Handover and Beam Tracking Algorithm Algorithm 1 describes our proposed joint handover and beam tracking algorithm along a trajectory with duration M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' If the current association cannot offer the required SNR level, the decision regarding handover or beam track is made based on a(i) as the output of the RL algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' In the case of the handover decision, the value of a(i) represents the target BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' The beam tracking algorithm based on small spatial mea- surement in time slot i is shown in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' In slot i, the algorithm starts by using the main beam of the same serving BS in the previous time slot i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' If the SNR value is lower than the threshold, then starts a small spatial measurement over the AoD direction of the main beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' To quantify the size of the spatial neighbourhood, we define ∆φ and ∆θ as the maximum absolute horizontal and vertical deviation from the main AoD direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' We define δφ and δθ as the measurement resolution in horizontal and vertical, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Inspired by [15], the spatial neighbourhood N surrounding the main AoD direction can be expressed using the horizontal neighbourhood Nφ and vertical neighbourhood Nθ as Nφ(∆φ, δφ) = � i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content='δφ : i ∈ � − �∆φ δφ � , �∆φ δφ ��� (8) Nθ(∆θ, δθ) = � j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content='δθ : j ∈ � − �∆θ δθ � , �∆θ δθ ��� (9) where ⌊.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content='⌋ is the floor operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' The complete neighbourhood is the Cartesian product of the horizontal and vertical neigh- bourhoods as N(∆φ, ∆θ, δφ, δθ) = Nφ(∆φ, δφ) × Nθ(∆θ, δθ) = {(φ, θ) : φ ∈ Nφ(∆φ, δφ), θ ∈ Nθ(∆θ, δθ)}(10) The spatial neighborhoods T (i) in time slot i surrounding the main AoD directions (φ(i−1) ℓ⋆ , θ(i−1) ℓ⋆ ) in previous time slot is T (i) = (φ(i−1) ℓ⋆ , θ(i−1) ℓ⋆ , ) + N(∆φ, ∆θ, δφ, δθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' (11) Now given the main AoD direction, we need to find the transmit direction from neighbourhoods T (i) that offers the SNR threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' We represent the sorted direction pairs as [T (i)]I, where I is the sorted indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' It means the directions in [T (i)]I increase in distance from the main AoD direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Starting from the main AoD direction, the SNR of each trans- mit direction in [T (i)]I is measured until a beam pair meets the required SNR level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Afterwards, no further measurements are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' If no direction meets the threshold, the entire (∆φ, ∆θ)-neighbourhood is measured to find the beam pairs that offer the SNR threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Note that in the worse scenario, if the selected target BS based on our proposed algorithm cannot offer the required SNR level due to very sudden blockage, the conventional handover methods based on searching over the candidate BSs in UEs vicinity can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' However, as shown in the numerical results, such extreme case is rare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Algorithm 1 Joint handover and beam tracking Input: Trajectory with duration M 1: Initialization: for i = 1 set j(1) S =1 2: for i ∈ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=', M do 3: if SNR(i) jS < SNRthr then 4: Choose the optimal action a(i) based on current s(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' 5: if a(i) ̸= 0 then.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' ▷ handover execution 6: Set j(i) S = a(i) and run the initial beam training process and compute the achieved throughput Γ(i) as (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' 7: else 8: Run Algorithm 2 and compute Γ(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' 9: end if 10: end if 11: end for Output: Γ(i) Algorithm 2 Beam tracking in time slot i at the BS j Input: [T (i)]I, SNRthr, duration of each beam pair testing (β), cnt(i) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' 1: for (φ, θ) ∈ [T ]I do 2: Set f (i) j = a(φ, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' 3: Measure SNR(i) j as (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' 4: Set cnt(i) = cnt(i) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' ▷ number of beam pair testing 5: if SNR(i) j >= SNRthr then 6: (φ(i) ℓ⋆ , θ(i) ℓ⋆ ) = (φBS, θBS) 7: τ (i) b = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content='cnt(i) 8: break;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' 9: end if 10: end for compute the achieved throughput Γ(i) as (3) IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' NUMERICAL RESULTS We evaluate the performance of the proposed method in an urban environment using the ray tracing tool in the MATLAB toolbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' The output of the ray tracing tool is the L available paths between a BS and a UE in a specific location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' The ray tracing maintains the spatial consistency of mmWave channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' As depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' 2, we extracted the building map of Kista in Stockholm city, Sweden and used it as the input data for the ray tracing simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' In our scenario, we assumed the building material is brick and the terrain material is concrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' We also add some random obstacles in the street with different heights (1 m and 3 m) and widths (2 m and 4 m) as the human bodies and various vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' These temporary obstacles are distributed randomly in the street with density 10−2 per m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' The material loss and the location of the temporary obstacles are chosen randomly in each realization of the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' The BSs are located on the wall of buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' The location of the BSs is chosen randomly while covering the entire trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' The BSs’ height is 6 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' We consider a pedestrian mobility Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' 2: Simulation area in Kista, Stockholm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' The yellow line shows the trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Stars show the location of the BSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' model with a speed of 1 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' We consider the different lengths of the trajectories as 100TA, 200TA, 300TA, 400TA, 500TA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' The main simulation parameters are listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' In the simulation, we consider the SNRthr = 2 dB and the throughput threshold Γthr = 1 bit/Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' The value of τc is 10 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' In the case of handover, we fix the initial beam training dura- tion as τb = 1 3τc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' In the case of beam tracking, τb is not fixed and equals the size of measuring neighbourhood multiplied by the duration of each beam pair testing (β = 10 µs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' We compare the performance of our proposed method with two baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' To have a fair comparison, we choose two baselines in which the target BS for the handover is pre-determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Hence, we do not take into account the discovery time of finding the target BS in the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Just like in our method, the handover is triggered if SNR < SNRthr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' As Baseline 1 we consider the multi-connectivity method [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' We implement a scenario where the UE maintains its connection with a nearby BS as a backup solution while being connected to the serving BS and once it experiences the blockage of the serving link, starts connecting to the backup solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' As Baseline 2 we select the learning-based handover in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' The method shows very good performance in maxi- mizing the achieved rate along the trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' In this baseline, the target BS during the handover process is determined by a learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Although the target BSs are selected based on the long-term effect on the achieved rate, still can cause frequent handovers and throughput degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' First, we fix the number of BSs to 10 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' We consider 104 different channel realization as the input of the RL algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' After getting the optimal policy, we test it over real-time measurements and report the average of the performance over 500 channel realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' 3 shows the average number of locations with unmet throughput thresholds along the trajectory with different lengths and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' 4 shows the average number of handovers needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' In comparison to the other two baselines, our method provides better throughput results by selecting to perform either beam tracking or a handover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Furthermore, we note that the two baselines have a higher number of handovers than our method due to only considering the handover solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Hence, by considering the joint handover and beam tracking problem our method pro- vides better-achieved throughput while decreasing the number of handovers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' 5 shows the average aggregate achieved Table I: Simulation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Parameters Values in Simulations BS transmit power 10 dBm Noise power level σ2=-174 dBm/Hz Signal bandwidth 100 MHz BS antenna 8 × 8 uniform planar array [11] Time interval duration TA = 1s Neighborhood size (∆φ, ∆θ) = (10◦, 10◦) Measurement resolution (δφ, δθ) = (5◦, 5◦) Discount factor η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content='99 λ 100 100 200 300 400 500 0 200 400 Trajectory length (m) Number of locations satisfying Γthr Our method Baseline 1 Baseline 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' 3: The average number of locations with unmet through- put threshold for different lengths of the trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' throughput along the trajectory with length 300 m for different numbers of BSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' By increasing the number of BSs the number of the locations satisfying the Γthr also increases hence the aggregate throughput along the trajectory increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Even with a small number of BSs, our method outperforms baselines in aggregate throughput along the trajectory by determining whether to use a handover or beam tracking solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' We consider 10000 iterations during the training in our method and Baseline 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' With the training machine MacBook Pro 2020 M1 with a memory of 16 GB, each iteration takes about 15 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Note that the absolute value of the training time per iteration depends on the running machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' CONCLUSIONS In this work, we proposed and studied a learning-based joint handover and beam tracking method in a mobile mmWave network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' The aim of our algorithm is to maximize the aggre- gate throughput of the UE along a trajectory and ensure the achieved throughput in each location is higher than the thresh- old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Our evaluation results showed that by making an optimal decision regarding handover execution or beam tracking, our method provides high achievable throughput and reduces the number of handovers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Considering different mobility models and studying the effect of neighbouring size can be valuable future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' REFERENCES [1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Rappaport, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Sun, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dE4T4oBgHgl3EQf3Q3I/content/2301.05305v1.pdf'} +page_content=' Mayzus, H.' metadata={'source': 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100644 index 0000000000000000000000000000000000000000..a92054ed9095042ec91d85f7175f3461fb08700d --- /dev/null +++ b/D9E4T4oBgHgl3EQffA2Y/content/tmp_files/2301.05104v1.pdf.txt @@ -0,0 +1,1218 @@ +Learning to compile smartly for program size reduction +Youwei Liang∗ 1 Kevin Stone∗ 2 Ali Shameli 2 Chris Cummins 2 Mostafa Elhoushi 2 Jiadong Guo 2 +Benoit Steiner 2 Pengtao Xie 1 Hugh Leather 2 Yuandong Tian 2 +Abstract +Compiler optimization passes are an important +tool for improving program efficiency and reduc- +ing program size, but manually selecting opti- +mization passes can be time-consuming and error- +prone. While human experts have identified a few +fixed sequences of optimization passes (e.g., the +Clang -Oz passes) that perform well for a wide +variety of programs, these sequences are not con- +ditioned on specific programs. In this paper, we +propose a novel approach that learns a policy to +select passes for program size reduction, allow- +ing for customization and adaptation to specific +programs. Our approach uses a search mecha- +nism that helps identify useful pass sequences +and a GNN with customized attention that se- +lects the optimal sequence to use. Crucially it +is able to generalize to new, unseen programs, +making it more flexible and general than previ- +ous approaches. We evaluate our approach on a +range of programs and show that it leads to size +reduction compared to traditional optimization +techniques. Our results demonstrate the potential +of a single policy that is able to optimize many +programs. +1. Introduction +Finding the right compiler optimization ordering for a given +program in order to execute them more efficiently with a +smaller amount of resources (e.g., memory, CPU and stor- +age) is an important yet challenging problem. Traditionally, +to tune configurations, human effort and expert knowledge +are needed, which is a time-consuming and error-prone +process that often yields sub-par results. +In recent years, machine learning-guided compiler optimiza- +tion has emerged as an interesting field to replace this labori- +ous process (Wang & O’Boyle, 2018). Along this line, many +*Equal contribution +1University of California, San Diego +2Meta AI. Correspondence to: +Kevin Stone +. +works show promising results using various machine learn- +ing (ML) techniques and optimizers (e.g., reinforcement +learning (Haj-Ali et al., 2020a), language modelling (Cum- +mins et al., 2017), evolutionary algorithms (Kulkarni & +Cavazos, 2012), etc). They focus on automatic decision- +making of specific components in compilation, e.g., op- +timizing computational graph of neural network for ML +compilers (Zhou et al., 2020), optimizing the loop structure +of neural network computations (Steiner et al., 2021), deter- +mination of a function to be inlining (MLGO) (Trofin et al., +2021), etc. There are only a few works targeting generic +optimization of compilation (Haj-Ali et al., 2020b; Almakki +et al., 2022). +In this work, we take a different path and explore the pos- +sibility of learning policies for more a general aspect of +compiler optimization. More specifically, we treat compila- +tion optimization as a general sequential decision process, +in which each step is to find an additional compiler pass +so that when combined with existing passes, can improve +specific metrics (e.g., the binary size / running speed of the +codebase). Our goal is to learn a policy, named pass policy, +to help quickly find the right compiler passes to be used, +given the current program and existing passes. We aim to +compare with existing handtuned policies in the compiler, +e.g., -Oz for code size reduction and “-O3” for running +speed optimization. Such policies have been tuned by the +domain experts for decades and used extensively in all the +computer systems, but are invariant to the program being +compiled. +As the first contribution, we propose a novel evaluation +protocol for compiler pass optimization, named zero-shot +generalizability (ZSG), to evaluate learned policies applied +to unseen programs. Here “zero-shot” means that each +unseen program has never been seen before in the training +process, and we allow a fixed number of optimization passes +(say 45). The policy is allowed to adjust the passes based +on previous outcomes (This includes backtracking if an +exploratory step turns out to be sub-optimal.) Compared +to previous works (e.g., Autophase (Haj-Ali et al., 2020b), +CompilerGym (Cummins et al., 2022), MLGO (Trofin et al., +2021), MLGoPerf (Ashouri et al., 2022)) that run adaptive +search algorithms to optimize a set of programs for many +hours, we argue that ZSG is a realistic setting for practical +arXiv:2301.05104v1 [cs.PL] 9 Jan 2023 + +Learning to compile smartly for program size reduction +deployment: for most programs except the critical ones, +there is a limited time budget to optimize them, and often +there is not enough computational resources to fine-tune +learned policies for each unseen task separately (e.g., fine- +tune setting in GO (Zhou et al., 2020)). While similar metric +has been used in previous works (e.g., MLGO and GO), we +are the first to test it in more general ML-guided compiler +optimizations. +With the ZSG metric, we then evaluate multiple existing +techniques applied to compiler optimization, with LLVM +compiler and on large-scale datasets provided by Compiler- +Gym (Cummins et al., 2022). We use code-size reduction +as the metric since it is a deterministic quality and relatively +easy/cheap to compute, and we leave program run-time +optimization for future work. +Given the metric, we discover a universal core subset of +LLVM pass sequences, of size 50, that leads to very strong +performance across multiple sets of programs from diverse +domains, ranging from Linux Kernel to BLAS library. The +total number of steps in the 50 pass sequences is 625 and +the average length of the sequences is 12.5. Specifically, +for one unseen program, there exists one of the 50 pass +sequences that leads to an average code size reduction of +5.8% compared to the default -Oz setting, across 10 diverse +codebases of over one million programs. In other words, +after running trying the 625 steps on an unseen program, the +smallest code size during the compilation is 5.8% smaller +than that of -Oz. Considering the huge search space of com- +piler flags (10104), this is a very surprising finding. We find +this coreset in a training set and it can generalize reasonably +well to the evaluation set including datasets not including in +the training set. +While it can be time-consuming to find such a compiler +flag configuration with an exhaustive enumeration of the +core subset, we find that the optimal pass can be directly +predicted with high accuracy via GNN with a customized +attention layer to control information flow with the Pro- +GraML (Cummins et al., 2021) graph as the input. The +prediction top-1 and top-5 accuracy is 75% and 95%. There- +fore, we can run a few sequences selected by the model on +an unseen program to obtain a good code size reduction. +This enables us to find a good flag configuration that leads +to 4% improvement on average, with just 45 compilation +passes (this is roughly 3 sequences since the average length +of the sequences in the coreset is 12.5). +We have compared our approach with extensive baselines, +including RL-based methods such as PPO and Q-learning +and black-box optimizers such as evolutionary algorithm. +It turns out that ML approaches often suffer from severe +overfitting to the training set and does not perform well +for new program categories, even combined with heuristic +search, and optimizers may get lost in the exponential action +space due to lack of domain knowledge. In comparison, our +approach is simple, effective and generalizable to unseen +programs. +2. Related Work +Graph structured data are present in numerous applications +and it has been shown that taking advantage of this data +can help us train very effective machine learning models. +(Brauckmann et al., 2020) use abstract syntax trees and con- +trol flow graphs for learning compiler optimization goals. +They show that using such graphs allows them to outper- +form state-of-the-art in the task of heterogeneous OpenCL +mapping. (Guo et al., 2020) uses a transformer based model +with a graph guided masked attention that incorporates the +data flow graph into the training. They achieve state of the +art performance in four tasks including code search, clone +detection, code translation, and code refinement. +As a contender to graph neural networks, (Mialon et al., +2021) uses transformers to process graphs. They show that +if we effectively encode positional and local sub-structures +of graphs and feed them to the transformer, then the trans- +former can outperform the classical GNN models. They +test their model on classification and regression tasks and +achieve state of the art performance. +In (Srinivas et al., +2020), they used an unsupervised model to learn embed- +dings of high dimensional pixel data using contrastive learn- +ing. They then use this embedding for downstream rein- +forcement learning tasks. +3. Methodology +3.1. Action space +The CompilerGym framework provides a convenient in- +terface for the compiler pass ordering problem. The de- +fault environment allows choosing one of 124 discrete ac- +tions at each step corresponding to running a sequence of +specific compiler pass. Given that our trajectories have a +length of 45 steps, this means we have 12445 ∼ 1.6 × 1094 +possible action trajectories to explore. To find an opti- +mal action sequence for a program, we can apply some +existing reinforcement learning methods including Q learn- +ing like DQN (Mnih et al., 2015) and policy gradient like +PPO (Schulman et al., 2017). +Action Sequences However for this problem it turns out +that certain action sequences are good at optimizing many +different programs (where “good” is defined as better than +the compiler default -Oz). We found that constraining the +action space to a learned set of action sequences enables +state of the performance and also significantly reduces the +challenge of exploration. This allows us to cast the prob- +lem as one of supervised learning over this set of action + +Learning to compile smartly for program size reduction +Figure 1. An example of a small action tree created by keeping +track of the common prefixes of the action sequences. +sequences. We use the following algorithm to find a good +set of action sequences. +• Random search We seed the list of candidate action +trajectories (action sequences) by running a random +policy on a subset of the training programs (N). For +each program we run M episodes and keep track of +the best action sequence for each program. +• Canonicalize During the search process we find that +trajectories often revisit the same state. Whenever this +happens we truncate all previous actions. On average +this reduces the trajectory length by a factor of 1/5. +• All-to-all We then test all N best action sequences +against all N programs. This gives us a N×N matrix +where each value is the cumulative reward (return). We +then normalize this matrix by the maximum return for +each program. A value of 1 represents that an action +sequences was optimal for a program (optimal in the +sense of this limited set of action sequences). +• Greedy assignment Many action sequences are opti- +mal for more than one program. We take advantage +of this to reduce the number of action sequences by +greedily picking action sequences that are optimal for +the the largest number of programs. This results in a +much smaller list of action sequences. We can visual- +ize this by creating a prefix tree that shows common +prefixes as single node. See Figure 1 for truncated tree +for illustration. +It is interesting to note that some actions are common across +the beginning of multiple action sequences. These popular +actions such as 27 (-reg2mem) as seen in Figure 1 are +pivotal for many programs. For a complete list of all LLVM +passes refer to Table 5. +3.2. Offline behavior cloning methods +Normalized Value Prediction After discovering the “good” +action sequences (i.e., the coreset), we can turn the problem +of the sequential decision-making on compiler passes into a +problem of supervised classification. The target is to train +a model to predict the best action sequence conditioned on +the program, where the label of the program is the index +of the action sequence that results in the highest code size +reduction. However, one important observation we have +is that there are typically multiple action sequences in the +coreset that all result in the highest code size reduction. +Therefore, instead of using the single-class classification +method with cross entropy loss, we leverage the fact we have +access to the values for all action sequences. We predict +the softmax normalized value of each action sequence with +a cross entropy loss detailed below. We call this approach +behavior cloning (BC) over the coreset. +For a program o, we roll out all the predefined action se- +quences on it, obtaining a return ro +i for the i-th sequence +(i.e., the highest cumulative reward observed during the +rollout of the action sequence), which forms a value vec- +tor ro = [ro +1, . . . , ro +n]. Then, the normalized values of the +action sequences are defined by +vo = Softmax(ro/T) +(1) +where T is temperature. +For an initial observation so +0 of a program, our model out- +puts a probability distribution, p = f(so +0), over the action +sequences. The target of the training is to make p close to +the normalized values of the action sequences. We use the +cross entropy loss to supervise the model +L(po, vo) = − +n +� +i +po +i log vo +i +(2) +3.3. Program Representation +We are considering LLVM optimization passes on that op- +erate on the Intermediate Representation (IR) of a program. +The IR contains very rich structures, while its size can be +quite large for large programs. It also contains informa- +tion (e.g., strings and constants) irrelevant to the task we +are considering. We found that working with compressed +representations made this problem tractable and run in a +reasonable amount of time. +ProGraML We leverage a graph based representation that +encodes semantic information of the program covering three +layers: control flow, data flow, and data types. This rep- +resentation has the advantage that it is not a fixed size - it +does oversimplify large programs - and yet it is still a more +compact format than the original IR format. +3.4. Network Architecture +One of the ways to model our policy function is to use a +graph neural network (GNN). To achieve this goal, we use + +53,122,31,36,111,10,97 +10 +64,31,10,52,111,116,36,40,48,54,30,53,114,29,120,10 +36,103,24,53,97,53,38,69,97,57,10,29 +39,64,55,53,38,122,31,111,64,10,39,21,105,36 +27 +104,55,57,26,103,10,29,31,36,120,102,53 +root +29 +55,39,61,27,41,36,25,103,10 +30,48,29,120,103,96,47,29,78,21,122,41,36,10 +72,55,103,36,122,59,30,65,53,10 +103,102,30,36,61,29,41,71,10,61,41,52Learning to compile smartly for program size reduction +1 +2 +5 +3 +𝑋!" +# +𝑋!$ +# +𝑋!% +# +𝑋"& +# +𝑋"! +𝑋%! +𝑋$! +𝑋&" +4 +Figure 2. Graph attention. Circles denote nodes and solid arrows +denote edges. Squares are the calculated features, and dash arrows +represent feature aggregation. The orange/green squares denote the +features to be aggregated in the target/source nodes of the edges. +the ProGraML graph structure proposed in (Cummins et al., +2021) in which individual statements are connected to other +statements through relational dependencies. Compared to +the approach of treating the program scripts as text and +encoding the programs with a language model, using the +ProGraML graph representations and encoding them with +GNNs has several advantages. Firstly, the long-range de- +pendencies of instructions are automatically captured by +the edges in the graphs, whereas the NLP approaches need +to use an LSTM or Transformer to capture the dependen- +cies. An LSTM could lose early memory when the program +is long, and a Transformer could cause out of memory is- +sues in such cases. Secoundly, changing the names of the +variables/constants/functions/classes in a program will not +affect its ProGraML representation but will change the rep- +resentations in texts. Our goal is to use the structure and +relational dependencies of this graph to learn an embed- +ding which allows us to learn a better policy. We experi- +mented with several different architectures such as Gated +Graph Convolutions (Li et al., 2015), Graph Attention Net- +works (Veliˇckovi´c et al., 2017), as well as our own custom +variation described bellow. +Edges types ProGraML supports multiple directed edge +types representing control flow, data flow, function call, and +data type definition. The other edge features include the +integer position of an edge among all the edges pointing +to the same node, a boolean feature indicating whether the +Notation +Meaning +E +The set of edges in the graph +(i, j) +Edge from node i pointing to node j +X(t) +i +Repr. of node i at layer t +E(t) +i→j +Repr. of the edge (i, j) at layer t +X(t) +ij +Repr. for node i associated with edge (i, j) +X′(t) +ij +Repr. for node i associated with edge (j, i) +a(t) +ij +Raw attention associated with repr. X(t) +ij +a′(t) +ij +Raw attention associated with repr. X′(t) +ij +α(t) +ij +Normalized attention associated with a(t) +ij +α′(t) +ij +Normalized attention associated with a′(t) +ij +Ti +Target neighbors of node i: {j|(i, j) ∈ E} +Si +Source neighbors of node i: {j|(j, i) ∈ E} +Table 1. The notations in GNN (“Repr.” means representation) +two nodes connected by the edge are in the same LLVM +basic block, and the distance of the two nodes if they are +in the same basic block. Equipped with the rich features of +the edges in ProGraML graphs, we propose a dynamic edge +encoding approach to capture the edge representations. +Dynamic edge representation Most existing GNNs that +exploit the edge feature basically use a static edge feature, +which means the same edge feature is repeatedly used for +all layers. It turns out that it is important to use a dynamic +edge representation during graph encoding, where the edge +representation gets updated in each GNN layer. The initial +edge representations are the concatenations of the embed- +ding of edge types, edge positions, and other edge features +discussed in the last paragraph. +Attention with edge features It was also helpful to modify +the default attention mechanism to support these custom +edge types. We propose to incorporate the edge features +by encoding a triplet containing the features of two nodes +and the feature of the edge that connects them. For clar- +ity, we show a table containing the notations used in the +GNN in Table 1. Then, the feature update process can be +mathematically defined by the following equations, where +Mi, i = 1, . . . , 5 is an encoding MLP. +X′(t+1) +ij += M1(X(t) +i , E(t) +i→j, X(t) +j ), +(3) +a′(t+1) +ij += M2(X(t) +i , E(t) +i→j, X(t) +j ), +(4) +X(t+1) +ji += M3(X(t) +i , E(t) +i→j, X(t) +j ), +(5) +a(t+1) +ji += M4(X(t) +i , E(t) +i→j, X(t) +j ), +(6) +E(t+1) +i→j += M5(X(t) +i , E(t) +i→j, X(t) +j ), +(7) +In words, the 3-tuple, (X(t) +i , E(t) +i→j, X(t) +j ), associated with + +Learning to compile smartly for program size reduction +edge (i, j), is encoded by MLPs to output 5 features, includ- +ing X′(t+1) +ij +and a′(t+1) +ij +(a representation and attention to +be aggregated in node i), and X(t+1) +ji +and a(t+1) +ji +(a repre- +sentation and attention to be aggregated in node j), and the +updated edge representation E(t+1) +i→j . Note that the features +to be aggregated to a target node are marked with the ′, and +those to a source node are without the ′. After the feature +encoding, we perform an attention-weighted neighborhood +aggregation for each node, which can be mathematically +described by the following equations. +� +{α(t+1) +ij +}j∈Ti ∪ {α′(t+1) +ij +}j∈Si +� += Softmax +� +{a(t+1) +ij +}j∈Ti ∪ {a′(t+1) +ij +}j∈Si +� +(8) +X(t+1) +i += +� +j∈Ti +α(t+1) +ij +X(t+1) +ij ++ +� +j∈Si +α′(t+1) +ij +X′(t+1) +ij +(9) +3.5. Dataset preparation +Overfitting issues could happen if training is performed on +a small subset of programs, or the set of programs is not +diverse enough. To mitigate this we found it helpful to +create an aggregate dataset that uses many different public +datasets as curated by CompilerGym. CompilerGym gives +us access to 14 different datasets constructed using two +different methods. +• Curated These are small collections of hand-picked +programs. They are curated to be distinct from one +another, so splitting curated suites can be challenging. +Typically programs are larger as they may comprise +multiple source files combined into a single program. +These are commonly used for evaluating compiler op- +timization improvements. +• Uncurated These are comprised of individual source +files scraped from open source repositories such as +Linux, Tensorflow, or synthetically generated pro- +grams, normally targeted for compiler testing (not op- +timization). They may not be as ”representative” of +human written test programs. +For our aggregate dataset we decided to hold-out the entirety +of the four curated datasets for use as an out-of-domain test +set. This is important because they represent the types of +programs we expect to see in the wild. We also split the +uncurated datasets into train, validaton, and test programs. +3.6. Evaluation +For all our metrics and rewards we leverage the IR instruc- +tion count as value we are trying to minimize. We also +report metrics on each CompilerGym dataset as well as +Type +Dataset +Splits +Uncurated +anghabench-v1 +train,val,test +blas-v0 +train,val,test +github-v0 +train,val,test +linux-v0 +train,val,test +opencv-v0 +train,val,test +poj104-v1 +train,val,test +tensorflow-v0 +train,val,test +clgen-v0 +train,val,test +csmith-v0 +train,val,test +llvm-stress-v0 +train,val,test +Curated +cbench-v1 +test +chstone-v0 +test +mibench-v1 +test +npb-v0 +test +Table 2. CompilerGym dataset types and training splits. +the mean over datasets to get a single number to compare +overall results. +• The mean percent improved over -Oz (MeanOverOz) +defined as following: +MeanOverOz := +1 +|P| +� +p +IOz +p +− Iπθ +p +IOz +p +, +(10) +where p is a specific program from the set of programs +P in the dataset. IOz +p +is the number of IR instructions +in program after running the default compiler pass -Oz. +Iπθ +p +is the number of IR instruction in the program after +applying the policy under consideration. We can think +of this as a simple average of the percent improvement +over -Oz. +• We +also +look +compare +the +geometric +mean +(GMeanOverOz) of final size relative to -Oz. +This metric is less sensitive to outliers and is used +by (Cummins et al., 2022). +GMeanOverOz := +�� +p +IOz +p +Iπθ +p +� +1 +|P| +(11) +4. Experiments +4.1. Baselines +• Oracle-All We consider a brute-force search over the +action tree in order to find the best action sequence for +a given program. This gives us an upper-bound of the +downstream policy network. In this case the action tree +has a total of 625 nodes. + +Learning to compile smartly for program size reduction +• Oracle-Top-45 We also consider how well we would +do if the oracle is only allowed to use the most popular +action sequences but limited to 45 steps. We use 45 +steps because this is maximum allowed for our all other +baselines and our proposed method. +• Autophase-RL We start with the strong baseline of us- +ing Autophase features. Autophase is a mapping from +a program to fixed size feature vector of 54 dimensions. +It contains integer counts of various program proper- +ties such as number of instructions, maximum loop +depth, etc. This is used in combined with a 2-layer +MLP model and trained with the PPO algorithm. This +is the approach presented in (Haj-Ali et al., 2020b). +• Autophase-BC We also consider the performance of +using the action sequences combined with a MLP +model using Autophase features to isolate the contribu- +tion of the GNN from the action sequences search. +• GNN-RL We compare with using ProGraML features +and our GNN model, but trained with the PPO algo- +rithm. This helps motivate the reason for performing +the search for action sequences in the first phase. +4.2. Results +In Table 3 we present the results of our experiments com- +paring our proposed model GNN-BC as compared to the +various baselines. The test programs were completely held +out during both data-driven learning phases (action sequence +search and model training). +The results show that our model achieves strong perfor- +mance over the prior method (Autophase-RL) proposed in +(Haj-Ali et al., 2020b). Additionally we can see that both +the GNN model and the action sequences were needed to +achieve our final performance. See Figure 3 for a visualiza- +tion of the improvement in program size over the 45 steps +on some of the programs from the holdout set. +The Oracle-All shows strong performance but requires a +large number of interactions with the compiler. But, this +shows that the action sequence search generalizes to new +unseen programs. This is somewhat unsurprising given that +the compilers built-in hand tuned pass list (-Oz) works +reasonably well for most programs. +The performance of Oracle-Top-45 by itself is weak show- +ing that in order to achieve good results in a reasonable +number of passes (45) we need to leverage a general pol- +icy and search to select the most likely candidate action +sequences to evaluate. +Both RL baselines using the original action space of single +passes Autophase-RL and GNN-RL performed poorly on +the generalization task. We hypothesis that this is partly +Method +Test MeanOverOz +Test GMeanOverOz +Compiler (-Oz) +0% +0 +Autophase-RL +-16.3% +0.960 +Autophase-BC +4.2% +1.056 +GNN-RL +-9.6% +1.005 +GNN-BC +4.7% +1.062 +Oracle-Top-45 +-7.5% +0.992 +Oracle-All +5.8% +1.075 +Table 3. Evaluation results on held out test set averaged over all +datasets. +due to the challenging exploration problem of the large +search space (12445). This is also a very hard setting for RL +because each program has its own cumulative reward upper +bound (which is unknown even for the training set). This +makes approximating the value function very difficult for +the baseline RL methods. +References +Almakki, M., Izzeldin, A., Huang, Q., Ali, A. H., and +Cummins, C. Autophase v2: Towards function level +phase ordering optimization. In ISCA 2022 Workshop on +MLArchSys, 2022. +Ashouri, A. H., Elhoushi, M., Hua, Y., Wang, X., Manzoor, +M. A., Chan, B., and Gao, Y. Mlgoperf: An ml guided +inliner to optimize performance, 2022. 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Proceedings of the IEEE, 106(11):1879– +1901, 2018. +Zhou, Y., Roy, S., Abdolrashidi, A., Wong, D., Ma, P., +Xu, Q., Liu, H., Phothilimtha, P., Wang, S., Goldie, A., +et al. Transferable graph optimizers for ml compilers. +Advances in Neural Information Processing Systems, 33: +13844–13855, 2020. + +Learning to compile smartly for program size reduction +Figure 3. Program optimization example over many steps comparing the Autophase-RL (blue) approach with our GNN-BC (orange) +approach. The dashed line represents the compiler default -Oz performance and higher is better. + +benchmark://cbench-v1/diikstra +benchmark://cbench-v1/stringsearch +1.5 +1.0 +0.5 +0.0 +benchmark://cbench-v1/blowifish +benchmark://cbench-v1/stringsearch2 +1.5 +1.0 +0.5 +0.0 +benchmark://cbench-vl/gsort +benchmark://cbench-v1/bitcount +1.5 +1.0 +0.5 +0.0 +benchmark://cbench-y1/rindae! +benchmark://cbench-v1/sha +1.5 +1.0 +0.5 +0.0 +i +0 +5 +10 +15 +20 +25 +30 +35 +40 +0 +5 +10 +15 +20 +25 +30 +35 +40Learning to compile smartly for program size reduction +Dataset +Oracle-All +Oracle-Top-45 +Autophase-RL +Autophase-BC +GNN-RL +GNN-BC +anghabench-v1 +0.7%/1.011 +-1.0%/0.996 +-15.9%/0.974 +-0.1%/1.002 +-2.5%/0.988 +-0.0%/1.003 +blas-v0 +2.6%/1.028 +-0.4%/0.997 +-1.7%/0.984 +1.2%/1.013 +-1.2%/0.989 +2.4%/1.026 +cbench-v1 +3.5%/1.041 +-2.4%/0.984 +-10.1%/0.925 +1.5%/1.021 +2.4%/1.030 +2.2%/1.028 +chstone-v0 +9.3%/1.106 +1.2%/1.016 +1.3%/1.018 +7.0%/1.079 +6.4%/1.071 +8.8%/1.101 +clgen-v0 +5.4%/1.060 +3.1%/1.034 +-0.5%/0.998 +4.5%/1.050 +2.2%/1.024 +5.0%/1.056 +csmith-v0 +21.2%/1.320 +-96.3%/0.851 +-116.0%/0.954 +21.1%/1.318 +-125.4%/0.994 +21.1%/1.320 +github-v0 +1.0%/1.011 +0.2%/1.002 +0.1%/1.001 +0.9%/1.009 +0.1%/1.002 +0.9%/1.010 +linux-v0 +0.6%/1.007 +-0.4%/0.998 +-0.5%/0.997 +0.6%/1.006 +-0.9%/0.996 +0.6%/1.007 +llvm-stress-v0 +6.3%/1.087 +-18.9%/0.885 +-67.0%/0.731 +1.6%/1.040 +-22.0%/0.872 +2.1%/1.045 +mibench-v1 +1.7%/1.020 +0.0%/1.003 +-2.8%/0.976 +-0.4%/0.999 +0.6%/1.008 +-0.1%/1.003 +npb-v0 +9.8%/1.159 +5.7%/1.085 +0.9%/1.035 +6.0%/1.088 +4.8%/1.074 +5.5%/1.085 +opencv-v0 +5.2%/1.061 +1.0%/1.013 +0.5%/1.007 +4.5%/1.054 +0.7%/1.009 +4.8%/1.057 +poj104-v1 +7.8%/1.105 +3.9%/1.055 +-17.5%/0.876 +5.7%/1.075 +0.1%/1.014 +6.3%/1.082 +tensorflow-v0 +6.1%/1.077 +-0.2%/0.998 +0.2%/1.004 +5.1%/1.063 +0.8%/1.011 +5.9%/1.075 +Average +5.8%/1.075 +-7.5%/0.992 +-16.3%/0.960 +4.2%/1.056 +-9.6%/1.005 +4.7%/1.062 +Table 4. Evaluation results on held out test set averaged over all datasets. + +Learning to compile smartly for program size reduction +Index +Flag +Index +Flag +Index +Flag +0 +-add-discriminators +42 +-globalsplit +84 +-lower-expect +1 +-adce +43 +-guard-widening +85 +-lower-guard-intrinsic +2 +-aggressive-instcombine +44 +-hotcoldsplit +86 +-lowerinvoke +3 +-alignment-from-assumptions +45 +-ipconstprop +87 +-lower-matrix-intrinsics +4 +-always-inline +46 +-ipsccp +88 +-lowerswitch +5 +-argpromotion +47 +-indvars +89 +-lower-widenable-condition +6 +-attributor +48 +-irce +90 +-memcpyopt +7 +-barrier +49 +-infer-address-spaces +91 +-mergefunc +8 +-bdce +50 +-inferattrs +92 +-mergeicmps +9 +-break-crit-edges +51 +-inject-tli-mappings +93 +-mldst-motion +10 +-simplifycfg +52 +-instsimplify +94 +-sancov +11 +-callsite-splitting +53 +-instcombine +95 +-name-anon-globals +12 +-called-value-propagation +54 +-instnamer +96 +-nary-reassociate +13 +-canonicalize-aliases +55 +-jump-threading +97 +-newgvn +14 +-consthoist +56 +-lcssa +98 +-pgo-memop-opt +15 +-constmerge +57 +-licm +99 +-partial-inliner +16 +-constprop +58 +-libcalls-shrinkwrap +100 +-partially-inline-libcalls +17 +-coro-cleanup +59 +-load-store-vectorizer +101 +-post-inline-ee-instrument +18 +-coro-early +60 +-loop-data-prefetch +102 +-functionattrs +19 +-coro-elide +61 +-loop-deletion +103 +-mem2reg +20 +-coro-split +62 +-loop-distribute +104 +-prune-eh +21 +-correlated-propagation +63 +-loop-fusion +105 +-reassociate +22 +-cross-dso-cfi +64 +-loop-guard-widening +106 +-redundant-dbg-inst-elim +23 +-deadargelim +65 +-loop-idiom +107 +-rpo-functionattrs +24 +-dce +66 +-loop-instsimplify +108 +-rewrite-statepoints-for-gc +25 +-die +67 +-loop-interchange +109 +-sccp +26 +-dse +68 +-loop-load-elim +110 +-slp-vectorizer +27 +-reg2mem +69 +-loop-predication +111 +-sroa +28 +-div-rem-pairs +70 +-loop-reroll +112 +-scalarizer +29 +-early-cse-memssa +71 +-loop-rotate +113 +-separate-const-offset-from-gep +30 +-early-cse +72 +-loop-simplifycfg +114 +-simple-loop-unswitch +31 +-elim-avail-extern +73 +-loop-simplify +115 +-sink +32 +-ee-instrument +74 +-loop-sink +116 +-speculative-execution +33 +-flattencfg +75 +-loop-reduce +117 +-slsr +34 +-float2int +76 +-loop-unroll-and-jam +118 +-strip-dead-prototypes +35 +-forceattrs +77 +-loop-unroll +119 +-strip-debug-declare +36 +-inline +78 +-loop-unswitch +120 +-strip-nondebug +37 +-insert-gcov-profiling +79 +-loop-vectorize +121 +-strip +38 +-gvn-hoist +80 +-loop-versioning-licm +122 +-tailcallelim +39 +-gvn +81 +-loop-versioning +123 +-mergereturn +40 +-globaldce +82 +-loweratomic +41 +-globalopt +83 +-lower-constant-intrinsics +Table 5. A list of LLVM compiler pass indices and their corresponding command line flag. + diff --git a/D9E4T4oBgHgl3EQffA2Y/content/tmp_files/load_file.txt b/D9E4T4oBgHgl3EQffA2Y/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cee943eb29787b902c2679bfbe876c6d79fafe81 --- /dev/null +++ b/D9E4T4oBgHgl3EQffA2Y/content/tmp_files/load_file.txt @@ -0,0 +1,957 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQffA2Y/content/2301.05104v1.pdf,len=956 +page_content='Learning to compile smartly for program size reduction Youwei Liang∗ 1 Kevin Stone∗ 2 Ali Shameli 2 Chris Cummins 2 Mostafa Elhoushi 2 Jiadong Guo 2 Benoit Steiner 2 Pengtao Xie 1 Hugh Leather 2 Yuandong Tian 2 Abstract Compiler optimization passes are an important tool for improving program efficiency and reduc- ing program size, but manually selecting opti- mization passes can be time-consuming and error- prone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQffA2Y/content/2301.05104v1.pdf'} +page_content=' While human experts have identified a few fixed sequences of optimization passes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQffA2Y/content/2301.05104v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQffA2Y/content/2301.05104v1.pdf'} +page_content=', the Clang -Oz passes) that perform well for a wide variety of programs, these sequences are not con- ditioned on specific programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQffA2Y/content/2301.05104v1.pdf'} +page_content=' In this paper, we propose a novel approach that learns a policy to select passes for program size reduction, allow- ing for customization and adaptation to specific programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQffA2Y/content/2301.05104v1.pdf'} +page_content=' Our approach uses a search mecha- nism that helps identify useful pass sequences and a GNN with customized attention that se- lects the optimal sequence to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQffA2Y/content/2301.05104v1.pdf'} +page_content=' Crucially it is able to generalize to new, unseen programs, making it more flexible and general than previ- ous approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQffA2Y/content/2301.05104v1.pdf'} +page_content=' We evaluate our approach on a range of programs and show that it leads to size reduction compared to traditional optimization techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQffA2Y/content/2301.05104v1.pdf'} +page_content=' Our results demonstrate the potential of a single policy that is able to optimize many programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQffA2Y/content/2301.05104v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQffA2Y/content/2301.05104v1.pdf'} +page_content=' Introduction Finding the right compiler optimization ordering for a given program in order to execute them more efficiently with a smaller amount of resources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQffA2Y/content/2301.05104v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQffA2Y/content/2301.05104v1.pdf'} +page_content=', memory, CPU and stor- age) is an important yet challenging problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQffA2Y/content/2301.05104v1.pdf'} +page_content=' Traditionally, to tune configurations, human effort and expert knowledge are needed, which is a time-consuming and error-prone process that often yields sub-par results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQffA2Y/content/2301.05104v1.pdf'} +page_content=' In recent years, machine learning-guided compiler optimiza- tion has emerged as an interesting field to replace this labori- ous process (Wang & O’Boyle, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQffA2Y/content/2301.05104v1.pdf'} +page_content=' Along this line, many Equal contribution 1University of California, San Diego 2Meta AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9E4T4oBgHgl3EQffA2Y/content/2301.05104v1.pdf'} +page_content=' Correspondence to: Kevin Stone 0, positive integers k, and i ∈ N \ {1}. Then, there exists a continuous function +q : (0, ∞) → [−∞, ∞) such that fn(x1, x2, . . . , xn) = q(x1x2 · · · xn) for all x1, . . . , xn > 0. +Proof. Suppose that fn fulfills assumption (1). First, we show that fn satisfies +fn(x, x2, . . . , xi−1, z/x, xi+1, . . . , xn) = fn(y, x2, . . . , xi−1, z/y, xi+1, . . . , xn) +(2) +for all i ∈ N \ {1} and x2, . . . , xi−1, xi+1, . . . , xn, x, y, z > 0. Assume without loss of generality that +i = 2; the proof for any other i ∈ {3, . . . , n} is analogous. Let x3, . . . , xn, z > 0 be fixed throughout. +Define h : (0, ∞) → [−∞, ∞) by h(x) := fn(x, z/x, x3, . . . , xn) for all x > 0. Note that h is +continuous due to the continuity of fn on (0, ∞)n. For any positive integer k and any x > 0, we have +h +�k + 1 +k +· x +� += fn +� +(k + 1) · x +k , k · +z +(k + 1)x, x3, . . . , xn +� += fn +� +k · x +k , (k + 1) · +z +(k + 1)x, x3, . . . , xn +� +(by (1)) += fn(x, z/x, x3, . . . , xn) += h(x), +so for any rational number r = a/b > 1, we have +h(rx) = h +� +a +a − 1 · a − 1 +a − 2 · · · · · b + 1 +b +· x +� += h +�a − 1 +a − 2 · · · · · b + 1 +b +· x +� += · · · = h(x). +Similarly, we have h(rx) = h(x) for any rational number 0 < r < 1, hence the same equation is true +for all positive rational numbers r. Since h is continuous and the positive rational numbers are dense in +(0, ∞), we can conclude that h is constant, and thus, fn(x, z/x, x3, . . . , xn) = fn(y, z/y, x3, . . . , xn) +for all x, y > 0. Hence, (2) is true for all i ∈ N \ {1} and x2, . . . , xi−1, xi+1, . . . , xn, x, y, z > 0. +Next, we prove by backward induction that for all integers 1 ≤ k ≤ n, there exists a continuous +function qk : (0, ∞)k → [−∞, ∞) such that fn(x1, . . . , xn) = qk(x1xk+1 · · · xn, x2, . . . , xk) for all +x1, . . . , xn > 0. Then, q := q1 gives the desired conclusion. +For the base case k = n, we have qn := fn|(0,∞)n. For the inductive step, let 2 ≤ k ≤ n be +given, and assume that such a function qk exists; we shall prove that qk−1 exists as well. Define qk−1 +by qk−1(y1, . . . , yk−1) := qk(y1, . . . , yk−1, 1) for all y1, . . . , yk−1 > 0. Note that qk−1 is continuous +on (0, ∞)k−1 due to the continuity of qk on (0, ∞)k. Let x1, . . . , xn > 0 be given. Then, by setting +x := x1 and y := z := x1xk, we have +fn(x1, . . . , xn) = fn(x, x2, . . . , xk−1, z/x, xk+1, . . . , xn) +2 + += fn(y, x2, . . . , xk−1, z/y, xk+1, . . . , xn) +(by (2)) += fn(x1xk, x2, . . . , xk−1, 1, xk+1, . . . , xn) += qk(x1xkxk+1 · · · xn, x2, . . . , xk−1, 1) +(by the inductive hypothesis) += qk−1(x1xk · · · xn, x2, . . . , xk−1), +establishing the inductive step and therefore the lemma. +We now state our characterization. Recall from Section 2 that a welfare function is assumed to be +non-decreasing on [0, ∞)n. +Theorem 2. Fix n ≥ 2. Let fn be a welfare function that is continuous2 and strictly increasing3 on +(0, ∞)n. Then, the following three statements are equivalent: +(a) For every profile that admits an allocation where every agent receives positive utility, every allo- +cation that can be chosen by the welfarist rule with function fn is EF1. +(b) For every profile that admits an allocation where every agent receives positive utility, there exists +an EF1 allocation that can be chosen by the welfarist rule with function fn. +(c) The following two statements hold for fn: +(i) There exists a strictly increasing and continuous function q : (0, ∞) → (−∞, ∞) such that +fn(x1, x2, . . . , xn) = q(x1x2 · · · xn) for all x1, . . . , xn > 0. +(ii) The inequality fn(x1, x2, . . . , xn) > fn(y1, y2, . . . , yn) holds for all x1, . . . , xn > 0 and +y1, . . . , yn ≥ 0 satisfying �n +i=1 yi = 0. +Note that if fn satisfies (c), then given any profile that admits an allocation where every agent +receives positive utility, an allocation can be chosen by the welfarist rule with function fn if and only +if it can be chosen by MNW, so the two rules are effectively equivalent.4 Hence, Theorem 2 provides a +characterization of MNW among all welfarist rules. +Proof of Theorem 2. The implication (a) ⇒ (b) is trivial. For the implication (c) ⇒ (a), if fn satisfies (c), +then given a profile that admits an allocation where every agent receives positive utility, every allocation +that can be chosen by the welfarist rule with function fn is also an allocation that can be chosen by MNW, +which is known to be EF1 [Caragiannis et al., 2019]; hence, fn also satisfies (a). It therefore remains +to prove the implication (b) ⇒ (c). Assume that fn satisfies (b); we will show that both statements (i) +and (ii) of (c) hold. +To prove (i), it suffices to show that fn satisfies (1) for all x1, . . . , xn > 0, positive integers k, and +i ∈ N \ {1}. Indeed, once this is shown, Lemma 1 provides a continuous function q : (0, ∞) → +[−∞, ∞) satisfying fn(x1, x2, . . . , xn) = q(x1x2 · · · xn) for all x1, . . . , xn > 0. Note that q must be +strictly increasing because fn is strictly increasing on (0, ∞)n, and −∞ cannot be in the range of q since +q is strictly increasing and its domain is an open set in R. +To show (1), suppose on the contrary that (1) is false for some x1, . . . , xn > 0, positive integer k, +and i ∈ N \ {1}; assume without loss of generality that i = 2, which means that +fn((k + 1)x1, kx2, x3, . . . , xn) ̸= fn(kx1, (k + 1)x2, x3, . . . , xn). +2In the prior characterization of additive welfarist rules, Suksompong [2023] made the stronger assumption that the welfare +function is differentiable. Here, we only assume that the function is continuous. +3The theorem does not hold without the assumption that fn is strictly increasing on (0, ∞)n: for example, if fn is a +constant function, then statement (b) holds but (a) does not. +4For profiles that do not admit an allocation where every agent receives positive utility, MNW requires an additional tie- +breaking specification in order to ensure EF1 [Caragiannis et al., 2019]. +3 + +Suppose that +fn((k + 1)x1, kx2, x3, . . . , xn) < fn(kx1, (k + 1)x2, x3, . . . , xn); +the case where the inequality goes in the opposite direction can be handled similarly. By the continuity +of fn, there exists ǫ ∈ (0, x1) such that +fn((k + 1)x1 − ǫ, kx2, x3 . . . , xn) < fn(kx1 − ǫ, (k + 1)x2, x3, . . . , xn). +(3) +Consider a profile with m = kn + 1 goods, where G′ := {g1, . . . , gkn} = G \ {gm}, such that +• for each g ∈ G′, we have uj(g) = xj for j ∈ {1, 2} and uj(g) = xj/k for j ∈ N \ {1, 2}; +• u1(gm) = x1 − ǫ, and uj(gm) = 0 for j ∈ N \ {1}. +Clearly, this profile admits an allocation where every agent receives positive utility. Let A be an EF1 +allocation chosen by the welfarist rule with function fn on this profile. Regardless of whom gm is +allocated to, each agent receives at most k goods from G′ in A—otherwise, if some agent j receives more +than k goods from G′, then some other agent receives fewer than k goods from G′ by the pigeonhole +principle and therefore envies j by more than one good, meaning that A is not EF1. Since |G′| = kn, +every agent receives exactly k goods from G′. Furthermore, gm must be allocated to agent 1; otherwise, +the allocation where gm is allocated to agent 1 (and all other goods are allocated as in A) has a higher +welfare than A, contradicting the fact that A is chosen by the welfarist rule with function fn. The welfare +of A must not be smaller than that of another allocation where agent 1 receives gm along with k − 1 +goods from G′, agent 2 receives k + 1 goods from G′, and every other agent receives k goods from G′ +each. This means that +fn((k + 1)x1 − ǫ, kx2, x3 . . . , xn) ≥ fn(kx1 − ǫ, (k + 1)x2, x3, . . . , xn), +contradicting (3). This establishes (i). +It remains to prove (ii). Consider any x1, . . . , xn > 0 and y1, . . . , yn ≥ 0 satisfying �n +i=1 yi = 0. +Let X := �n +i=1 xi > 0. Without loss of generality, assume that y1 = · · · = yk = 0 and Y := +�n +i=k+1 yi > 0 for some k ∈ {1, . . . , n} (if k = n, the empty product �n +i=k+1 yi is taken to be 1). +Define z1, . . . , zn by zi := (X/2Y )1/k for all i ∈ {1, . . . , k} and zi := yi for all i ∈ {k + 1, . . . , n}. +Then, +fn(y1, . . . , yn) ≤ fn(z1, . . . , zn) +(since fn is non-decreasing) += q(z1 · · · zk · zk+1 · · · zn) +(by (i) and since all zi’s are positive) += q((X/2Y ) · yk+1 · · · yn) += q(X/2) +< q(X) +(since q is strictly increasing) += q(x1 · · · xn) += fn(x1, . . . , xn), +(by (i) and since all xi’s are positive) +completing the proof of the theorem. +Acknowledgments +This work was partially supported by the Singapore Ministry of Education under grant number MOE- +T2EP20221-0001 and by an NUS Start-up Grant. +4 + +References +Ioannis Caragiannis, David Kurokawa, Herv´e Moulin, Ariel D. Procaccia, Nisarg Shah, and Junxing +Wang. The unreasonable fairness of maximum Nash welfare. ACM Transactions on Economics and +Computation, 7(3):12:1–12:32, 2019. +Herv´e Moulin. Fair division in the internet age. Annual Review of Economics, 11:407–441, 2019. +Warut Suksompong. A characterization of maximum Nash welfare for indivisible goods. Economics +Letters, 222:110956, 2023. +5 + diff --git a/FdE2T4oBgHgl3EQfTAdd/content/tmp_files/load_file.txt b/FdE2T4oBgHgl3EQfTAdd/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7edb31ef53b6281fb3b4ab52a834031ab8a9280b --- /dev/null +++ b/FdE2T4oBgHgl3EQfTAdd/content/tmp_files/load_file.txt @@ -0,0 +1,331 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf,len=330 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content='03798v1 [econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content='TH] 10 Jan 2023 Extending the Characterization of Maximum Nash Welfare Sheung Man Yuen and Warut Suksompong National University of Singapore Abstract In the allocation of indivisible goods, the maximum Nash welfare rule has recently been charac- terized as the only rule within the class of additive welfarist rules that satisfies envy-freeness up to one good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' We extend this characterization to the class of all welfarist rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' 1 Introduction The fair allocation of indivisible goods—be it artwork, furniture, school supplies, or electronic devices— is a ubiquitous problem in society and has attracted significant interest in economics [Moulin, 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Among the plethora of methods that one may use to allocate indivisible goods fairly, the method that has arguably received the most attention in recent years is the maximum Nash welfare (MNW) rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' For instance, MNW is used to allocate goods on the popular fair division website Spliddit,1 which has served hundreds of thousands of users since its launch in 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' MNW selects from each profile an allocation that maximizes the product of the agents’ utilities, or equivalently, the sum of their logarithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' In an influential work, Caragiannis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' [2019] showed that every allocation output by MNW satisfies envy-freeness up to one good (EF1): given any two agents, if the first agent envies the second agent, then this envy can be eliminated by removing some good in the second agent’s bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Recently, Suksompong [2023] provided the first characterization of MNW by showing that it is the unique additive welfarist rule that guarantees EF1—an additive welfarist rule selects an allocation maximizing a welfare notion that can be expressed as the sum of some function of the agents’ utilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Suksompong’s characterization raises an obvious question: Is MNW also the unique (not necessarily additive) welfarist rule that guarantees EF1, where a welfarist rule selects an allocation maximizing a welfare notion that can be expressed as some function of the agents’ utilities?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' In this note, we answer the above question in the affirmative, by extending the characterization of Suksompong [2023] to the class of all welfarist rules (whether additive or not).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' This further solidifies the “unreasonable fairness” of MNW established by Caragiannis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' [2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' 2 Preliminaries Let N = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , n} be the set of agents, and G = {g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , gm} be the set of goods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Each agent i ∈ N has a utility function ui : 2G → R≥0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' for simplicity, we write ui(g) instead of ui({g}) for a single good g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' We assume that the utility functions are additive, that is, ui(G′) = � g∈G′ ui(g) for all i ∈ N and G′ ⊆ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' A profile consists of N, G and (ui)i∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' An allocation A = (A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , An) is an ordered partition of G into n bundles such that bundle Ai is allocated to agent i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' An agent i ∈ N receives utility ui(Ai) from allocation A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' An allocation A is EF1 if for every pair i, j ∈ N such that Aj ̸= ∅, there exists a good g ∈ Aj with the property that ui(Ai) ≥ ui(Aj \\ {g}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' A rule maps any given profile to an allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' 1http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content='spliddit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content='org 1 Given n ≥ 2, a welfare function is a non-decreasing function fn : [0, ∞)n → [−∞, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' The welfarist rule with (welfare) function fn chooses from each profile an allocation A that maximizes the welfare fn(u1(A1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , un(An));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' if there are multiple such allocations, the rule may choose one arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' 3 The Result Before proceeding to our characterization, we first establish a technical lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Fix n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Let fn : [0, ∞)n → [−∞, ∞) be a function that is continuous on (0, ∞)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Suppose that fn((k + 1)x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xi−1, kxi, xi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn) = fn(kx1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xi−1, (k + 1)xi, xi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn) (1) for all x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn > 0, positive integers k, and i ∈ N \\ {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Then, there exists a continuous function q : (0, ∞) → [−∞, ∞) such that fn(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn) = q(x1x2 · · · xn) for all x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Suppose that fn fulfills assumption (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' First, we show that fn satisfies fn(x, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xi−1, z/x, xi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn) = fn(y, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xi−1, z/y, xi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn) (2) for all i ∈ N \\ {1} and x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xi−1, xi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn, x, y, z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Assume without loss of generality that i = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' the proof for any other i ∈ {3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , n} is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Let x3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn, z > 0 be fixed throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Define h : (0, ∞) → [−∞, ∞) by h(x) := fn(x, z/x, x3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn) for all x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Note that h is continuous due to the continuity of fn on (0, ∞)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' For any positive integer k and any x > 0, we have h �k + 1 k x � = fn � (k + 1) · x k , k · z (k + 1)x, x3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn � = fn � k · x k , (k + 1) · z (k + 1)x, x3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn � (by (1)) = fn(x, z/x, x3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn) = h(x), so for any rational number r = a/b > 1, we have h(rx) = h � a a − 1 · a − 1 a − 2 · · · · · b + 1 b x � = h �a − 1 a − 2 · · · · · b + 1 b x � = · · · = h(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Similarly, we have h(rx) = h(x) for any rational number 0 < r < 1, hence the same equation is true for all positive rational numbers r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Since h is continuous and the positive rational numbers are dense in (0, ∞), we can conclude that h is constant, and thus, fn(x, z/x, x3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn) = fn(y, z/y, x3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn) for all x, y > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Hence, (2) is true for all i ∈ N \\ {1} and x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xi−1, xi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn, x, y, z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Next, we prove by backward induction that for all integers 1 ≤ k ≤ n, there exists a continuous function qk : (0, ∞)k → [−∞, ∞) such that fn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn) = qk(x1xk+1 · · · xn, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xk) for all x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Then, q := q1 gives the desired conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' For the base case k = n, we have qn := fn|(0,∞)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' For the inductive step, let 2 ≤ k ≤ n be given, and assume that such a function qk exists;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' we shall prove that qk−1 exists as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Define qk−1 by qk−1(y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , yk−1) := qk(y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , yk−1, 1) for all y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , yk−1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Note that qk−1 is continuous on (0, ∞)k−1 due to the continuity of qk on (0, ∞)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Let x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn > 0 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Then, by setting x := x1 and y := z := x1xk, we have fn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn) = fn(x, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xk−1, z/x, xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn) 2 = fn(y, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xk−1, z/y, xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn) (by (2)) = fn(x1xk, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xk−1, 1, xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn) = qk(x1xkxk+1 · · · xn, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xk−1, 1) (by the inductive hypothesis) = qk−1(x1xk · · · xn, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xk−1), establishing the inductive step and therefore the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' We now state our characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Recall from Section 2 that a welfare function is assumed to be non-decreasing on [0, ∞)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Fix n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Let fn be a welfare function that is continuous2 and strictly increasing3 on (0, ∞)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Then, the following three statements are equivalent: (a) For every profile that admits an allocation where every agent receives positive utility, every allo- cation that can be chosen by the welfarist rule with function fn is EF1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' (b) For every profile that admits an allocation where every agent receives positive utility, there exists an EF1 allocation that can be chosen by the welfarist rule with function fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' (c) The following two statements hold for fn: (i) There exists a strictly increasing and continuous function q : (0, ∞) → (−∞, ∞) such that fn(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn) = q(x1x2 · · · xn) for all x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' (ii) The inequality fn(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn) > fn(y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , yn) holds for all x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn > 0 and y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , yn ≥ 0 satisfying �n i=1 yi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Note that if fn satisfies (c), then given any profile that admits an allocation where every agent receives positive utility, an allocation can be chosen by the welfarist rule with function fn if and only if it can be chosen by MNW, so the two rules are effectively equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content='4 Hence, Theorem 2 provides a characterization of MNW among all welfarist rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' The implication (a) ⇒ (b) is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' For the implication (c) ⇒ (a), if fn satisfies (c), then given a profile that admits an allocation where every agent receives positive utility, every allocation that can be chosen by the welfarist rule with function fn is also an allocation that can be chosen by MNW, which is known to be EF1 [Caragiannis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=', 2019];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' hence, fn also satisfies (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' It therefore remains to prove the implication (b) ⇒ (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Assume that fn satisfies (b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' we will show that both statements (i) and (ii) of (c) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' To prove (i), it suffices to show that fn satisfies (1) for all x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn > 0, positive integers k, and i ∈ N \\ {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Indeed, once this is shown, Lemma 1 provides a continuous function q : (0, ∞) → [−∞, ∞) satisfying fn(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn) = q(x1x2 · · · xn) for all x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Note that q must be strictly increasing because fn is strictly increasing on (0, ∞)n, and −∞ cannot be in the range of q since q is strictly increasing and its domain is an open set in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' To show (1), suppose on the contrary that (1) is false for some x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn > 0, positive integer k, and i ∈ N \\ {1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' assume without loss of generality that i = 2, which means that fn((k + 1)x1, kx2, x3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn) ̸= fn(kx1, (k + 1)x2, x3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' 2In the prior characterization of additive welfarist rules, Suksompong [2023] made the stronger assumption that the welfare function is differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Here, we only assume that the function is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' 3The theorem does not hold without the assumption that fn is strictly increasing on (0, ∞)n: for example, if fn is a constant function, then statement (b) holds but (a) does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' 4For profiles that do not admit an allocation where every agent receives positive utility, MNW requires an additional tie- breaking specification in order to ensure EF1 [Caragiannis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' 3 Suppose that fn((k + 1)x1, kx2, x3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn) < fn(kx1, (k + 1)x2, x3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' the case where the inequality goes in the opposite direction can be handled similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' By the continuity of fn, there exists ǫ ∈ (0, x1) such that fn((k + 1)x1 − ǫ, kx2, x3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn) < fn(kx1 − ǫ, (k + 1)x2, x3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' (3) Consider a profile with m = kn + 1 goods, where G′ := {g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , gkn} = G \\ {gm}, such that for each g ∈ G′, we have uj(g) = xj for j ∈ {1, 2} and uj(g) = xj/k for j ∈ N \\ {1, 2};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' u1(gm) = x1 − ǫ, and uj(gm) = 0 for j ∈ N \\ {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Clearly, this profile admits an allocation where every agent receives positive utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Let A be an EF1 allocation chosen by the welfarist rule with function fn on this profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Regardless of whom gm is allocated to, each agent receives at most k goods from G′ in A—otherwise, if some agent j receives more than k goods from G′, then some other agent receives fewer than k goods from G′ by the pigeonhole principle and therefore envies j by more than one good, meaning that A is not EF1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Since |G′| = kn, every agent receives exactly k goods from G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Furthermore, gm must be allocated to agent 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' otherwise, the allocation where gm is allocated to agent 1 (and all other goods are allocated as in A) has a higher welfare than A, contradicting the fact that A is chosen by the welfarist rule with function fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' The welfare of A must not be smaller than that of another allocation where agent 1 receives gm along with k − 1 goods from G′, agent 2 receives k + 1 goods from G′, and every other agent receives k goods from G′ each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' This means that fn((k + 1)x1 − ǫ, kx2, x3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn) ≥ fn(kx1 − ǫ, (k + 1)x2, x3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn), contradicting (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' This establishes (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' It remains to prove (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Consider any x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn > 0 and y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , yn ≥ 0 satisfying �n i=1 yi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Let X := �n i=1 xi > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Without loss of generality, assume that y1 = · · · = yk = 0 and Y := �n i=k+1 yi > 0 for some k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , n} (if k = n, the empty product �n i=k+1 yi is taken to be 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Define z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , zn by zi := (X/2Y )1/k for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , k} and zi := yi for all i ∈ {k + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Then, fn(y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , yn) ≤ fn(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , zn) (since fn is non-decreasing) = q(z1 · · · zk · zk+1 · · · zn) (by (i) and since all zi’s are positive) = q((X/2Y ) · yk+1 · · · yn) = q(X/2) < q(X) (since q is strictly increasing) = q(x1 · · · xn) = fn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' , xn), (by (i) and since all xi’s are positive) completing the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Acknowledgments This work was partially supported by the Singapore Ministry of Education under grant number MOE- T2EP20221-0001 and by an NUS Start-up Grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' 4 References Ioannis Caragiannis, David Kurokawa, Herv´e Moulin, Ariel D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Procaccia, Nisarg Shah, and Junxing Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' The unreasonable fairness of maximum Nash welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' ACM Transactions on Economics and Computation, 7(3):12:1–12:32, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Herv´e Moulin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Fair division in the internet age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Annual Review of Economics, 11:407–441, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Warut Suksompong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' A characterization of maximum Nash welfare for indivisible goods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' Economics Letters, 222:110956, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} +page_content=' 5' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE2T4oBgHgl3EQfTAdd/content/2301.03798v1.pdf'} diff --git a/HtE3T4oBgHgl3EQfuQsq/content/2301.04682v1.pdf 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Learning On Heterogeneous Feature Spaces +Alain Rakotomamonjy * 1 Maxime Vono * 1 Hamlet Jesse Medina Ruiz 1 Liva Ralaivola 1 +Abstract +Most personalised federated learning (FL) ap- +proaches assume that raw data of all clients are +defined in a common subspace i.e. +all clients +store their data according to the same schema. +For real-world applications, this assumption is +restrictive as clients, having their own systems +to collect and then store data, may use heteroge- +neous data representations. We aim at filling this +gap. To this end, we propose a general frame- +work coined FLIC that maps client’s data onto a +common feature space via local embedding func- +tions. The common feature space is learnt in a +federated manner using Wasserstein barycenters +while the local embedding functions are trained +on each client via distribution alignment. We in- +tegrate this distribution alignement mechanism +into a federated learning approach and provide +the algorithmics of FLIC. We compare its per- +fomances against FL benchmarks involving het- +erogeneous input features spaces. In addition, we +provide theoretical insights supporting the rele- +vance of our methodology. +1. Introduction +Federated learning (FL) is a machine learning paradigm +where models are trained from multiple isolated data sets +owned by individual agents (coined clients), without re- +quiring to move raw data into a central server, nor even +share them in any way (Kairouz et al., 2021). +This +framework has lately gained a strong traction from both +industry and academic research. +Indeed, it avoids the +communication costs entailed by data transfer, allows all +clients to benefit from participating to the learning co- +hort, and finally, it fulfills first-order confidentiality guar- +antees, which can be further enhanced by resorting to so- +called privacy-enhancing technologies such as differential +privacy (Dwork and Roth, 2014) or secure multi-party com- +*Equal contribution 1Criteo AI Lab, Paris, France. Correspon- +dence to: Maxime Vono , Alain Rako- +tomamonjy . +putation (Bonawitz et al., 2017). As core properties, FL +ensures data ownership, and structurally incorporates the +principle of data exchange minimisation by only transmit- +ting the required updates of the models being learnt. De- +pending on the data partitioning and target applications, nu- +merous FL approaches have been proposed, such as hori- +zontal FL (McMahan et al., 2017) and vertical FL (Hardy +et al., 2017; Yang et al., 2019). The latter paradigm consid- +ers that clients hold disjoint subsets of features correspond- +ing to the same users while the former assumes that clients +have data samples from different users. Recently, horizon- +tal FL works have focused on personalised FL to tackle +statistical heterogeneity by using local models to fit client- +specific data (Hanzely and Richt´arik, 2020; Jiang et al., +2019; Khodak et al., 2019; Tan et al., 2022). +Existing horizontal personalised FL works assume that the +raw data on all clients share the same structure and are de- +fined in a common feature space. Yet, in practice, data col- +lected by clients may use differing structures. For instance, +clients may not collect exactly the same information, some +features may be missing or not stored, or some might have +been transformed (e.g. via normalisation, scaling, or linear +combinations). To address the key issue of implementing +FL when the clients’ feature spaces are heterogeneous, in +the sense that they have different dimensionalites or that the +semantics of given vector coordinates are different, we in- +troduce the first — to the best of our knowledge — person- +alised FL framework dedicated to this learning situation. +Proposed Approach. The framework and algorithm de- +scribed in this paper rest on the idea that before performing +efficient FL training, a key step is to map the raw data into a +common subspace. This is a prior necessary step before FL +since it allows to define a relevant aggregation scheme on +the central server for model parameters (e.g. via weighted +averaging) as the latter become comparable. Thus, we map +clients’ raw data into a common low-dimensional latent +space, via local and learnable feature embedding functions. +In order to ease subsequent learning steps, data related to +the same semantic information (e.g. label) have to be em- +bedded in the same region of the latent space. To ensure +this property, we align clients’ embedded feature distribu- +tions via a latent anchor distribution that is shared across +clients. +The learning of this anchor distribution is per- +arXiv:2301.11447v1 [cs.LG] 26 Jan 2023 + +Personalised Federated Learning On Heterogeneous Feature Spaces +2 +formed in a federated manner i.e. by updating it locally on +each client before aggregation on the central server. More +precisely, each client updates her local version of the an- +chor distribution by aligning it, i.e. making it closer, to +the embedded feature distribution. Then, the central server +aims at finding the mean element, i.e. barycenter, of these +local anchor distributions (Banerjee et al., 2005; Veldhuis, +2002). Once this distribution alignment mechanism (based +on local embedding functions and anchor distribution) is +defined, it can be seamlessly integrated into a personalised +FL framework; the personalisation part aiming at tackling +residual statistical heterogeneity. In this paper, without loss +of generality, we have embedded this alignment framework +into a personalised FL approach similar to the one proposed +in Collins et al. (2021). +Related Ideas. Ideas that we have built on for solving the +task of FL from heterogeneous feature spaces have been +partially explored in related literature. From the theoretical +standpoint, works on the Gromov-Wasserstein distance or +variants seek at comparing distributions from incomparable +spaces in a (non-FL) centralised manner (Alaya et al., 2022; +Bunne et al., 2019; M´emoli, 2011). Other methodological +works on autoencoders (Xu et al., 2020), word embeddings +(Alvarez-Melis and Jaakkola, 2018; Alvarez-Melis et al., +2019) or FL under high statistical heterogeneity (Luo et al., +2021; Makhija et al., 2022; Zhou et al., 2022) use simi- +lar ideas of distribution alignment for calibrating feature +extractors and classifiers. A detailed literature review and +comparison with the proposed methodology is postponed +to Section 2. +Contributions. In order to help the reader better grasp the +differences of our approach with respect to the existing lit- +erature, we spell out our contributions: +1. We are the first to formalise the problem of person- +alised horizontal FL on heterogeneous clients’ feature +spaces. In contrast to existing approaches, the pro- +posed general framework, coined FLIC, allows each +client to leverage other clients’ data even though they +do not have the same raw representation. +2. We introduce a distribution alignment framework and +an algorithm that learns the feature embedding func- +tions along with the latent anchor distribution in a lo- +cal and global federated manner, respectively. We also +show how those essential algorithmic pieces are inte- +grated into a personalised FL algorithm, easing adop- +tion by practitioners. +3. We provide algorithmic and theoretical support to the +proposed methodology. In particular, we show that for +an insightful simpler learning scenario, FLIC is able +to recover the true latent subspace underlying the FL +problem. +4. Experimental analyses on toy data sets and real-world +problems illustrate the accuracy of our theory and +show that FLIC provides better performance than +competing FL approaches. +Conventions and Notations. The Euclidean norm on Rd +is ∥ · ∥, we use |S| to denote the cardinality of the set S +and N∗ = N \ {0}. For n ∈ N∗, we refer to {1, . . . , n} +with the notation [n]. We denote by N(m, Σ) the Gaussian +distribution with mean vector m and covariance matrix Σ +and use the notation X ∼ ν to denote that the random vari- +able X has been drawn from the probability distribution +ν. We define the Wasserstein distance of order 2 for any +probability measures µ, ν on Rd with finite 2-moment by +W2(µ, ν) = (infζ∈T (µ,ν) +� +Rd×Rd ∥θ − θ′∥2dζ(θ, θ′))1/2, +where T (µ, ν) is the set of transference plans of µ and ν. +2. Proposed Methodology +Problem Formulation. We are considering a centralised +and horizontal FL framework involving b ∈ N∗ clients +and a central entity (Kairouz et al., 2021; Yang et al., +2019). Under this paradigm, the central entity orchestrates +the collaborative solving of a common machine learning +problem by the b clients; without requiring raw data ex- +changes. For the sake of simplicity, we consider the setting +where all clients want to solve a multi-class classification +task with C ∈ N∗ classes. In Appendix, we also high- +light how regression tasks could be encompassed in the +proposed framework. The b clients are assumed to pos- +sess local data sets {Di}i∈[b] such that, for any i ∈ [b], +Di = {(x(j) +i , y(j) +i )}j∈[ni] where x(j) +i +stands for a feature +vector, y(j) +i +is a label and ni = |Di|. +A core assump- +tion of FL is that the local data sets {Di}i∈[b] are statis- +tically heterogeneous i.e. for any i ∈ [b] and j ∈ [ni], +(x(j) +i , y(j) +i ) +i.i.d. +∼ νi where νi is a local probability measure +defined on an appropriate measurable space. Existing hor- +izontal FL approaches typically assume that the raw input +features of the clients are defined on a common subspace +so that their marginal distributions admit the same support. +In contrast, we suppose here that these features live in +heterogeneous spaces. +Our main goal is to cope with +this new type of heterogeneity in horizontal FL. More pre- +cisely, for any i ∈ [b] and j ∈ [ni], we assume that +x(j) +i +∈ Xi ⊆ Rki such that {Xi}i∈[b] are not part of a +common ground metric. This setting is challenging since +standard FL approaches (Li et al., 2020; McMahan et al., +2017) and even personalised FL ones (Collins et al., 2021; +Hanzely et al., 2021) cannot be applied directly. In addi- +tion, we also assume a specific type of prior probability +shift where, for any i ∈ [b] and j ∈ [ni], y(j) +i +∈ Yi ⊆ [C]. +For instance, a client might only see digits 1 and 7 from the + +Personalised Federated Learning On Heterogeneous Feature Spaces +3 +AB7HicdVDLSgNBEOz1GeMr6tHLYBA8LbNroskt6MVjBPOAZAmzk9lkyOzsMjMrhJBv8OJBE +a9+kDf/xslDUNGChqKqm+6uMBVcG4w/nJXVtfWNzdxWfntnd2+/cHDY1EmKGvQRCSqHRLNBJesYbgRrJ0 +qRuJQsFY4up75rXumNE/knRmnLIjJQPKIU2Ks1OjKrOf1CkXsVqu4VCoj7Jax7/sVS/C5X6l6yHPxHEVYot +4rvHf7Cc1iJg0VROuOh1MTIgynAo2zXczVJCR2TAOpZKEjMdTObHTtGpVfoSpQtadBc/T4xIbHW4zi0n +TExQ/3bm4l/eZ3MRJVgwmWaGSbpYlGUCWQSNPsc9bli1IixJYQqbm9FdEgUocbmk7chfH2K/idN3/Uu3NJt +qVi7WsaRg2M4gTPw4BJqcAN1aAFDg/wBM+OdB6dF+d10briLGeO4Aect0/ytI7N⌫1 +AB7HicdVDLSgNBEOyNrxhfUY9eBoPgKeyEkMct6MVjBPOAZAmzk9lkyOzsMjMrhCXf4MWDIl79I +G/+jbNJBUtaCiqunu8mPBtXHdDye3sbm1vZPfLeztHxweFY9PujpKFGUdGolI9X2imeCSdQw3gvVjxUjoC9 +bzZ9eZ37tnSvNI3pl5zLyQTCQPOCXGSp2hTEaVUbHkl3XxRijOB6zbWk2WxUcAPhzLIowRrtUfF9OI5oEjJp 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+feICRxQIUmHCvVdVCk+wmWmhFO01wvVjTCZIJHtGuowAFV/WR+ewqPjTKAw1CaEhrO1e8TCQ6Umga+6QywHqvf3kz8y+vGeljtJ0xEsaCLBYNYw51CGdBwAGTl +Gg+NQTycytkIyxESbuHImhK9P4f+k5drOmV2+LhfrF8s4suAQHIEScMA5qIMr0ABNQMA9eABP4NlKrUfrxXpdtGas5cwB+AHr7RPJTJRM +⌫(1) +�1 +Client 1 with +28 x 28 images +Client 2 with +16 x 16 images +Client 3 with +32 x 32 images +Figure 1. Illustration of part of the proposed methodology for b = 3 clients with heterogeneous digit images coming from three different +data sets namely MNIST (Deng, 2012), USPS (Hull, 1994) and SVHN (Netzer et al., 2011). The circles with digits inside stand for a +group of samples, of a given class, owned by a client and the size of the circles indicates their probability mass. In the subspace Φ, +{µi}i∈[b] (and their level sets) refer to some learnable reference measures to which we seek to align the transformed version νφi of νi. +Personalised FL then occurs in the space Φ and aims at learning local models {θi}i∈[b] for each clients as well as {φi, µi}i∈[b]. +MNIST data set while another one only has access to USPS +digits 1, 2 and 9, see Figure 1. +Methodology. To address the feature space heterogene- +ity issue, we propose to map clients’ features into a fixed- +dimension common subspace Φ ⊆ Rk by resorting to local +embedding functions {φi : Xi → Φ}i∈[b]1. Our proposal +for learning those local functions is illustrated in Figure 1. +In order to preserve some semantical information (such as +the class associated to a feature vector) on the original data +distribution, we seek at learning the functions {φi}i∈[b] +such that they are aligned with (i.e. close to) some learnable +latent anchor distribution that is shared across all clients. +This anchor distribution must be seen an universal “calibra- +tor” for clients that avoids similar semantical information +from different client being scattered across the subspace +Φ, impeding then a proper subsequent federated learning +procedure of the classification model. As depicted in Fig- +ure 1, we propose to learn the feature embedding functions +by aligning their probability distributions conditioned on +the class c ∈ [C], denoted by ν(c) +φi , via C learnable anchor +measures {µc}c∈[C] (Kollias et al., 2021; Tschannen et al., +2020; Xu et al., 2020; Zhou et al., 2021). In the literature, +several approaches have been considered to align probabil- +ity distributions ranging from mutual information maximi- +sation (Tschannen et al., 2020), maximum mean discrep- +ancy (Zellinger et al., 2017) to the usage of other probabil- +ity distances such as Wasserstein or Kullback-Leibler ones +(Shen et al., 2018). +Once data from the heterogeneous spaces are embedded +in the same latent subspace Φ, we can deploy a federated +1Note that we could also have considered push-forward opera- +tors acting on the marginals associated to the clients’ features, see +Peyr´e and Cuturi (2019, Remark 2.5). +learning methodology for training from this novel repre- +sentation space. At this step, we need to choose which of +standard FL approaches, e.g. FedAvg (McMahan et al., +2017), or personalised one are more appropriate. Since the +proposed distribution alignment training procedure via the +use of an anchor distribution might not be perfect, some +statistical heterogeneity may still appear in the common la- +tent subspace Φ. Therefore, we aim at solving a person- +alised FL problem where each client has a local model tai- +lored to her specific data distribution in Φ (Tan et al., 2022). +By considering an empirical risk minimisation formulation, +the resulting data-fitting term we want to minimise writes +f(θ1:b, φ1:b) = +b +� +i=1 +ωifi(θi, φi) , +(1) +where φi is the aforementioned local embedding function, +θi ∈ Rdi stands for a local model parameter and {ωi}i∈[b] +are non-negative weights associated to each client such that +�b +i=1 ωi = 1; and for any i ∈ [b], +fi(θi, φi) = 1 +ni +ni +� +j=1 +ℓ +� +y(j) +i , g(i) +θi +� +φi +� +x(j) +i +��� +. +(2) +In the local objective function defined in (2), ℓ(·, ·) stands +for a classification loss function between the true label y(j) +i +and the predicted one g(i) +θi [φi(x(j) +i )] where g(i) +θi is the lo- +cal model that admits a personalised architecture parame- +terised by θi and taking as input an embedded feature vec- +tor φi(x(j) +i ) ∈ Φ. +Objective Function. At this stage, we are able to inte- +grate the FL paradigm and the local embedding function +learning into a global objective function we want to op- +timise, see (1). Remember that we want to learn the pa- +rameters {θi}i∈[b] of personalised FL models, in conjuction + +29Personalised Federated Learning On Heterogeneous Feature Spaces +4 +with some local embedding functions {φi}i∈[b] and shared +anchor distributions {µc}. In particular, the latter have to +be aligned with class-conditional distributions {ν(c) +φi }. We +propose to perform this alignement via a Wasserstein reg- +ularisation term leading to consider a regularised version +of the empirical risk minimisation problem defined in (1), +namely +θ⋆ +1:b, φ⋆ +1:b, µ⋆ +1:C = +arg min +θ1:b,φ1:b,µ1:C +b +� +i=1 +Fi(θi, φi, µ1:C) , +where for any i ∈ [b], +Fi(θi, φi, µ1:C) = ωifi(θi, φi) + λ1ωi +� +c∈Yi +W2 +2 +� +µc, ν(c) +φi +� ++ λ2ωi +� +c∈Yi +1 +J +J +� +j=1 +ℓ +� +c, g(i) +θi +� +Z(j) +c +�� +, +(3) +where {Z(j) +c ; j ∈ [J]}c∈[C] stand for samples drawn from +{µc}c∈[C], and λ1, λ2 > 0 are regularisation parameters. +The second term in (3) aims at aligning the conditional +probability measures of the transformed features. The third +one is an optional term aspiring to calibrate the reference +measures with the classifier in cases where two or more +classes are still ambiguous after mapping onto the common +feature space; it has also some benefits to tackle covariate +shift in standard FL (Luo et al., 2021). +Design Choices and Justifications. In the sequel, we con- +sider the Gaussian anchor measures µc = N(vc, Σc) where +vc ∈ Rk and c ∈ [C]. +Note that, under this choice, +the samples {Z(j) +c ; j ∈ [J]}c∈[C] can be written Z(j) +c += +vc+Lc ξ(j) +c +where ξ(j) +c +∼ N(0k, Ik) and Lc ∈ Rk×k is such +that Σc = LcL⊤ +c by exploiting the positive semi-definite +property of Σc. Invertibility of Lc is ensured by adding a +diagonal matrix εIk with small positive diagonal elements. +One of the key advantages of this Gaussian assumption is +that, under mild assumptions, it guarantees the existence +of a transport map T (i) such that T (i) +# (νi) = µ, owing to +Brenier’s theorem (Santambrogio, 2015) as a mixture of +Gaussians admits a density with respect to the Lebesgue +measure. Hence, in our case, learning the local embedding +functions boils down to approximating this transport map +by φi. In addition, sampling from a Gaussian probabil- +ity distribution can be performed efficiently (Gilavert et al., +2015; Parker and Fox, 2012; Vono et al., 2022), even in +high dimension. We also consider approximating the con- +ditional probability measures {ν(c) +φi ; c ∈ Yi}i∈[b] by using +Gaussian measures {ˆν(c) +φi += N( ˆm(c) +i , ˆΣ(c) +i ); c ∈ Yi}i∈[b] +such that for any i ∈ [b] and c ∈ [C], ˆm(c) +i +and ˆΣ(c) +i +stand +for empirical mean vector and covariance matrix. The rel- +evance of this approximation is detailed in Appendix S1.2. +These two Gaussian choices (for the anchor distribution +and the class-conditional distributions) allow us to have a +closed-form expression for the Wasserstein distance of or- +der 2 which appears in (3), see e.g. Dowson and Landau +(1982); Gelbrich (1990). More precisely, we have for any +i ∈ [b] and c ∈ [C], +W2 +2 +� +µc, ν(c) +φi +� += +���vc − m(c) +i +��� +2 ++ B2 � +Σc, Σ(c) +i +� +, +(4) +where B(·, ·) denotes the Bures distance between two pos- +itive definite matrices (Bhatia et al., 2019). +In addition +to yield the closed-form expression (4), the choice of the +Wasserstein distance is motivated by two other important +properties. First, it is always finite no matter how degener- +ate the Gaussian distributions are, contrary to other diver- +gences such as the Kullback-Leibler one (Vilnis and Mc- +Callum, 2015). Being able to output a meaningful distance +value when supports of distribution do not overlap is a key +benefit of the Wasserstein distance, since when initialising +φi, we do not have any guarantee on such overlapping (see +illustration given in Figure S4). Second, its minimisation +can be handled using efficient algorithms proposed in the +optimal transport literature (Muzellec and Cuturi, 2018). +Related Work. As pointed out in Section 1, several ex- +isting works can be related to the proposed methodology. +Loosely speaking, we can divide these related approaches +into three categories namely (i) heterogeneous-architecture +personalised FL, (ii) vertical FL and (iii) federated transfer +learning. +Compared to traditional horizontal personalised FL (PFL) +approaches, so-called heterogeneous-architecture ones are +mostly motivated by local heterogeneity regarding resource +capabilities of clients e.g. computation and storage (Collins +et al., 2021; Diao et al., 2021; Hong et al., 2022; Makhija +et al., 2022; Shamsian et al., 2021; Zhang et al., 2021). +Nevertheless, they never consider features defined on het- +erogeneous subspaces, which is our main motivation. In +vertical federated learning (VFL), clients hold disjoint sub- +sets of features. However, a restrictive assumption is that a +large number of users are common across the clients (An- +gelou et al., 2020; Hardy et al., 2017; Romanini et al., 2021; +Yang et al., 2019). In addition, up to our knowledge, no ver- +tical personalised FL approach has been proposed so far, +which is restrictive if clients have different business objec- +tives and/or tasks. Finally, some works have focused on +adapting standard tranfer learning approaches with hetero- +geneous feature domains under the FL paradigm. These +federated transfer learning (FTL) approaches (Gao et al., +2019; Liu et al., 2020; Mori et al., 2022; Sharma et al., +2019) stand for FL variants of heterogeneous-feature trans- +fer learning where there are b source clients and 1 target +client with a target domain. However, these methods do not +consider the same setting as ours and assume that clients +share a common subset of features. We compare the most +relevant approaches among the previous ones in Table 1. + +Personalised Federated Learning On Heterogeneous Feature Spaces +5 +Table 1. Related works. PFL refers to horizontal personalised FL, VFL to vertical FL and FTL to federated transfer learning. +METHOD +TYPE +̸= FEATURE SPACES +MULTI-PARTY +NO SHARED ID +NO SHARED FEATURE +(ZHANG ET AL., 2021) +PFL + +✓ +✓ + +(DIAO ET AL., 2021) +PFL + +✓ +✓ + +(COLLINS ET AL., 2021) +PFL + +✓ +✓ + +(SHAMSIAN ET AL., 2021) +PFL + +✓ +✓ + +(HONG ET AL., 2022) +PFL + +✓ +✓ + +(MAKHIJA ET AL., 2022) +PFL + +✓ +✓ +✓ +FLIC (THIS PAPER) +PFL +✓ +✓ +✓ +✓ +(HARDY ET AL., 2017) +VFL +✓ + + +✓ +(YANG ET AL., 2019) +VFL +✓ + + +✓ +(GAO ET AL., 2019) +FTL +✓ +✓ +✓ + +(SHARMA ET AL., 2019) +FTL + + +✓ + +(LIU ET AL., 2020) +FTL +✓ + + +✓ +(MORI ET AL., 2022) +FTL +✓ +✓ + + +3. Algorithm +As detailed in Equation (3), we perform personalisation +under the FL paradigm by considering local model ar- +chitectures {g(i) +θi }i∈[b] and local weights θ1:b. As an ex- +ample, we could resort to federated averaging with fine- +tuning (e.g. +FedAvg-FT, see Collins et al. (2022)), +model interpolation (e.g. L2GD, see Hanzely and Richt´arik +(2020); Hanzely et al. (2020)) or partially local models (e.g. +FedRep, see Collins et al. (2021); Oh et al. (2022); Sing- +hal et al. (2021)). Table 2 details how these methods can be +embedded into the proposed methodology. +Table 2. Current personalised FL techniques that can be embed- +ded in the proposed framework. The parameters α, βi stand for +model weights while ω ∈ [0, 1]. +Algorithm +Local model +Local weights +FedAvg-FT +g(i) +θi = gθi +θi +L2GD +g(i) +θi = gθi +θi = ωα + (1 − ω)βi +FedRep +g(i) +θi = g(i) +βi ◦ gα +θi = [α, βi] +In Algorithm 1, we detail the pseudo-code associated to +a specific instance of the proposed methodology when +FedRep is resorted to learn model parameters {θi = +[α, βi]}i∈[b] under the FL paradigm. In this setting, α stand +for the shared weights associated to the first layers of a neu- +ral network architecture and βi for local ones aiming at +performing personalised classification. Besides these two +learnable parameters, the algorithm also learns the local +embedding functions φ1:b and the anchor distribution µ1:C. +In practice, at a given epoch t of the algorithm, a subset +At+1 ⊆ [b] of clients are selected to participate to the train- +ing process. Those clients receive the current latent anchor +distribution µ(t) +1:C and the current shared representation α(t). +Then, each client locally updates φi, βi and her local ver- +sions of α(t) and µ(t) +1:C. Afterwards, clients send back to +the server an updated version of α(t) and µ(t) +1:C. Updated +global parameters α(t+1) and µ(t+1) +1:C +are then obtained by +weighted averaging of client updates on appropriate man- +ifolds. +The use of the Wasserstein loss in (3) naturally +leads to perform averaging of the local anchor distributions +via a Wasserstein barycenter; algorithmic details are pro- +vided in the next paragraph. In Algorithm 1, we use for +the sake of simplicity the notation DescStep(F (t,m) +i +, ·) +to denote a (stochastic) gradient descent step on the func- +tion F (t,m) +i += Fi(β(t,m) +i +, φ(t,m) +i +, α(t), µ(t) +1:C) with respect +to a subset of parameters in (θi, φi, µ1:C). This subset is +specified in the second argument of DescStep. An ex- +plicit version of Algorithm 1 is provided in Appendix, see +Algorithm S2. +Note that we take into account key inherent challenges +to federated learning namely partial participation and +communication bottleneck. +Indeed, we cope with the +client/server upload communication issue by allowing each +client to perform multiple steps (here M ∈ N∗) so that +communication is only required every M local steps. This +allows us to consider updating global parameters, locally, +via only one stochastic gradient descent step and hence +avoiding the client drift phenomenon (Karimireddy et al., +2020). +Averaging Anchor Distributions. In this paragraph, we +provide algorithmic details regarding steps 14 and 20 in +Algorithm 1. +For any c ∈ [C], the anchor distribution +µc involves two learnable parameters namely the mean +vector vc and the covariance matrix Σc. +Regarding the +former, step 14 stands for a (stochastic) gradient descent +step aiming to obtain a local version of vc denoted by +v(t+1) +i,c +and step 20 boils down to compute v(t+1) +c += +(b/|At+1|) � +i∈At+1 ωiv(t+1) +i,c +. +To enforce the positive + +Personalised Federated Learning On Heterogeneous Feature Spaces +6 +Algorithm 1 FLIC +Require: initialisation α(0), µ(0) +1:C, φ(0,0) +1:b , β(0,0) +1:b . +1: for t = 0 to T − 1 do +2: +Sample a set of At+1 of active clients. +3: +for i ∈ At+1 do +4: +The central server sends α(t) and µ(t) +1:C to At+1. +5: +// Update local parameters +6: +for m = 0 to M − 1 do +7: +φ(t,m+1) +i +← DescStep +� +F (t,m) +i +, φ(t,m) +i +� +. +8: +β(t,m+1) +i +← DescStep +� +F (t,m) +i +, β(t,m) +i +� +. +9: +end for +10: +φ(t+1,0) +i += φ(t,M) +i +. +11: +β(t+1,0) +i += β(t,M) +i +. +12: +// Update global parameters +13: +α(t+1) +i +← DescStep +� +F (t,M) +i +, α(t)� +. +14: +µ(t+1) +i,1:C ← DescStep +� +F (t,M) +i +, µ(t) +1:C +� +. +15: +// Communication with the server +16: +Send α(t+1) +i +and µ(t+1) +i,1:C to central server. +17: +end for +18: +// Averaging global parameters +19: +α(t+1) = +b +|At+1| +� +i∈At+1 wiα(t+1) +i +20: +µ(t+1) +1:C +← WassersteinBarycenter({µ(t+1) +i,1:C }) +21: end for +Ensure: parameters α(T ), µ(T ) +1:C, φ(T,0) +1:b +, β(T,0) +1:b +. +semi-definite constraint of the covariance matrix, we +rewrite it as Σc = LcL⊤ +c where Lc ∈ Rk×k and optimise +in step 14 with respect to the factor Lc instead of Σc. We +can handle the gradient computation of the Bures distance +in step 14 using the work of Muzellec and Cuturi (2018); +and obtain a local factor L(t+1) +i,c +at iteration t. In step 20, +we compute L(t+1) +c += (b/|At+1|) � +i∈At+1 ωiL(t+1) +i,c +and +set Σ(t+1) +c += L(t+1) +c +[L(t+1) +c +]⊤. +When λ2 = 0 in (3), +these mean vector and covariance matrix updates exactly +boil down to perform one stochastic (because of partial par- +ticipation) gradient descent step to solve the Wasserstein +barycenter problem arg minµc +�b +i=1 ωiW2 +2(µc, ν(c) +φi ). +4. Non-Asymptotic Convergence Guarantees +in a Simplified Setting +Deriving non-asymptotic convergence bounds for Algo- +rithm 1 in the general case is challenging since the con- +sidered C-class classification problem leads to jointly solv- +ing personalised FL and federated Wasserstein barycenter +problems. Regarding the latter, obtaining non-asymptotic +convergence results is still an active research area in the +centralised learning framework (Altschuler et al., 2021). +As such, we propose to analyse a simpler regression frame- +Figure 2. Red dashed line indicates that the two embedded fea- +tures φ⋆ +i (x(j) +i ) and ˆφi(x(j) +i ) come from the same initial raw +feature x(j) +i . +On test data, mean prediction errors for both +FedRep operating on φ⋆ +i (x(j) +i ) and Algorithm S3 (referred to as +FLIC-FedRep) are similar (≈ 4.98 × 10−5). +work where the anchor distribution is known beforehand +and not learnt under the FL paradigm. +More precisely, we assume that x(j) +i +∼ N(mi, Σi) with +mi ∈ Rki and Σi ∈ Rki×ki for i ∈ [b], j ∈ [ni]. In +addition, we consider that the continuous scalar labels are +generated via the oracle model y(j) +i += (A⋆β⋆ +i )⊤φ⋆ +i (x(j) +i ) +where A⋆ ∈ Rk×d, β⋆ +i ∈ Rd and φ⋆ +i (·) are ground-truth pa- +rameters and feature transformation function, respectively. +We make the following assumptions on the ground truth. +H1. (i) For any i ∈ [b], j ∈ [ni], embedded features +φ⋆ +i (x(j) +i ) are distributed according to N(0k, Ik). +(ii) Ground-truth model parameters satisfy ∥β⋆ +i ∥2 = +√ +d +for i ∈ [b] and A⋆ has orthonormal columns. +(iii) For any t ∈ {0, . . . , T − 1}, |At+1| = b′ with 1 ≤ +b′ ≤ b, and if we select b′ clients, their ground-truth head +parameters {β⋆ +i }i∈At+1 span Rd. +(iv) In (2), ℓ(·, ·) is the ℓ2 norm, ωi = 1/b, θi = [A, βi] +and g(i) +θi (x) = (Aβi)⊤x for x ∈ Rk. +Under H1-(i), Delon et al. (2022, Theorem 4.1) show +that φ⋆ +i can be expressed as a non-unique affine map with +closed-form expression. To recover the true latent distri- +bution µ = N(0k, Ik), we propose to estimate ˆφi by lever- +aging this closed-form mapping between N(mi, Σi) and µ. +Because of the non-unicity of φ⋆ +i , we show in Theorem 1 +that we can only recover it up to a matrix multiplication. In- +terestingly, Theorem 1 shows that the global representation +A(T ) learnt via FedRep (see Algorithm S3 in Appendix) +is able to correct this feature mapping indetermination. As- +sociated convergence behavior is illustrated in Figure 2 on +a toy example whose details are postponed to Appendix S2. +Theorem 1. Assume H1. Then, for any xi ∈ Rki, we +have ˆφi(xi) = Qφ⋆ +i (xi) where Q ∈ Rk×k is of the form +diagk(±1). Under additional technical assumptions de- +tailed in Appendix S2, we have for any t ∈ {0, . . . , T − 1} +and with high probability, +dist(A(t+1), QA⋆) ≤ (1 − κ)(t+1)/2dist(A(0), QA⋆) , + +100 +I angle distance +dist(A(t),A*) with FedRep +10 +dist(A(t),A*) with FLIC-FedRep +dist(A (t), QA*) with FLIC-FedRep +Principal +10-6 +10-8 +Estimated projected features +Oracle +0 +100 +200 +300 +400 +500 +Epoch tPersonalised Federated Learning On Heterogeneous Feature Spaces +7 +Table 3. Performance over 3 runs of our FLIC model and the competitors on some real-data problems (Digits and TextCaps data set). +Data sets (setting) +Local +FedHeNN +FLIC-Class +FLIC-HL +Digits (b = 100, 3 Classes/client) +97.49 +97.45 +97.83 +97.70 +Digits (b = 100, 5 Classes/client) +96.16 +96.15 +96.46 +96.31 +Digits (b = 200, 3 Classes/client) +93.33 +93.40 +94.50 +94.51 +Digits (b = 200, 5 Classes/client) +86.50 +87.22 +90.66 +90.63 +TextCaps (b = 100, 2 Classes/client) +84.19 +83.99 +89.14 +89.68 +TextCaps (b = 100, 3 Classes/client) +76.04 +75.39 +81.27 +81.50 +TextCaps* (b = 200, 2 Classes/client) +83.78 +83.89 +87.73 +87.74 +TextCaps* (b = 200, 3 Classes/client) +74.95 +74.77 +79.08 +78.49 +where κ ∈ (0, 1) is detailed explicitly in Theorem S3 and +dist denotes the principal angle distance. +5. Numerical Experiments +In this section, we aim at illustrating how our algorithm +FLIC, when using FedRep as FL approach, works in prac- +tice and showcasing its numerical performances. We con- +sider several toy problems with different characteristics of +heterogeneity; as well as experiments on real data namely a +digit classification problem from images of different sizes +and an object classification problem from either images or +text captioning on clients. +Baselines. Since the problem we are addressing is novel, +there exists no FL competitor that can serve as a baseline +beyond local learning. However, we propose to modify the +methodology proposed in Makhija et al. (2022) to make +it applicable to clients with heterogeneous feature spaces. +This latter approach can handle local representation models +with different architectures and the key idea, coined Repre- +sentation Alignement Dataset (RAD), is to calibrate those +models by matching the latent representation of some fixed +data inputs shared by the server to all clients. In our case, +we can not use the same RAD accross all clients due to the +different dimensionality of the local models. A simple al- +ternative that we consider in our experiments is to build a +RAD given the largest dimension space among all clients +and then prune it to obtain a lower-dimensional RAD suit- +able to each client. We refer to the corresponding algorithm +as FedHeNN. +We are going to consider the same architecture networks +for all baselines. As Makhija et al. (2022) considers all but +the last layer of the network as the representation learning +module, for a fair comparison, we also assume the same +for our approach. Hence, in our case, the last layer is the +classifier layer and the alignment with the latent reference +distribution applies on the penultimate layer. This model +is referred to as FLIC-Class, in which all weights are thus +local (α is empty and βi refers to the last layer). In addi- +tion, we also have a model, coined FLIC-HL, which has +an additional trainable global hidden layer, which α being +the parameter of linear layer and βi the parameter of the +classification layer. +Data Sets. We consider three different classification prob- +lems to assess the performances of our approach. First, we +are considering a toy classification problem with C = 20 +classes and where each class-conditional distribution is a +Gaussian with random mean. Covariance matrices of all +classes are the same and considered fixed. Using this toy +data set, we are considering two sub-experiments. For the +first one, named noisy features (and labelled toy NF in +figures), we consider a 5-dimensional problem (k = 5) +and for each client add some random spurious features +which dimensionality goes up to 10. Hence, in this case +ki ∈ [5, 15]. For the second sub-experiment, denoted as +linear mapping (and labelled toy LM in figures), we apply a +Gaussian random linear mapping to the original data which +are of dimension 30. The output dimension of the mapping +is uniformly drawn from 5 to 30 leading to ki ∈ [5, 30]. +More details are provided in Appendix. +The second problem we consider is a digit classification +problem with the original MNIST and USPS data sets +which are respectively of dimension 28 × 28 and 16 × 16 +and we assume that a client hosts either a subset of MNIST +or USPS dataset. Finally, the last classification problem +is associated to a subset of the TextCaps data set (Sidorov +et al., 2020), which is an image captioning data set, that +we convert into a 4-class classification problem, with about +12, 000 and 3, 000 examples for respectively training and +testing, either based on the caption text or the image. Some +examples of image/caption pairs as well as as more de- +tails on how the dataset has been obtained are shown in +the Figure S5. The caption has been embedded into a 768- +dimensional vector using a pre-trained Bert embedding and +the image into a 512-dimensional ones using a pre-trained +ResNet model. We further generated some heterogeneity +by randomly pruning 10% of these features on each client. +Again, we assume that a client hosts either some image + +Personalised Federated Learning On Heterogeneous Feature Spaces +8 +Figure 3. Performance of FLIC and competitors on the toy data +sets with respect to the number of clients. (left) Gaussian classes +in dimension k = 5 with added noisy feature. (right) Gaussian +classes in dimension k = 30, transformed by a random map. Only +3 classes are present on each client among the 20 possible ones. +embeddings or text embeddings. For all simulations, we +assume prior probability shift e.g each client will have ac- +cess to data of only specific classes. +Experimental Setting. For the experimental analysis, we +use the codebase of Collins et al. (2021) with some modi- +fications to meet our setting. For all experiments, we con- +sider T = 50 communication rounds for all algorithms; and +at each round, a client participation rate of r = 0.1. The +number of local epochs for training has been set to M = +10. As optimisers, we have used an Adam optimiser with a +learning rate of 0.001 for all problems and approaches. Fur- +ther details are given in Appendix S3.3. For each compo- +nent of the latent anchor distribution, we consider a Gaus- +sian with learnable mean vectors and fixed Identity covari- +ance matrix. As such, the Wasserstein barycenter compu- +tation boils down to simply average the mean of client up- +dates and for computing the third term in Equation (3), we +just sample from the Gaussian distribution. Accuracies are +computed as the average accuracy over all clients after the +last epoch in which all local models are trained. +Results on Toy Data Sets. Figure 3 depicts the perfor- +mance, averaged over 5 runs, of the different algorithms +with respect to the number of clients and when only 3 +classes are present in each client. For both data sets, we can +note that for the noisy feature setting, FLIC improves on +FedHeNN of about 3% of accuracy across the setting and +performs better than local learning. For the linear mapping +setting, FLIC achieves better than other approaches with +a gain of performance of about 4% while the gap tends to +decrease as the number of clients increases. Interestingly, +FLIC-HL performs slightly better than FLIC-Class show- +ing the benefit of using a shared representation layer α. +Results on Digits and TextCaps Data Sets. Performance, +averaged over 3 runs, of all algorithms on the real-word +problems are reported in Table 3. For the Digits data set +problem, we first remark that in all situations, FL algo- +rithms performs a bit better than local learning. In addi- +tion, both variants of FLIC achieve better accuracy than +Figure 4. . 2D t-sne projection of 5 classes partially shared by 3 +clients for the toy LM dataset after learning the local transfor- +mation functions for (left) 10 epochs, (middle) 50 epochs, (right) +100 epochs. The three different markers represent the different +clients. the ⋆ marker represents the class-conditional mean of the +reference distribution. We note that training set converges towards +those means. +competitors. Difference in performance in favor our FLIC +reaches 3% for the most difficult problem. For the TextCaps +data set, gains in performance of FLIC-HL reach about 4% +across settings. While FedHeNN and FLIC algorithms fol- +low the same underlying principle (alignment of represen- +tation in a latent space), we believe that our framework ben- +efits from the use of the latent anchor distributions, avoid- +ing the need of sampling from the original space. Instead, +FedHeNN may fail as the sampling strategy of their RAD +approach suffers from the curse of dimensionality and does +not properly lead to a successful feature alignment. +Additional Experiments in Appendix. Due to the limited +number of pages, additional experiments are postponed to +the Appendix. In particular, we investigate the impact of +pre-training the local embedding functions for a fixed ref- +erence distribution as in Section 4, before running the pro- +posed algorithm detailed in Algorithm 1. The main mes- +sage is that pre-training helps in enhancing performance +but may lead to overfitting if too many epochs are consid- +ered. We also analyse the impact of client participation rate +at each communication round reaching the conclusion that +our model is robust to participation rate. +6. CONCLUSION +We have introduced a new framework for personalised FL +when clients have heterogeneous feature spaces. We pro- +posed a novel FL algorithm involving two key components: +(i) a local feature embedding function; and (ii) a latent an- +chor distribution which allows to match similar semantical +information from each client. Experiments on relevant data +sets have shown that FLIC achieves better performances +than competing approaches. Finally, we provided theoret- +ical support to the proposed methodology, notably via a +non-asymptotic convergence result. + +85.0 +Local +82.5 +FedHeNN +FLic Class +80.0 +FLic HL +(% +77.5 +Accuracy ( +75.0 +72.5 +70.0 +67.5 +25 +50 +75 +100 +125 +150 +175 +200 +# Clients74 +Local +FedHeNN +72 +FLic Class +FLic HL +(%) +70 +Accuracy ( +68 +66 +64 +62 +25 +50 +75 +100 +125 +150 +175 +200 +# ClientsClass 1 +Class 1 +Class 1 +KK +Class 3 +Class 3 +Class 3 +Class 5 +Class 5 +Class 5 +Class 6 +Class 6 +Class 6 +Class 11 +Class 11 +Class 11 +Mean vector of μ1Personalised Federated Learning On Heterogeneous Feature Spaces +9 +REFERENCES +Mokhtar Z Alaya, Maxime B´erar, Gilles Gasso, and Alain +Rakotomamonjy. +Theoretical guarantees for bridging +metric measure embedding and optimal transport. Neu- +rocomputing, 468:416–430, 2022. +Jason Altschuler, Sinho Chewi, Patrik Robert Gerber, and +Austin J Stromme. 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FedFA: Fed- +erated Learning with Feature Anchors to Align Feature +and Classifier for Heterogeneous Data. arXiv preprint +arXiv: 22211.09299, 2022. + +Personalised Federated Learning On Heterogeneous Feature Spaces +13 +Appendix +Table of Contents +S1 Algorithmic and Theoretical Insights +14 +S1.1 Some Limited but Common Alternatives to Cope with Feature Space Heterogeneity . . . . . . . . . . . . +14 +S1.2 Use of Wasserstein Losses Involving Empirical Probability Distributions . . . . . . . . . . . . . . . . . . +14 +S1.3 Detailed Pseudo-Code for Algorithm 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +14 +S1.4 Additional Algorithmic Insights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +15 +S2 Proof of Theorem 1 +16 +S2.1 Estimation of the Feature Transformation Functions +. . . . . . . . . . . . . . . . . . . . . . . . . . . . +16 +S2.2 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +16 +S2.3 Technical Lemmata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +18 +S3 Experimental Details +23 +S3.1 Reference Distribution for Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +23 +S3.2 Data Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +23 +S3.3 Models and Learning Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +24 +S3.4 Ablating Loss Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +24 +S3.5 On Importance of Alignment Pre-Training and Updates. . . . . . . . . . . . . . . . . . . . . . . . . . . +25 +S3.6 On the Impact of the Participation Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +25 +SUPPLEMENTARY MATERIAL +Notations and conventions. +We denote by B(Rd) the Borel σ-field of Rd, M(Rd) the set of all Borel measurable func- +tions f on Rd and ∥·∥ the Euclidean norm on Rd. For µ a probability measure on (Rd, B(Rd)) and f ∈ M(Rd) a +µ-integrable function, denote by µ(f) the integral of f with respect to (w.r.t.) µ. Let µ and ν be two sigma-finite measures +on (Rd, B(Rd)). Denote by µ ≪ ν if µ is absolutely continuous w.r.t. ν and dµ/dν the associated density. We say that +ζ is a transference plan of µ and ν if it is a probability measure on (Rd × Rd, B(Rd × Rd)) such that for all measurable +set A of Rd, ζ(A × Rd) = µ(A) and ζ(Rd × A) = ν(A). We denote by T (µ, ν) the set of transference plans of µ and ν. +In addition, we say that a couple of Rd-random variables (X, Y ) is a coupling of µ and ν if there exists ζ ∈ T (µ, ν) such +that (X, Y ) are distributed according to ζ. We denote by P1(Rd) the set of probability measures with finite 1-moment: for +all µ ∈ P1(Rd), +� +Rd ∥x∥dµ(x) < ∞. We denote by P2(Rd) the set of probability measures with finite 2-moment: for all +µ ∈ P2(Rd), +� +Rd ∥x∥2dµ(x) < ∞. We define the squared Wasserstein distance of order 2 associated with ∥ · ∥ for any +probability measures µ, ν ∈ P2(Rd) by +W2 +2(µ, ν) = +inf +ζ∈T (µ,ν) +� +Rd×Rd ∥x − y∥2dζ(x, y) . +By Villani (2008, Theorem 4.1), for all µ, ν probability measures on Rd, there exists a transference plan ζ⋆ ∈ T (µ, ν) +such that for any coupling (X, Y ) distributed according to ζ⋆, W2(µ, ν) = E[∥x − y∥2]1/2. This kind of transference +plan (respectively coupling) will be called an optimal transference plan (respectively optimal coupling) associated with +W2. By Villani (2008, Theorem 6.16), P2(Rd) equipped with the Wasserstein distance W2 is a complete separable metric +space. For the sake of simplicity, with little abuse, we shall use the same notations for a probability distribution and its + +Personalised Federated Learning On Heterogeneous Feature Spaces +14 +associated probability density function. For n ≥ 1, we refer to the set of integers between 1 and n with the notation +[n]. The d-multidimensional Gaussian probability distribution with mean µ ∈ Rd and covariance matrix Σ ∈ Rd×d is +denoted by N(µ, Σ). Given two matrices M, N ∈ Rk×d, the principal angle distance between the subspaces spanned +by the columns of M and N is given by dist(M, N) = ∥ ˆ +M † +⊥ ˆN∥2 = ∥ ˆN † +⊥ ˆ +M∥2 where ˆ +M, ˆN are orthonormal bases of +Span(M) and Span(N), respectively. Similarly, ˆ +M⊥, ˆN⊥ are orthonormal bases of orthogonal complements Span(M)⊥ +and Span(N)⊥, respectively. This principal angle distance is upper bounded by 1, see Jain et al. (2013, Definition 1). +Outline. This supplementary material aims at providing the interested reader with a further understanding of the statements +pointed out in the main paper. More precisely, in Appendix S1, we support the proposed methodology FLIC with algorith- +mic and theoretical details. In Appendix S2, we prove the main results stated in the main paper. Finally, in Appendix S3, +we provide further experimental design choices and show complementary numerical results. +S1. Algorithmic and Theoretical Insights +In this section, we highlight alternative but limited ways to cope with feature space heterogeneity; and justify the usage, +in the objective function (3) of the main paper, of Wasserstein distances with empirical probability distributions instead of +true ones. In addition, we detail the general steps depicted Algorithm 1. +S1.1. Some Limited but Common Alternatives to Cope with Feature Space Heterogeneity +Depending on the nature of the spaces {Xi}i∈[b], the feature transformation functions {φi}i∈[b] can be either known be- +forehand or more difficult to find. As an example, if for any i ∈ [b], Xi ⊆ X, then we can set mask functions as feature +transformation functions in order to only consider features that are shared across all the clients. Besides, we could consider +multimodal embedding models to perform feature transformation on each client (Duquenne et al., 2021). For instance, if +clients own either pre-processed images or text of titles, descriptions and tags, then we can use the Contrastive Language- +Image Pre-Training (CLIP) model as feature transformation function (Radford et al., 2021). These two examples lead to +the solving of a classical (personalised) FL problem which can be performed using existing state-of-the-art approaches. +However, when the feature transformation functions cannot be easily found beforehand, solving the FL problem at stake +becomes more challenging and has never been addressed in the federated learning literature so far, up to the authors’ +knowledge. +S1.2. Use of Wasserstein Losses Involving Empirical Probability Distributions +Since the true probability distributions {ν(c) +φi ; c ∈ Yi}i∈[b] are unknown a priori, we propose in the main paper to estimate +the latter using {ˆν(c) +φi ; c ∈ Yi}i∈[b] and to replace W2 +2 +� +µc, ν(c) +φi +� +by W2 +2 +� +µc, ˆν(c) +φi +� +in the objective function (3) in the main +paper. As shown in the following result, this assumption is theoretically grounded when the marginal distributions of the +input features are Gaussian. +Theorem S2. For any i ∈ [b] and c ∈ [C], let n(c) +i += |D(c) +i | where D(c) +i +denotes the subset of the local data set Di only +involving observations associated to the label c. Besides, assume that ν(c) +φi is Gaussian with mean vector m(c) +i +∈ Rk and +full-rank covariance matrix Σ(c) +i +∈ Rk×k. Then, we have in the limiting case n(c) +i +→ ∞, +� +n(c) +i +� +W2 +2 +� +µc, ˆν(c) +φi +� +− W2 +2 +� +µc, ν(c) +φi +�� +in distribution +−−−−−−−−−−→ Z(c) +i +, +where Z(c) +i +∼ N(0, s(c) +i ) and s(c) +i += 4(m(c) +i +− vc)⊤Σ(c) +i (m(c) +i +− vc) + 2Tr(Σ(c) +i Σc) − 4 �k +j=1 κ1/2 +j +r⊤ +j Σ−1/2 +c +Σ(c) +i Σ1/2 +c +rj, +with {κj, rj}j∈[k] standing for (eigenvalue, eigenvector) pairs of the symmetric covariance matrix Σ(c) +i . +Proof. The proof follows from Rippl et al. (2016, Theorem 2.1) with the specific choices µ1 = ν(c) +φi , µ2 = µc and ˆµ1 = ˆν(c) +φi +which are defined in Section 2 in the main paper. +S1.3. Detailed Pseudo-Code for Algorithm 1 +In Algorithm S2, we provide algorithmic support to Algorithm 1 in the main paper by detailing how to perform each step. +Note that we use the decomposition Σ = LL⊤ to enfore the positive semi-definite constraint for the covariance matrix Σ. + +Personalised Federated Learning On Heterogeneous Feature Spaces +15 +Algorithm S2 Detailed version of FLIC when using FedRep +Require: initialisation α(0), µ(0) +1:C = [Σ(0) +1:C, v(0) +1:C] with Σ(0) +c += L(0) +c [L(0) +c ]⊤, φ(0,0) +1:b , β(0,0) +1:b +and step-size η ≤ ¯η for some +¯η > 0. +1: for t = 0 to T − 1 do +2: +Sample a set of At+1 of active clients. +3: +for i ∈ At+1 do +4: +The central server sends α(t) and µ(t) +1:C to At+1. +5: +// Update local parameters +6: +for m = 0 to M − 1 do +7: +Sample a fresh batch I(i,m) +t+1 +of n′ +i samples with n′ +i ∈ [ni]. +8: +Sample Z(j,t,m) +c +∼ µ(t) +c +for j ∈ I(i,m) +t+1 +and c ∈ Yi via Z(j,t,m) +c += v(t) +c ++ L(t) +c ξ(t,m) +i +where ξ(t,m) +i +∼ N(0k, Ik). +9: +φ(t,m+1) +i += φ(t,m) +i +− η +ni +|I(i,m) +t+1 | +� +j∈I(i,m) +t+1 +∇φiℓ +� +y(j) +i +, g(i) +[α(t),β(t,m) +i +] +� +φ(t,m) +i +� +x(j) +i +��� +− ηλ1 +� +c∈Yi +∇φiW2 +2 +� +µ(t) +c , ν(c) +φ(t,m) +i +� +. +10: +β(t,m+1) +i +← β(t,m) +i +−η +ni +|I(i,m) +t+1 | +� +j∈I(i,m) +t+1 +� +� +�∇βiℓ +� +y(j) +i +, g(i) +[α(t),β(t,m) +i +] +� +φ(t,m) +i +� +x(j) +i +��� +− ηλ2 +� +c∈Yi +∇βiℓ +� +y(j) +i +, g(i) +[α(t),β(t,m) +i +] +� +Z(j,t,m) +c +��� +� +�. +11: +end for +12: +φ(t+1,0) +i += φ(t,M) +i +. +13: +β(t+1,0) +i += β(t,M) +i +. +14: +// Update global parameters +15: +α(t+1) +i +← α(t)−η +ni +|I(i,M) +t+1 | +� +j∈I(i,M) +t+1 +� +� +�∇αℓ +� +y(j) +i +, g(i) +[α(t),β(t,M) +i +] +� +φ(t,M) +i +� +x(j) +i +��� +− ηλ2 +� +c∈Yi +∇αℓ +� +y(j) +i +, g(i) +[α(t),β(t,M) +i +] +� +Z(j,t,M) +c +��� +� +�. +16: +for c = 1 to C do +17: +Update ˆm(c,t) +i +, ˆΣ(c,t) +i +using φ(t,M) +i +. +18: +v(t+1) +i,c += v(t) +c +− ηλ1∇vc +���v(t) +c +− ˆm(c,t) +i +��� +2 +− ηλ2 +� +c∈Yi +ni +|I(i,m) +t+1 | +� +j∈I(i,m) +t+1 +∇vcℓ +� +y(j) +i +, g(i) +[α(t),β(t,M) +i +] +� +Z(j,t,M) +c +�� +. +19: +L(t+1) +i,c += L(t) +c −ηλ1∇LcB2 � +L(t) +c [L(t) +c ]⊤, ˆΣ(c,t) +i +� +−ηλ2 +� +c∈Yi +ni +|I(i,m) +t+1 | +� +j∈I(i,m) +t+1 +∇Lcℓ +� +y(j) +i +, g(i) +[α(t),β(t,M) +i +] +� +Z(j,t,M) +c +�� +. +20: +end for +21: +// Communication with the server +22: +Send α(t+1) +i +, v(t+1) +i,1:C and L(t+1) +i,1:C to central server. +23: +end for +24: +// Averaging global parameters +25: +α(t+1) = +b +|At+1| +� +i∈At+1 wiα(t+1) +i +. +26: +for c = 1 to C do +27: +v(t+1) +c += (b/|At+1|) � +i∈At+1 ωiv(t+1) +i,c +. +28: +L(t+1) +c += (b/|At+1|) � +i∈At+1 ωiL(t+1) +i,c +and set Σ(t+1) +c += L(t+1) +c +[L(t+1) +c +]⊤. +29: +end for +30: end for +Ensure: parameters α(T ), µ(T ) +1:C, φ(T,0) +1:b +, β(T,0) +1:b +. +S1.4. Additional Algorithmic Insights +Scalability. When the number of classes C is large, both local computation and communication costs are increased. In this +setting, we propose to partition all the classes into Cmeta ≪ C meta-classes and consider reference measures {µc}c∈[Cmeta] +associated to these meta-classes. As an example, if we are considering a dataset made of features associated to animals, +the meta-class refers to an animal (e.g. a dog) and the class refers to a specific breed (e.g. golden retriever). +Privacy Consideration. As other standard (personalised) FL algorithms, FLIC satisfies first-order privacy guarantees by +not allowing raw data exchanges but rather exchanges of local Gaussian statistics. Note that FLIC stands for a post-hoc + +Personalised Federated Learning On Heterogeneous Feature Spaces +16 +approach and can be combined with other privacy/confidentiality techniques such as differential privacy (Dwork and Roth, +2014), secure aggregation via secure multi-party computation (Chen et al., 2022) or trusted execution environments (Mo +et al., 2021). +Inference on New Clients. When a client who has not participated to the training procedure appears, there is no need to +re-launch a potentially costly federated learning procedure. Instead, the server sends the shared parameters {α(T ), µ(T ) +1:C} +to the new client and the latter only needs to learn the local parameters {φi, βi}. +S2. Proof of Theorem 1 +This section aims at proving Theorem 1 in the main paper. To this end, we first provide in Appendix S2.1 a closed-form +expression for the estimated embedded features based on the features embedded by the oracle. Then, in Appendix S2.3, +we show technical lemmata that will be used in Appendix S2.2 to show Theorem 1. +To prove our results, we consider the following set of assumptions. +H1. (i) For any i ∈ [b], j ∈ [ni], ground-truth embedded features φ⋆ +i (x(j) +i ) are distributed according to N(0k, Ik). +(ii) Ground-truth model parameters satisfy ∥β⋆ +i ∥2 = +√ +d for i ∈ [b] and A⋆ has orthonormal columns. +(iii) For any t ∈ {0, . . . , T − 1}, |At+1| = ⌊rb⌋ with 1 ≤ ⌊rb⌋ ≤ b, and if we select ⌊rb⌋ clients, their ground-truth head +parameters {β⋆ +i }i∈At+1 span Rd. +(iv) In (2) in the main paper, ℓ(·, ·) is the ℓ2 norm, ωi = 1/b, θi = [A, βi] and g(i) +θi (x) = (Aβi)⊤x for x ∈ Rk. +S2.1. Estimation of the Feature Transformation Functions +As in Section 4 in the main paper, we assume that x(j) +i +∼ N(mi, Σi) with mi ∈ Rki and Σi ∈ Rki×ki for i ∈ [b], j ∈ [ni]. +In addition, we consider that the continuous scalar labels are generated via the oracle model y(j) +i += (A⋆β⋆ +i )⊤φ⋆ +i (x(j) +i ) where +A⋆ ∈ Rk×d, β⋆ +i ∈ Rd and φ⋆ +i (·) are ground-truth parameters and feature transformation function, respectively. Under +H1-(i), the oracle feature transformation functions {φ⋆ +i }i∈[b] are assumed to map ki-dimensional Gaussian distributions +N(mi, Σi) to a common k-dimension Gaussian N(0k, Ik). As shown in Delon et al. (2022, Theorem 4.1), there exist closed- +form expressions for {φ⋆ +i }i∈[b], which can be shown to stand for solutions of a Gromov-Wasserstein problem restricted to +Gaussian transport plans. More precisely, these oracle feature transformation stand for affine maps and are of the form, for +any i ∈ [b], +φ⋆ +i +� +x(j) +i +� += +� +˜I(i,⋆) +k +(D(k) +i +)−1/2 +0k,ki−k +� � +x(j) +i +− mi +� +, +where ˜I(i,⋆) +k += diagk(±1) is a k-dimensional diagonal matrix with diagonal elements in {−1, 1}, Σi = PiDiP ⊤ +i +is the +diagonalisation of Σi and D(k) +i +stands for the restriction of Di to the first k components. In the sequel, we assume that all +oracle feature transformation functions share the same randomness, that is ˜I(i,⋆) +k += ˜I⋆ +k = diagk(±1). +For the sake of simplicity, we assume that we know the true latent distribution of φ⋆ +i (x(j) +i ) and as such consider a pre- +fixed reference latent distribution that equals the latter, that is µ = N(0k, Ik). Since we know from Delon et al. (2022, +Theorem 4.1) that there exist mappings between Gaussian distributions with supports associated to different metric spaces, +we propose an estimate for the ground-truth feature transformation functions defined by for any i ∈ [b], +ˆφi +� +x(j) +i +� += +� +˜Ik(D(k) +i +)−1/2 +0k,ki−k +� � +x(j) +i +− mi +� +, +where ˜Ik = diagk(±1). By noting that ˜Ik = Q˜I⋆ +k, where Q ∈ Rk×k is a diagonal matrix of the form diagk(±1), it follows +that +ˆφi +� +x(j) +i +� += Qφ⋆ +i +� +x(j) +i +� +. +(S1) +In Appendix S2.2, the equation (S1) will allow us to relate the ground-truth labels y(j) +i += (A⋆β⋆ +i )⊤φ⋆ +i (x(j) +i ) with estimated +predictions ˆy(j) +i += (A(T )β(T ) +i +)⊤ ˆφi(x(j) +i ) via Algorithm S3 starting from the same embedded features. +S2.2. Proof of Theorem 1 + +Personalised Federated Learning On Heterogeneous Feature Spaces +17 +Algorithm S3 FLIC-FedRep for linear regression and Gaussian features +Require: step size η, number of outer iterations T, participation rate r ∈ (0, 1), diagonalizations Σi = PiDiP ⊤ +i +sorting +eigenvalues in decreasing order. +1: // Estimation of embedded features +2: For each client i ∈ [b], set ˆφi +� +x(j) +i +� += +� +˜Ik(D(k) +i +)−1/2 +0k,ki−k +� � +x(j) +i +− mi +� +. +3: // Initialisation A(0) +4: Each client i ∈ [b] sends Zi = (1/ni) �ni +j=1(y(j) +i )2 ˆφi +� +x(j) +i +� +[ˆφi +� +x(j) +i +� +]⊤ to the central server. +5: The central server computes UDU ⊤ ← rank−d SVD +� +(1/b) �b +i=1 Zi +� +. +6: The central server initialises A(0) = U. +7: for t = 0 to T − 1 do +8: +Sample a set of At+1 of active clients such that |At+1| = ⌊rb⌋. +9: +for i ∈ At+1 do +10: +The central server sends A(t)to At+1. +11: +// Update local parameters +12: +β(t+1) +i += arg minβi +�ni +j=1 +� +y(j) +i +− β⊤ +i [A(t)]⊤ ˆφi +� +x(j) +i +��2 +. +13: +// Update global parameters +14: +A(t+1) +i += A(t) − η∇A +�ni +j=1 +� +y(j) +i +− [β(t+1) +i +]⊤A⊤ ˆφi +� +x(j) +i +��2 +. +15: +// Communication with the server +16: +Send A(t+1) +i +to the central server. +17: +end for +18: +// Averaging and orthogonalisation of global parameter +19: +¯A(t+1) = +1 +⌊rb⌋ +� +i∈At+1 A(t+1) +i +. +20: +A(t+1), R(t+1) ← QR +� ¯A(t+1)� +. +21: end for +Ensure: parameters A(T ), β(T ) +1:b . +Let B ∈ Rb×d the matrix having local model parameters {βi}i∈[b] as columns and denote by BAt+1 ∈ R⌊rb⌋×d its +restriction to the row set defined by At+1 where |At+1| = ⌊rb⌋ for some r ∈ (0, 1]. For the sake of simplicity, we assume +in the sequel that all clients have the same number of data points that is for any i ∈ [b], ni = n. For random batches of +samples {(x(j) +i , y(j) +i ), j ∈ [n]}i∈[⌊rb⌋], we define similarly to Collins et al. (2021); Jain et al. (2013), the random linear +operator A : R⌊rb⌋×d → R⌊rb⌋n for any M ∈ R⌊rb⌋×d as A(M) = [⟨ei(φ⋆ +i (x(j) +i ))⊤, M⟩]1≤i≤⌊rb⌋,1≤j∈n, where ei stands +for the i-th standard vector of R⌊rb⌋. Using these notations, it follows from Algorithm S3 that for any t ∈ {0, . . . , T − 1}, +the model parameters θ(t+1) +i += [A(t+1), β(t+1) +i +] are computed as follows: +B(t+1) +At+1 = arg min +BAt+1 +1 +⌊rb⌋ n +���A(t+1) � +B⋆ +At+1[A⋆]⊤ − BAt+1[A(t)]⊤Q +���� +2 +, +(S2) +¯A(t+1) = ¯A(t) − +η +⌊rb⌋ n +� +(A(t+1))†A(t+1) � +B⋆ +At+1[A⋆]⊤ − B(t+1) +At+1 [A(t)]⊤Q +��⊤ +QB(t+1) +At+1 , +A(t+1), R(t+1) ← QR +� +¯A(t+1)� +, +(S3) +where A(t+1) stands for a specific instance of A depending on the random subset of active clients available at each round +and A† is the adjoint operator of A defined by A†(M) = � +i∈[⌊rb⌋] +�n +i=1[⟨ei(φ⋆ +i (x(j) +i ))⊤, M⟩]ei(φ⋆ +i (x(j) +i )). +The update in (S2) admits a closed-form expression as shown in the following lemma. +Lemma S1. For any t ∈ . . . 0, . . . , T − 1, we have +B(t+1) +At+1 = B⋆ +At+1[A⋆]⊤QA(t) − F (t) , + +Personalised Federated Learning On Heterogeneous Feature Spaces +18 +where F (t) is defined in (S12), A(t) is defined in (S3) and B(t) +At is defined in (S2). +Proof. The proof follows from the same steps as in Collins et al. (2021, Proof of Lemma 1) using (S2). +Under H1, we have the following non-asymptotic convergence result. +Theorem S3. Assume H1. Then, for any xi ∈ Rki, we have ˆφi(xi) = Qφ⋆ +i (xi) where Q ∈ Rk×k is of the form diagk(±1). +Define E0 = dist(A(0), QA⋆). Assume that n ≥ c(d3 log(⌊rb⌋))/E2 +0 + d2k/(E2 +0 ⌊rb⌋) for some absolute constant c > 0. +Then, for any t ∈ {0, . . . , T − 1}, η ≤ 1/(4¯σ2 +max,⋆) and with high probability at least 1 − e−110k − e−110d2 log(⌊rb⌋), we +have +dist(A(t+1), QA⋆) ≤ (1 − κ)(t+1)/2dist(A(0), QA⋆) , +where A(t) is computed via Algorithm S3, dist denotes the principal angle distance and κ ∈ (0, 1) is defined as +κ = 1 − ηE0¯σ2 +min,⋆/2. +Proof. The proof follows first by plugging Lemma S3, Lemma S8 and Lemma S9 into Lemma S2. Then, we use the same +technical arguments and steps as in Collins et al. (2021, Proof of Lemma 6). +S2.3. Technical Lemmata +In this section, we provide a set of useful technical lemmata to prove our main result in Appendix S2.2. +Notations. We begin by defining some notations that will be used in the sequel. For any t ∈ {0, . . . , T − 1}, we define +Z(t+1) = B(t+1) +At+1 [A(t)]⊤Q − B⋆ +At+1[A⋆]⊤ . +(S4) +In addition, let +G(t) = +� +��� +G(t) +11 +· · · +G(t) +1d +... +... +... +G(t) +d1 +· · · +G(t) +dd +� +��� , C(t) = +� +��� +C(t) +11 +· · · +C(t) +1d +... +... +... +C(t) +d1 +· · · +C(t) +dd +� +��� , D(t) = +� +��� +D(t) +11 +· · · +D(t) +1d +... +... +... +D(t) +d1 +· · · +D(t) +dd +� +��� , +where for p, q ∈ [d], +G(t) +pq = 1 +n +� +i∈At+1 +n +� +j=1 +ei +� +φ⋆ +i (x(j) +i ) +�⊤ +Qa(t) +p [a(t) +q ]⊤Qφ⋆ +i (x(j) +i )e⊤ +i , +(S5) +C(t) +pq = 1 +n +� +i∈At+1 +n +� +j=1 +ei +� +φ⋆ +i (x(j) +i ) +�⊤ +Qa(t) +p [a⋆ +q]⊤Qφ⋆ +i (x(j) +i )e⊤ +i , +(S6) +D(t) +pq = ⟨a(t) +p , a⋆ +q⟩I⌊rb⌋ , +(S7) +with a(t) +p +∈ Rk standing for the p-th column of A(t) ∈ Rk×d; and a⋆ +p ∈ Rk standing for the p-th column of A⋆ ∈ Rk×d. +Finally, we define for any i ∈ At+1, +Πi = 1 +n +n +� +j=1 +φ⋆ +i (x(j) +i )[φ⋆ +i (x(j) +i )]⊤ , +(S8) +(G(t))i = [A(t)]⊤QΠiQA(t) , +(S9) +(C(t))i = [A(t)]⊤QΠiQA⋆ , +(S10) +(D(t))i = [A(t)]⊤QA⋆ . +(S11) + +Personalised Federated Learning On Heterogeneous Feature Spaces +19 +Using these notations, we also define ˜β⋆ = [(β⋆ +1)⊤, . . . , (β⋆ +d)⊤]⊤ ∈ R⌊rb⌋d and +F (t) = [([G(t)]−1(G(t)D(t) − C(t))˜β⋆)1, . . . , ([G(t)]−1(G(t)D(t) − C(t))˜β⋆)d] . +(S12) +Technical results. To prove our main result in Theorem S3, we begin by providing a first upper bound on the quantity of +interest namely dist +� +A(t+1), QA⋆� +. This is the purpose of the next lemma. +Lemma S2. For any t ∈ {0, . . . , T − 1} and η > 0, we have +dist +� +A(t+1), QA⋆� +≤ C1 + C2, , +where +C1 = +����[A⋆ +⊥]⊤QA(t) +� +Id − +η +⌊rb⌋[B(t+1) +At+1 ]⊤B(t+1) +At+1 +����� +2 +���� +� +R(t+1)�−1���� +2 +, +(S13) +C2 = +η +⌊rb⌋ +����� +� 1 +n[A⋆ +⊥]⊤(QA(t+1))†A(t+1) � +Z(t+1)� +Q − Z(t+1) +�⊤ +B(t+1) +At+1 +����� +2 +���� +� +R(t+1)�−1���� +2 +, +(S14) +where A(t) is defined in (S3), B(t) +At is defined in (S2), Z(t) is defined in (S4) and R(t) comes from the QR factorisation of +¯A(t), see step 20 in Algorithm S3. +Proof. The proof follows from the same steps as in Collins et al. (2021, Proof of Lemma 6) and by noting that +dist(A(t), QA⋆) = dist(QA(t), A⋆) for t ∈ {0, . . . , T − 1}. +We now have to control the terms C1 and C2. For the sake of clarity, we split technical results aiming to upper bound of +C1 and C2 in two different paragraphs. +Control of C1. +Lemma S3. Assume H1. +Let δd = cd3/2� +log(⌊rb⌋)/n1/2 for some absolute constant c > 0. +Then, for any +t ∈ {0, . . . , T − 1}, with probability at least 1 − e−111k2 log(⌊rb⌋), we have for δd ≤ 1/2 and η ≤ 1/(4¯σ2 +max,⋆) +C1 ≤ +� +≤ 1 − η +� +1 − dist +� +A(0), QA⋆�� +¯σ2 +min,⋆ + 2η +δd +1 − δd +¯σ2 +max +� +dist +� +A(t), QA⋆� ���� +� +R(t+1)�−1���� +2 +, +where ¯σ2 +min, ¯σ2 +max are defined in (S15)-(S16), C1 is defined in (S13), A(t) is defined in (S3) and R(t) comes from the QR +factorisation of ¯A(t), see step 20 in Algorithm S3. +Proof. Using Cauchy-Schwarz inequality, we have +C1 ≤ +���(A⋆ +⊥)⊤QA(t)��� +2 +����Id − +η +⌊rb⌋[B(t+1) +At+1 ]⊤B(t+1) +At+1 +���� +2 +���� +� +R(t+1)�−1���� +2 += dist +� +A(t), QA⋆� ����Id − +η +⌊rb⌋[B(t+1) +At+1 ]⊤B(t+1) +At+1 +���� +2 +���� +� +R(t+1)�−1���� +2 +. +Define the following minimum and maximum singular values: +¯σ2 +min,⋆ = +min +A⊆[b],|A|=⌊rb⌋ σmin +� +1 +� +⌊rb⌋ +B⋆ +A +� +(S15) +¯σ2 +max,⋆ = +min +A⊆[b],|A|=⌊rb⌋ σmax +� +1 +� +⌊rb⌋ +B⋆ +A +� +. +(S16) +Using Collins et al. (2021, Proof of Lemma 6, equations (67)-(68)), we have for δd ≤ 1/2 where δd is defined in Lemma S4 +and η ≤ 1/(4¯σ2 +max,⋆), +����Id − +η +⌊rb⌋[B(t+1) +At+1 ]⊤B(t+1) +At+1 +���� +2 +≤ 1 − η +� +1 − dist +� +A(0), QA⋆�� +¯σ2 +min,⋆ + 2η +δd +1 − δd +¯σ2 +max,⋆ , +with probability at least 1 − e−111k2 log(⌊rb⌋) The proof is concluded by combining the two previous bounds. + +Personalised Federated Learning On Heterogeneous Feature Spaces +20 +Control of C2. We begin by showing four intermediary results gathered in the next four lemmata. +Lemma S4. Assume H1. +Let δd = cd3/2� +log(⌊rb⌋)/n1/2 for some absolute constant c > 0. +Then, for any +t ∈ {0, . . . , T − 1}, with probability at least 1 − e−111k3 log(⌊rb⌋), we have +���[G(t)]−1��� +2 ≤ +1 +1 − δd +, +where G(t) is defined in (S5). +Proof. The proof stands as a straightforward extension of Collins et al. (2021, Proof of Lemma 2) by noting that the random +variable Qφ⋆ +i (x(j) +i ) = ˆφi(x(j) +i ) is sub-Gaussian under H1-(i); and as such is omitted. +Lemma S5. Assume H1. +Let δd = cd3/2� +log(⌊rb⌋)/n1/2 for some absolute constant c > 0. +Then, for any +t ∈ {0, . . . , T − 1}, with probability at least 1 − e−111k2 log(⌊rb⌋), we have +���(G(t)D(t) − C(t))B⋆ +At +��� +2 ≤ δd +��B⋆ +At +�� +2 dist +� +A(t), QA⋆� +, +where G(t) is defined in (S5), D(t) is defined in (S7), C(t) is defined in (S6) and A(t) in (S3). +Proof. Without loss of generality and to ease notation, we remove the superscript (t) in the proof and re-index the indexes +of clients in At+1. Let H = GD − C. From (S8), (S9), (S10) and (S11), it follows, for any i ∈ [⌊rb⌋], that +Hi = GiDi − Ci = A⊤QΠiQ(AA⊤ − Ik)QA⋆ . +Hence, by using the definition of H, we have +∥(GD − C)β⋆∥2 +2 = +⌊rb⌋ +� +i=1 +��Hiβ⋆ +i +��2 +2 ≤ +⌊rb⌋ +� +i=1 +��Hi��2 +2 ∥β⋆ +i ∥2 ≤ +d +⌊rb⌋ ∥B⋆∥2 +2 +⌊rb⌋ +� +i=1 +��Hi��2 +2 , +where the last inequality follows almost surely from H1-(iii). As in Collins et al. (2021, Proof of Lemma 3), we then define +for any j ∈ [n], the vectors +u(j) +i += +1 +√n[A⋆]⊤(AA⊤ − Ik)Qφ⋆ +i (x(j) +i ) , +v(j) +i += +1 +√nA⊤Qφ⋆ +i (x(j) +i ) . +Let Sd−1 denotes the d-dimensional unit spheres. Then, by Vershynin (2018, Corollary 4.2.13), we can define Nd, the +1/4-net over Sd−1 such that |Nd| ≤ 9d. Therefore, by using Vershynin (2018, Equation (4.13)), we have +��Hi��2 +2 ≤ 2 max +z,y∈Nd +n +� +j=1 +⟨z, u(j) +i ⟩⟨v(j) +i , y⟩ . +Since φ⋆ +i (x(j) +i ) is a standard Gaussian vector, it is sub-Gaussian and therefore ⟨z, u(j) +i ⟩ and ⟨v(j) +i , y⟩ are sub-Gaussian with +norms ∥ 1 +√n[A⋆]⊤(AA⊤ − Ik)Q∥2 = (1/√n)dist(A, QA⋆) and (1/√n), respectively. In addition, we have +E +� +⟨z, u(j) +i ⟩⟨v(j) +i , y⟩ +� += 1 +nE +� +z⊤ 1 +√n[A⋆]⊤(AA⊤ − Ik)Qφ⋆ +i (x(j) +i )[φ⋆ +i (x(j) +i )]⊤QAy +� += 1 +nz⊤ 1 +√n[A⋆]⊤(AA⊤ − Ik)Ay += 0, +where we have used the fact that E[φ⋆ +i (x(j) +i )[φ⋆ +i (x(j) +i )]⊤] = 1, Q2 = Ik and (AA⊤ − Ik)A = 0. The rest of the proof is +concluded by using the Bernstein inequality by following directly the steps detailed in Collins et al. (2021, Proof of Lemma +3, see equations (35) to (39)). + +Personalised Federated Learning On Heterogeneous Feature Spaces +21 +Lemma S6. Assume H1. Let δd = cd3/2� +log(⌊rb⌋)/n1/2 for some absolute constant c > 0. Then, for any t ∈ [T], with +probability at least 1 − e−111k2 log(⌊rb⌋), we have +���F (t)��� +F ≤ +δd +1 − δd +��B⋆ +At +�� +2 dist +� +A(t), QA⋆� +, +where F (t) is defined in (S12) and A(t) in (S3). +Proof. By the Cauchy-Schwarz inequality, we have +��F (t)�� +F = +��[G(t)]−1(G(t)D(t) − C(t))B⋆ +At +�� +2 ≤ δd +��B⋆ +At +�� +2 ≤ +��[G(t)]−1�� +2 +��(G(t)D(t) − C(t))B⋆ +At +�� +2 ≤ δd +��B⋆ +At +�� +2. The proof is concluded by combining the upper bounds given in +Lemma S4 and Lemma S5. +Lemma S7. Assume H1 and let δ′ +d = cd +√ +k/ +� +⌊rb⌋ n for some absolute positive constant c. For any t ∈ [T] and whenever +δ′ +d ≤ d, we have with probability at least 1 − e−110k − e−110d2 log(⌊rb⌋) +1 +⌊rb⌋ +����� +� 1 +nQ(A(t))†A(t) � +Z(t)� +Q − Z(t) +�⊤ +B(t) +At +����� +2 +≤ δ′ +d dist +� +A(t), QA⋆� +, +where B(t) +At is defined in (S2) and Z(t) is defined in (S4). +Proof. Let t ∈ [T]. Note that we have +� 1 +nQ(A(t))†A(t) � +Z(t)� +Q − Z(t) +�⊤ +B(t) +At = 1 +n +� +i∈At +m +� +j=1 +⟨Qφ⋆ +i (x(j) +i ), z(t) +i ⟩Qφ⋆ +i (x(j) +i ) +� +β(t) +i +�⊤ +− z(t) +i +� +β(t) +i +�⊤ +. +Let Sk−1 and Sd−1 denote the k-dimensional and d-dimensional unit spheres, respectively. Then, by Vershynin (2018, +Corollary 4.2.13), we can define Nk and Nd, 1/4-nets over Sk−1 and Sd−1, respectively, such that |Nk| ≤ 9k and +|Nd| ≤ 9d. Therefore, by using Vershynin (2018, Equation (4.13)), we have +����� +� 1 +nQ(A(t))†A(t) � +Z(t)� +Q − Z(t) +�⊤ +B(t) +At +����� +2 +2 += 2 +max +u∈Nd,v∈Nk u⊤ +� +� 1 +n +� +i∈At +m +� +j=1 +⟨Qφ⋆ +i (x(j) +i ), z(t) +i ⟩Qφ⋆ +i (x(j) +i ) +� +β(t) +i +�⊤ +− z(t) +i +� +β(t) +i +�⊤ +� +� v += 2 +max +u∈Nd,v∈Nk +1 +n +� +i∈At +m +� +j=1 +⟨Qφ⋆ +i (x(j) +i ), z(t) +i ⟩⟨u, Qφ⋆ +i (x(j) +i )⟩⟨β(t) +i , v⟩ − ⟨u, z(t) +i ⟩⟨β(t) +i , v⟩ . +(S17) +In order to control (S17) using Bernstein inequality as in Lemma S5, we need to characterise, in particular, the sub- +Gaussianity of ⟨u, z(t) +i ⟩ and ⟨β(t) +i , v⟩ which require a bound on ∥z(t) +i ∥ and ∥β(t) +i ∥, respectively. From Lemma S1, we have +[β(t) +i ]⊤ = (β⋆ +i )⊤(A⋆)⊤A(t) − (z(t) +i )⊤ which leads to +���z(t) +i +��� +2 += +���QA(t)(A(t))⊤QA⋆β⋆ +i − QA(t)f (t) +i +− A⋆β⋆ +i +��� +2 +2 += +���(QA(t)(A(t))⊤Q − Id)A⋆β⋆ +i − QA(t)f (t) +i +��� +2 +2 +≤ 2 +���(QA(t)(A(t))⊤Q − Id)A⋆��� +2 +2 ∥β⋆ +i ∥2 + 2 +���f (t) +i +��� +2 +≤ 2d dist2(A(t), QA⋆) + 2 +���f (t) +i +��� +2 +. +Using (S12) and the Cauchy-Schwarz inequality, we have +���f (t) +i +��� +2 += +���[Gi,(t)]−1(Gi,(t)Di,(t) − Ci,(t))β⋆ +i +��� +2 + +Personalised Federated Learning On Heterogeneous Feature Spaces +22 +≤ +���[Gi,(t)]−1��� +2 +2 +���Gi,(t)Di,(t) − Ci,(t)��� +2 +2 ∥β⋆ +i ∥2 +≤ d +���[Gi,(t)]−1��� +2 +2 +���Gi,(t)Di,(t) − Ci,(t)��� +2 +2 , +(S18) +where the last inequality follows from H1-(ii). +Using Lemma S4 and Lemma S5 and similarly to Collins et al. (2021, Equation (45)), it follows for any i ∈ At that +���z(t) +i +��� +2 +2 ≤ 4d dist(A(t), QA⋆) , +with probability at least 1 − e110d2 log(⌊rb⌋). +Similarly, using Lemma S1 and (S18), we have with probability at least 1 − e110d2 log(⌊rb⌋) and for any i ∈ At, that +���β(t) +i +��� +2 +≤ 2 +���[A(t)]⊤QA⋆β⋆ +i +��� +2 ++ 2 +���f (t) +i +��� +2 +≤ 4d . +Besides, note we have +E +� +⟨Qφ⋆ +i (x(j) +i ), z(t) +i ⟩⟨u, Qφ⋆ +i (x(j) +i )⟩⟨β(t) +i , v⟩ +� += ⟨u, z(t) +i ⟩⟨β(t) +i , v⟩ . +The proof is then concluded by applying the Bernstein inequality following the same steps as in the final steps of Collins +et al. (2021, Proof of Lemma 5). +We are now ready to control C2. +Lemma S8. Assume H1 and let δ′ +d = cd +√ +k/ +� +⌊rb⌋ n for some absolute positive constant c. For any t ∈ {0, . . . , T − 1}, +η > 0 and whenever δ′ +d ≤ d, we have with probability at least 1 − e−110k − e−110d2 log(⌊rb⌋) +C2 ≤ ηδ′ +d dist +� +A(t), QA⋆� ���� +� +R(t+1)�−1���� +2 +, +where C2 is defined in (S14), A(t) is defined in (S3) and R(t) comes from the QR factorisation of ¯A(t), see step 20 in +Algorithm S3. +Proof. Let t ∈ {0, . . . , T − 1} and η > 0. Then, whenever δ′ +d ≤ d, we have with probability at least 1 − e−110k − +e−110d2 log(⌊rb⌋), we have +C2 = +η +⌊rb⌋ +����� +� 1 +n[A⋆ +⊥]⊤(QA(t+1))†A(t+1) � +Z(t+1)� +Q − Z(t+1) +�⊤ +B(t+1) +At+1 +����� +2 +���� +� +R(t+1)�−1���� +2 +≤ +η +⌊rb⌋ +����� +� 1 +n(QA(t+1))†A(t+1) � +Z(t+1)� +Q − Z(t+1) +�⊤ +B(t+1) +At+1 +����� +2 +���� +� +R(t+1)�−1���� +2 +≤ ηδ′ +d dist +� +A(t), QA⋆� ���� +� +R(t+1)�−1���� +2 +, +where we used the Cauchy-Schwarz inequality in the second inequality and Lemma S7 for the last one. +Control of ∥ +� +R(t+1)�−1 ∥2. To finalise our proof, it remains to bound ∥ +� +R(t+1)�−1 ∥2. The associated result is depicted +in the next lemma. +Lemma S9. Define ¯δd = δd + δ′ +d where δd and δ′ +d are defined in Lemma S4 and Lemma S5, respectively. Assume H1. +Then, we have with probability at least 1 − e−110k − e−110d2 log(⌊rb⌋), +���� +� +R(t+1)�−1���� +2 +≤ +� +1 − 4η +¯δd +(1 − ¯δd)2 ¯σ2 +max,⋆ +�−1/2 +. +Proof. The proof follows from Collins et al. (2021, Proof of Lemma 6). + +Personalised Federated Learning On Heterogeneous Feature Spaces +23 +S3. Experimental Details +S3.1. Reference Distribution for Regression +For regression problem, our goal is to map all samples for all clients into a common latent subspace, in which some +structural information about regression problem is preserved. As such, in order to reproduce the idea of using a Gaussian +mixture model as a anchor distribution, we propose to use an infinite number of Gaussian mixtures in which the distribution +of x associated to a response y is going to be mapped on a unit-variance Gaussian distribution whose mean depends +uniquely on y. Formally, we define the anchor distribution as +µy = N(m(y), I) +where m(y) is a vector of dimension d that is uniquely defined. In practice, we consider as m(y) = ya + (1 − y)b where a +and b are two vectors in Rd. +When training FLIC, this means that for a client i, we can compute W2 +2 +� +µy, ν(y) +φi +� +based on the set of training samples +{x, y}. In practice, if for a given batch of samples we have a single sample x, then the Wasserstein distance boils to +∥φi(x) − m(y)∥2 +2. +S3.2. Data Sets +We provide some details about the datasets we used for our numerical experiments +S3.2.1. TOY DATA SETS +The first toy dataset, denoted as noisy features, is a 20-class classification problem in which the features for a given class +is obtained by sampling a Gaussian distribution of dimension 5, with random mean and Identity covariance matrix. For +building the training set, we sample 2000 examples for each class and equally share those examples among clients who +hold that class. Then, in order to generate some class imbalances on clients, we randomly subsample examples on all +clients. For instance, with 100 clients and 2 classes per clients, this results in a problem with a total of about 16k samples +with a minimal number of samples of 38 and a maximal one of 400. In order to get different dimensionality, we randomly +append on each client dataset some Gaussian random noisy features with dimensionality varying from 1 to 10. +The second toy dataset, denoted as linear mapping, is a 20-class classification problem where each class-conditional +distribution is Gaussian distribution of dimension 5, with random mean and random diagonal covariance matrix. As above, +we generate 2000 samples per class and distribute and subsample them across clients in the similar way, leading to a total +number of samples of about 15k. The dimensionality perturbation is modelled by a random (Gaussian)linear transformation +that maps the original samples to a space which dimension goes up to 50. +S3.2.2. MNIST-USPS +We consider a digit classification problem with the original MNIST and USPS data sets which are respectively of dimension +28 × 28 and 16 × 16 and we assume that a client hosts either a subset of MNIST or USPS data set. We use the natural +train/test split of those datasets and randomly share them accross clients. +S3.2.3. TEXTCAPS DATA SET +The TextCaps data set (Sidorov et al., 2020) is an Image captioning dataset for which goal is to develop a model able to +produce a text that captions the image. The dataset is composed of about 21k images and 110k captions and each image +also comes with an object class. For our purpose, we have extracted pair of 14977 images and captions from the following +four classes Bottle, Car, Food and Book. At each run, those pairs are separated in 80% train and 20% test sets. Examples +from the TextCaps datasets are presented in Figure S5. Images and captions are represented by vectors by feeding them +respectively to a pre-trained ResNet18 and a pretrained Bert, leading to vectors of size 512 and 768. +Each client holds either the image or the text representation of subset of examples and the associated vectors are randomly +pruned of up to 10% coordinates. As such, all clients hold dataset with different dimensionality. + +Personalised Federated Learning On Heterogeneous Feature Spaces +24 +Figure S1. Evolution of the local loss curve of three different clients for three different learning situations. See text for details. +S3.3. Models and Learning Parameters +For the toy problems and the TextCaps data set, as a local transformation functions we used a fully connected neural +network with one input, one hidden layer and one output layers. The number of units in hidden layer has been fixed to 64 +and the dimension of latent space as been fixed to 64. ReLU activation has been applied after the input and hidden layers. +For the digits dataset, we used a CNN model with 2 convolutional layers followed by a max-pooling layer and a sigmoid +activation function. Once flattened, we have a one fully-connected layer and ReLU activation. The latent dimension is +fixed to 64. +For all datasets, as for the local model gθi, in order to be consistent with competitors, we first considered a single layer linear +model implementing the local classifier as well as a model with one input layer (linear units followed by a LeakyReLU +activation funcion) denoting the shared representation layer and an output linear layer. +For training, all methods use Adam with a default learning rate of 0.001 and a batch size of 100. Other hyperparameters +have been set as follows. Unless specified, the regularization strength λ1 and λ2 have been fixed to 0.001. Local sample +batch size is set to 100 and the participation rate r to 0.1. For all experiments, we have set the number of communication +round T to 50 and the number of local epochs to respectively 10 and 100 for the real-world and toy datasets. For FLIC, as +in FedRep those local epochs is followed by one epoch for representation learning. We have trained the local embedding +functions for 100 local epochs and a batch size of 10 for toy datasets and TextCaps and while of 100 for MNIST-USPS. +Reported accuracies are computed after local training for all clients. +S3.4. Ablating Loss Curves +In order to gain some understanding on the learning mechanism that involves local and global training respectively due to +the local embedding functions, the local classifier and the global representation learning, we propose to look at local loss +curves across different clients. +Here, we have considered the linear mapping toy dataset as those used in the toy problem analysis. However, the learning +parameters we have chosen are different from those we have used to produce the results so as to highlight some specific +features. The number of epochs (communication rounds) is set to 100 with a client activation ration of 0.1. Those local +epochs are shared for either training the local parameters or the global ones (note that in our reference Algorithm 1, the +global parameter is updated only once for each client) Those latter are trained starting after the 20-th communication round +and in this case, the local epochs are equally shared between local and global parameter updates. Note that because of the +randomness in the client selection at each epoch, the total number of local epochs is different from client to client. We have +evaluated three learning situations and plotted the loss curves for each client. +• the local embedding functions and the global models are kept fixed, and only the local classifier is trained. Examples of +loss curves for 3 clients are presented in the left plot of Figure S1. For this learning situation, there is no shared global +parameters that are trained locally. Hence, the loss curve is typical of those obtained by stochastic gradient descent +with a smooth transition, at multiple of 100 local epochs, when a given client is retrained after a communication +rounds. +• the local embedding functions are kept fixed, while the classifier and global parameters are updated using half of the + +3.0 +clients 10 +2.625 +clients 50 +2.9 +clients 87 +2.600 +180 +200 +220 +2.8 +S 2.7 +S +2.3795 +2.6 +2.3790 +2.5 +780 +800 +820 +2.4 +2.3 +0 +200 +400 +600 +800 +#local epochs3.0 +clients 10 +2.65 +2.9 +2.60 +clients 50 +2.55 +2.8 +clients 87 +180 +200 +220 +2.7 +S +S +2.6 +0 +2.4 +2.5 +2.3 - +2.4 +800 +820 +2.3 +2.2 +0 +200 +400 +600 +800 +#local epochs30 +clients 10 +6.0 +clients 50 +25 +clients 87 +5.5 +180 +200 +220 +20 +ss +0 +15 +3.5 +3.4 +10 +780 +800 +820 +5 +0 +200 +400 +600 +800 +#local epochsPersonalised Federated Learning On Heterogeneous Feature Spaces +25 +Figure S2. Impact of epochs used for pretraining φi on the model accuracy as well as updating those functions during the training. +Results for three different datasets are reported. Plain and dashed curves are respectively related to local training with and without +updates φi. +local epochs each. This situation is interesting and reported in middle plot in Figure S1. We can see that for some +rounds of 100 local epochs, a strong drop in the loss occurs at starting at the 50th local epoch because the global +parameters are being updated. Once the local update of a client is finished the global parameter is sent back to the +server and all updates of global parameters are averaged by the server. When a client is selected again for local +updates, it is served with a new global parameter (hence a new loss value ) which causes the discontinuity in the loss +curve at the beggining of each local update. +• all the part (local embedding functions, global parameter and the classifier) of the models are trained. Note at first that +the loss value for those curves (right plot in Figure S1) is larger than for the two first most left plots as the Wasserstein +distance to the anchor distribution is now taken into account and tends to dominate the loss. The loss curves are +globally decreasing with larger drops in loss at the beginning of local epochs. +S3.5. On Importance of Alignment Pre-Training and Updates. +We have analyzed the impact of pretraining the local transformation functions and their updates during learning for fixed +reference distribution. We have considered two learning situations : one in which they are updated during local training +(as usual) and another one in they are kept fixed all along the training. We have chosen the setting with 100 users and have +kept the same experimental settings as for the performance figure and made only varied the number of epochs considered +for pretraining from 1 to 200. Results, averaged over 5 runs are shown in Figure S2. We remark that for the three datasets, +increasing the number of epochs up to a certain number tends to increase performance, but overfitting may occur. The latter +is mostly reflected in the toy linear mapping dataset for which 10 to 50 epochs is sufficient for good pretraining. Examples +of how classes evolves during pretraining are illustrated in Figure 4, through t-sne projection. We also illustrate cases of +how pretraining impact on the test set and may lead to overfitting as shown in the supplementary Figure S4. +S3.6. On the Impact of the Participation Rate +We have analyzed the effect of the participation rate of each client into our federated learning approach. Figure S3 reports +the accuracies, averaged over 3 runs, of our approach for the toy datasets and the TextCaps problem with respect to the +partication rate at each round. We can note that the proposed approach is rather robust to the participation rate but may +rather suffer from overfitting due to overtraining of local models. On the left plot, performances, measured after the last +communication round, for TextCaps is stable over participation rate while those performances tend to decrease for the +toy problems. We associate these decrease to overfitting since when we report (see right plot) the best performance over +communication rounds (and not the last one), they are stable for all problems. This suggests that number of local epochs +may be dependent to the task on each client and the client participation rate. + +90 +TextCaps +Toy NF +Toy LM +85 +TextCaps -fixed- +Toy NF -fixed- +Accuracy (%) +80 +Toy LM -fixed- +75 +70 +65 +60 +0 +25 +50 +75 +100 +125 +150 +175 +200 +#pretraining epochsPersonalised Federated Learning On Heterogeneous Feature Spaces +26 +Figure S3. Evolution of the performance of our FLIC-Class algorithm with respects to the participation rate of clients, using the same +experimental setting as in Figure 3. (left) evaluating performance after last communication rounds, (right) best performance across +communication rounds. +Class 1 +Class 3 +Class 5 +Class 6 +Class 11 +Class 1 +Class 3 +Class 5 +Class 6 +Class 11 +Class 1 +Class 3 +Class 5 +Class 6 +Class 11 +Class 1 +Class 3 +Class 5 +Class 6 +Class 11 +Class 1 +Class 3 +Class 5 +Class 6 +Class 11 +Class 1 +Class 3 +Class 5 +Class 6 +Class 11 +Figure S4. . 2D t-sne projection of 5 classes partially shared by 3 clients for the toy linear mapping dataset after learning the local +embedding functions for (left) 10 epochs, (middle) 50 epochs, (right) 100 epochs. Original dimensions on clients vary from 5 to 50. Top +row shows the projection the training set while bottom row plots show both training and test set. Star ⋆ markers represent the projection +of the mean of each class-conditional. The three different marker styles represent the different clients. Classes are denoted by colors +and similar tones of color distinguish train and test sets. We see that each class from the training set from each client converges towards +the mean of its anchor distribution, represented by the star marker. Interestingly, we also remark that unless convergence is reached, +empirical class-conditional distributions on each clients are not equal making necessary the learning of a joint representation. From the +bottom plots, we can understand that distribution alignment impacts mostly the training set but this alignment does not always generalize +properly to the test sets. + +90 +85 +Accuracy (%) +TextCaps +80 +Toy NF +Toy LM +75 +70 +0.05 +0.15 +0.20 +0.25 +0.00 +0.10 +0.30 +Participation rate90.0 +87.5 +85.0 +Accuracy (%) +82.5 +TextCaps +Toy NF +80.0 +Toy LM +77.5 +75.0 +72.5 +70.0 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +Participation ratePersonalised Federated Learning On Heterogeneous Feature Spaces +27 +Figure S5. Examples of some TextCaps pairs of image/caption from the 4 classes we considered of (top-left) Food, (top-right) Bottle, +(bottom-left) Book (bottom-right) Car. We can see how difficult some examples can be, especially from the caption point of view since +few hint about the class is provided by the text. + +Apansitsonahobwitha lidbearing +thelogoHamiltonBeachStaynGo +STAYEGODecidingonwhethertodrinkspringwaterora7UP +iana +NATURALSPRINGWATER +rSvetiaJiuta +0251Thebookshelf consists ofabout2odifferenttypesof books.TheyellowconvertibleisondisplaysomewhereinCalifornia \ No newline at end of file diff --git a/JNFJT4oBgHgl3EQfGSzR/content/tmp_files/load_file.txt b/JNFJT4oBgHgl3EQfGSzR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4abd3b1348ea6bbfb325a5b77ba8f5f0ef89031c --- /dev/null +++ b/JNFJT4oBgHgl3EQfGSzR/content/tmp_files/load_file.txt @@ -0,0 +1,1846 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfGSzR/content/2301.11447v1.pdf,len=1845 +page_content='Personalised Federated Learning On Heterogeneous Feature Spaces Alain Rakotomamonjy * 1 Maxime Vono * 1 Hamlet Jesse Medina Ruiz 1 Liva Ralaivola 1 Abstract Most personalised federated learning (FL) ap- proaches assume that raw data of all clients are defined in a common subspace i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfGSzR/content/2301.11447v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfGSzR/content/2301.11447v1.pdf'} +page_content=' all clients store their data according to the same schema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfGSzR/content/2301.11447v1.pdf'} +page_content=' For real-world applications, this assumption is restrictive as clients, having their own systems to collect and then store data, may use heteroge- neous data representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfGSzR/content/2301.11447v1.pdf'} +page_content=' We aim at filling this gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfGSzR/content/2301.11447v1.pdf'} +page_content=' To this end, we propose a general frame- work coined FLIC that maps client’s data onto a common feature space via local embedding func- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfGSzR/content/2301.11447v1.pdf'} +page_content=' The common feature space is learnt in a federated manner using Wasserstein barycenters while the local embedding functions are trained on each client via distribution alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfGSzR/content/2301.11447v1.pdf'} +page_content=' We in- tegrate this distribution alignement mechanism into a federated learning approach and provide the algorithmics of FLIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfGSzR/content/2301.11447v1.pdf'} +page_content=' We compare its per- fomances against FL benchmarks involving het- erogeneous input features spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfGSzR/content/2301.11447v1.pdf'} +page_content=' In addition, we provide theoretical insights supporting the rele- vance of our methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfGSzR/content/2301.11447v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfGSzR/content/2301.11447v1.pdf'} +page_content=' Introduction Federated learning (FL) is a machine learning paradigm where models are trained from multiple isolated data sets owned by individual agents (coined clients), without re- quiring to move raw data into a central server, nor even share them in any way (Kairouz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfGSzR/content/2301.11447v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfGSzR/content/2301.11447v1.pdf'} +page_content=' This framework has lately gained a strong traction from both industry and academic research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfGSzR/content/2301.11447v1.pdf'} +page_content=' Indeed, it avoids the communication costs entailed by data transfer, allows all clients to benefit from participating to the learning co- hort, and finally, it fulfills first-order confidentiality guar- antees, which can be further enhanced by resorting to so- called privacy-enhancing technologies such as differential privacy (Dwork and Roth, 2014) or secure multi-party com- Equal contribution 1Criteo AI Lab, Paris, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNFJT4oBgHgl3EQfGSzR/content/2301.11447v1.pdf'} +page_content=' Correspon- dence to: Maxime Vono 2, and close to the non-illuminated surface of the +semi-infinite atmosphere. +We recover easily, using our +new numerical procedures, the “overpopulation” of f2 at +large u’s already put in evidence by Borsenberger et al. +(1987). +Such a material is only accessible using the Oxenius- +like formalism that we adopted. Note also that the de +facto neglected potential effects of velocity-changing elas- +tic collisions can only be addressed, and studied in that +theoretical frame (see details in PP21 and §7 hereafter). +More generally, such additional information could be very +valuable for the detailed coupling between the radiation +transfer problem and any other physical processes that +would take place within an atmosphere, and for which the +very knowledge of the various VDF’s of contributing ele- + +0.05 +0.25 +0.5 +1 +4 +17 +68 +0.4 +270 +e +profile +Emission +0.3 +E +0.2 +0.1 +0.0 +0 +1 +2 +m +4 +5 +x0.05 +0.25 +25 +1 +4 +17 +68 +20 +270 +normalized f2 +15 +10 +5 +0 +0 +1 +2 +m +4 +5 +u6 +FIG. 4. Variation with frequency of the normalized source +function at τ ≈ 1, for different values of the velocity-changing +elastic collision parameter ζ (indicated in the top-left frame). +The model-atmosphere is the same as the one considered in §6 +and Hummer (1969). As expected, successive solutions range +between standard PRD-RI−A values (blue) and the limit of +the CRD solution (dashed line) for increasing values of ζ. +ments, at different excitation and ionization stages would +be critical. +VII. +NON-ZERO VELOCITY-CHANGING +ELASTIC COLLISIONS +To the best of our knowledge, such preliminary compu- +tations were only addressed by Atanackoviˇc et al. (1987) +so far. +We now go a bit further by illustrating in Fig. (4) +the dependence of the source function at τ ≈ 1 on +the velocity-changing elastic collisions parameterized as +ζ. The model-atmosphere used for this computation is +identical to the one adopted in §6, but we have made +vary ζ from 0 to 50. As expected, the source function +S(x, τ) ranges between the standard PRD-RI−A solutions +when ζ = 0, and the frequency independent CRD values +(shown as black dashed line) for increasing values of ζ. +As also expected, the numerical problem becomes eas- +ier for increasing values of ζ. Effects of velocity-changing +elastic collisions will be discussed in more details in an- +other devoted study. +FIG. 5. +Deviations from Maxwellian illustrated by the +changes of the normalized f2 at various optical depths (men- +tioned in the top-left frame) across an asymmetrically and +strongly illuminated finite slab of total depth τ += 103. +Largest amplitudes are for these two values around τ = 1, +while smaller but significant deviations are still noticeable at +midslab (τ = 500). +VIII. +A FINITE SLAB CASE +Among numerous applications, we are particularly in- +terested in the radiative modelling of isolated and illumi- +nated finite slabs. As an example of expected effects, we +simulated a 1D plane parallel horizontal slab strongly ir- +radiated asymmetrically, only from below. This mimics +the radiative modelling of a “cold filament” suspended +above a stellar disk (see e.g., Paletou 1997). +We used a 33 points optical depth grid, logarithmically +spaced away from both open surfaces (using an initial +δτ = 0.01), and symmetric around a midslab depth set +at τ1/2 = 500. For this example we also set ε = 10−4 and +ζ = 0. External illumination is applied only at the bot- +tom surface, using a flat profile of normalized intensity +Iext. = 3. +Figure (5) displays values of the normalized VDF +f2(u, τ)/f M(u) for optical depth values, counted from +the top surface, of τ = 0.58 and τ = 1.15 i.e., around +the critical value of 1, and at midslab. The quite strong +external illumination that we applied generates very sig- +nificant departures of f2 from Maxwellian, especially at +u > 2 again, but larger than what we already identified +for the semi-infinite atmosphere case. + +0.6 +0.25 +0.5 +0.8 +1 +3 +5 +10 +-1.0 +30 +50 +log(S) +-1.2 +-1.4 +1.6 +1.8 +0 +1 +2 +3 +4 +5 +X70 +0.6 +1.15 +500 +60 +50 + normalized +40 +30 +20 +10 +0 +0 +1 +2 +m +4 +5 +u7 +IX. +DISCUSSION AND CONCLUSION +A first critical evaluation of new numerical procedures +in radiative transfer for the solution of the “full non– +LTE” problem have been conducted. +They were vali- +dated by reproducing both CRD and standard PRD with +Hummer’s RI−A (Hummer 1962, 1969). This will allow +us to now move forward in several directions. +After the very first computations shown here, we shall +also be able to evaluate further, in more details, the ad- +ditional effects of potential velocity-changing elastic col- +lisions with different set of “classic” parameters. +Also +finite slab models, with different conditions of external +illumination will be considered. Such cases lead to more +significant departures from Maxwellian than what hap- +pens for semi-infinite slabs, as indicated by our prelim- +inary filament-like computation. Relevant astrophysical +“objects” should range from solar prominences to circum- +stellar environements for instance. +The next obvious step is to modify the atomic ab- +sorption description for the more realistic case of natural +broadening of the upper level of the transition. It would +go beyond the previous studies of Borsenberger et al. +(1987) and Atanackoviˇc et al. (1987) which were limited +to “pure” Doppler broadening. Therefore, a Lorentzian +profile for atomic absorption will be used. This will also +require to implement modifications to the original formal- +ism of Oxenius for that case, following rather Bommier’s +(1997) approach, as discussed in PP21. This will be suit- +able for a preliminary study of resonance lines such as Ly- +man α of H i for instance. The successive computations +of Voigt-like profiles at every iterative step, according to +the departures of f2 to Maxwellian, and throughout the +whole atmosphere will certainly benefit from the numer- +ical scheme proposed by Paletou et al. (2020). This may +also require us to implement and validate a more robust +iterative scheme, more likely inspired by the so-called +FBF scheme proposed by Paletou & Auer (1995). +Then we shall proceed with the consideration of at +least an additional distribution, either for another excited +state or for free electrons. This should lead to an alter- +native, less heuristic, approach to the so-called “cross- +redistribution” model for the multi-level atoms case (see +e.g., Milkey et al. +1975, Hubeny & Lites 1995, Sam- +poorna & Nagendra 2017). +ACKNOWLEDGMENTS +M.S. acknowledges the support from the Science +and +Engineering +Research +Board +(SERB), +Depart- +ment of Science and Technology, Government of India +via a SERB-Women Excellence Award research grant +WEA/2020/000012. +[1] Atanackoviˇc, O., Borsenberger, J., Oxenius, J., Simon- +neau, E. 1987, JQSRT, 38, 427 +[2] Bommier, V. 1997, A&A, 328, 706 +[3] Borsenberger, J., Oxenius, J., Simonneau, E. 1986, +JQSRT, 35, 303 +[4] Borsenberger, J., Oxenius, J., Simonneau, E. 1987, +JQSRT, 37, 331 +[5] Hubeny, I. & Lites, B. 1995, ApJ, 455, 376 +[6] Hubeny, +I. & Mihalas, +D. 2014, +Theory of Stellar +Atmospheres: +An Introduction to Astrophysical Non- +equilibrium Quantitative Spectroscopic Analysis, Prince- +ton University Press +[7] Hummer, D.G. 1962, MNRAS, 125, 21 +[8] Hummer, D.G. 1969, MNRAS, 145, 95 +[9] Lambert, J., Paletou, F., Josselin, E., Glorian, J.-M. +2016, Eur. J. Phys, 37, 015603 +[10] Milkey, R.W., Shine, R.A., Mihalas, D. 1975, ApJ, 199, +718 +[11] Oxenius, J. 1986, Kinetic theory of particles and pho- +tons – Theoretical foundations of non–LTE plasma spec- +troscopy, Springer +[12] Paletou, F. 1997, A&A, 317, 244 +[13] Paletou, F., Auer, L.H. 1995, A&A, 297, 771 +[14] Paletou, F., Bommier, V., Faurobert, M. 1999, in So- +lar polarization, K.N. Nagendra and J.O. Stenflo eds., +Kluwer +[15] Paletou, F., Peymirat, C. 2021, A&A, 649, A165 +[16] Paletou, F., Peymirat, C., Anterrieu, E., B¨ohm, T. 2020, +A&A, 633, A111 +[17] Sampoorna, M., Nagendra, K.N. 2017, ApJ, 838, 95 +[18] Sampoorna, M., Nagendra, K.N., Frisch, H. 2011, A&A, +527, A89 + diff --git a/LtE4T4oBgHgl3EQfJwyq/content/tmp_files/load_file.txt b/LtE4T4oBgHgl3EQfJwyq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f76570d6b1dfd7a2d89309ebef985407b63f3ff0 --- /dev/null +++ b/LtE4T4oBgHgl3EQfJwyq/content/tmp_files/load_file.txt @@ -0,0 +1,351 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf,len=350 +page_content='Full non–LTE spectral line formation II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Two–distribution radiation transfer with coherent scattering in the atom’s frame Fr´ed´eric Paletou∗ Universit´e de Toulouse,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Observatoire Midi–Pyr´en´ees,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Cnrs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Cnes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Irap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' F–31400 Toulouse,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' France Malali Sampoorna† Indian Institute of Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Koramangala,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Bengaluru 560034,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Karnataka,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' India Christophe Peymirat‡ Universit´e de Toulouse,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Facult´e des Sciences et d’Ing´enierie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' F–31062 Toulouse cedex 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' France (Dated: January 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' 2023) In the present article,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' we discuss a numerical method of solution for the so-called “full non-LTE” radiation transfer problem,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' basic formalism of which was revisited by Paletou & Peymirat (2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' see also Oxenius 1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' More specifically, usual numerical iterative methods for non-LTE radiation transfer are coupled with the above-mentioned formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' New numerical additions are explained in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' We benchmark the whole process with the standard non-LTE transfer problem for a two-level atom with Hummer’s (1962, 1969) RI−A partial frequency redistribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' We finally display new quantities such as the spatial distribution of the velocity distribution function of excited atoms, that can only be accessed to by adopting this more general frame for non-LTE radiation transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' INTRODUCTION Oxenius (1986) formulated the so-called “full non- LTE” radiation transfer problem, wherein the distribu- tion of photons as well as the massive particles in a stel- lar atmosphere may in general both deviate from their equilibrium distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Paletou & Peymirat (2021, hereafter PP21) revisited this formalism and rediscussed some basic elements using standard notations prevalent in this field of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' In PP21, we however, lim- ited ourselves to the more detailed statistical equilibrium equations for the simplest case of a “two-distribution” model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' The present study is devoted to the coupling of that formalism with usual numerical methods used for radia- tion transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' The set and sequence of the new quantities required for our computations are outlined in §2 here- after, in the continuation and sometimes the generaliza- tion of relationships previously discussed in PP21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' The validation of these new elements, dealing directly with the coupling of the frequency dependence of the ra- diation and the velocities of the atoms scattering light, is first made for the case of coherent scattering in the atom’s frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' We can indeed show – see §3 – that the latter assumption for our two-distribution problem makes it equivalent to a two-level atom problem with the standard angle-averaged partial frequency redistribution (PRD) function RI−A for isotropic scattering introduced by Hummer (1962).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Useful details about the very numerical implementa- tion of our new computations are exposed in §4 and a ∗ frederic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='paletou@univ-tlse3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='fr † sampoorna@iiap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='in ‡ christophe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='peymirat@univ-tlse3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='fr simple iterative scheme is described in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' We fully rely on a nowadays quite usual short-characteristics formal solver, mostly used for iterative methods such as accel- erated Λ-iteration (hereafter ALI), whose material was developed and made available by us1 in Python (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=', Lambert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' 2016 and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' A first, indispensable validation is presented in §6, by reproducing results for standard PRD-RI−A of Hummer (1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' It is however shown that, the two-distribution model provides additional physical quantities, which can- not be accessed to using the classical non-LTE frame- work, such as the spatial distribution of the velocity dis- tribution function (hereafter VDF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' see §6 and 8) of the excited atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Sections 7 and 8 therefore present, respectively, a pre- liminary exploration of the effects of velocity-changing elastic collisions, and new computations for a strongly illuminated finite slab case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' We finally discuss the various perspectives of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' COUPLING TO RADIATION TRANSFER The implementation of our kinetic approach together with radiation transfer requires, in a first place, two ma- jor modifications of existing numerical radiation transfer tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' In the following, and after Paletou & Peymirat (2021), we shall adopt both the reduced frequency x, and the normalized atomic velocity u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' The reduced frequency is usually defined in radiation transfer as x = (ν −ν0)/∆νD 1 https://hal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='archives-ouvertes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='fr/hal-02546057 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='04924v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='IM] 12 Jan 2023 2 that is the frequency shift from the line center normal- ized to the Doppler width ∆νD = (ν0/c)vth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=', where c is the speed of light;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' while u is the modulus of the atomic velocity normalized to the “most probable ve- locity” vth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' = � 2kT/M, with k being the Boltzmann constant, T the temperature, and M the mass of the atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' The first requirement is the computation of the scat- tering integral J12 defined as: J12(⃗u, τ) = � dΩ 4π � δ(x − ⃗u · ⃗Ω)I(x, ⃗Ω, τ)dx , (1) in the case of coherent scattering in the atomic frame, where δ is the Dirac distribution for the a priori known absorption profile and I(x, ⃗Ω, τ) the usual specific inten- sity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' The latter is computed at every frequency x, pho- ton propagation direction ⃗Ω, and optical depth τ using a “classic” short characteristic based formal solver (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Lambert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' 2016, and associated resources).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Once this quantity is available, our modified formal solver calls a specific function which performs the angular integration over the Dirac distribution δ(x − ⃗u · ⃗Ω), adapted from Sampoorna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' (2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' see also §4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' In a second step, one computes ¯J12(⃗u, τ) = J12(⃗u, τ)/BW with BW denoting the Planck function in the Wien limit, and: J12(τ) = � ⃗u ¯J12(⃗u, τ)f M(⃗u)d3⃗u , (2) with f M(⃗u) = e−⃗u·⃗u/π3/2 for the Maxwellian veloc- ity distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Here d3⃗u = u2dudΩu wherein dΩu = sin θudθudφu, with θu and φu denoting the polar angles of the normalized atomic velocity vector ⃗u about the at- mospheric normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Once we have computed these two quantities, we can evaluate the velocity distribution function of the first ex- cited state of the atom2 using, as defined in PP21: f2(⃗u, τ) = � ζ 1 + ζ + � 1 1 + ζ � ε + (1 − ε) ¯J12(⃗u, τ) ε + (1 − ε)J12 � f M(⃗u) , (3) at every depth in the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' In this expression, ε is the usual collisional destruction probability of standard non–LTE radiation transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' However, another quan- tity ζ appears now, which characterizes the amount of velocity-changing elastic collisions defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' (27) of PP21 (note again that in PP21 the depth dependence of the different quantities such as ¯J12, f2 etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' were omitted, as radiative transfer was not considered in detail at this stage).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' The next critical quantity is the computation of the emission profile, which is defined, at a given optical depth τ in our (1D) atmosphere as: ψ(x, τ) = � dΩ 4π � ⃗u δ(x − ⃗u · ⃗Ω)f2(⃗u, τ)d3⃗u , (4) where f2 is the VDF of the excited atoms scattering light, computed using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' At this stage, we apply Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='1) of PP21 to first perform the integral over dΩ and then perform the integral over dΩu to give: ψ(x, τ) = 2 √π � ∞ |x| � ζ 1 + ζ + � 1 1 + ζ � ε + (1 − ε) ˜J12(u, τ) ε + (1 − ε)J12 � ue−u2du , (5) 2 In the present study, following PP21 and Oxenius (1986) we do not consider departure of the VDF of the fundamental state of the atom from Maxwellian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' This is a realistic assumption in the so-called “weak radiation field regime” when stimulated emission can be neglected, leading to a “natural population” of the lower level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' This is also the case for an atomic ground level of infinite lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' where: ˜J12(u, τ) = � ¯J12(⃗u, τ)dΩu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' (6) Clearly, the emission profile given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' (5) above rep- resents the generalization of the same analytical angular integration result in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' (42) of PP21 which however was written only for the ζ = 0 and ε = 0 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Once all the quantities defined by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' (1)–(5) have been evaluated, one may then compute the source func- 3 tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Hereafter we shall use the below expression, not given explicitly in PP21 though, for the source function: S(x, τ) = [ε + (1 − ε)J12(τ)] �ψ(x, τ) ϕ(x) � , (7) where ϕ is the so-called Doppler absorption profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' It is quite straightforward to establish, and this is indeed the very expression of the source function that we used in the present study (note that this expression remains unchanged when ζ ̸= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' EQUIVALENCE WITH HUMMER’S RI−A REDISTRIBUTION It is easy to show that a “two-distribution” model as- suming a Maxwellian distribution for f1, characterizing the fundamental level, and coherent scattering in the atom’s frame is equivalent to a standard PRD model for a two–level atom problem considering Hummer’s angle- averaged (and isotropic scattering) RI−A redistribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' It therefore provides a critical benchmark for our modi- fied numerical tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' A way to verify this equivalence is by using the de- veloped expression of the emission profile given by the combination of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' (3) and (4) together with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' This should also be considered for ζ = 0 since this new parameter is not relevant to standard PRD using Hum- mer’s redistribution functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Then, one may split in two the expression of the emission profile with a first part which goes like εf M(⃗u), and another one which implies (1−ε) ¯J12(⃗u, τ)f M(⃗u) – besides the common denominator which is independent from ⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' The first part, after inte- grations, will lead to the Doppler absorption profile ϕ, since it is the convolution of the Dirac function charac- terizing coherent scattering in the atom’s frame with the Maxwellian VDF f M(⃗u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' But most interesting is how- ever the second part involving ¯J12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' For this second part we go back to real frequencies ξ (in the atomic frame) and ν (in the observer’s frame), from x and the Doppler transform (see also notations adopted in PP21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Now the Dirac function appearing in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' (4) becomes δ(ξ−ν0), while that in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' (1) when substituted in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' (4) becomes δ(ξ′ − ν0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Thus, the second part of the emission profile contains the product of two Dirac distributions, namely δ(ξ−ν0)δ(ξ′−ν0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' One can also rewrite this combination as: δ(ξ − ν0)δ(ξ′ − ξ) which is indeed the atomic redis- tribution function rI (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=', Hubeny & Mihalas, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' This has also been discussed in §4 of Borsenberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' After integration over velocities along the Maxwellian VDF, and after angular integration, we finally recover the more usual form of the standard PRD, frequency- dependent source function (normalized to the Planckian): S(x, τ) = ε + (1 − ε) � dΩ′ 4π � x′ �RI−A(x′, x) ϕ(x) � I(x′, Ω′, τ)dx′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' (8) And the solution of such a problem can easily be com- puted using the methods proposed by Paletou & Auer (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' NUMERICAL IMPLEMENTATION Our numerical implementation of the new, full non- LTE problem relies on modifications brought to a nowa- days classic short-characteristics (hereafter SC) based formal solver (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Lambert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' 2016, and ref- erences therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' This, together with various iterative schemes which can be set on that basis constitutes a more efficient way, both fast and accurate, as well as easy to develop on, to address the problem than what was made previously by Borsenberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' (1986, 1987) and Atanackoviˇc et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' The main numerical problem now, as compared to the standard non-LTE problem, is to properly evaluate ex- pressions given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' (1) and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' First is the angular integration for making J12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' It consists in using a quadra- ture in polar angle and azimuth (θ, φ) both for the ray direction and for the atomic velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Then, one may write: ⃗u · ⃗Ω = γu where: γ = cos(θr) cos(θu) + sin(θr) sin(θu) cos(φr − φu) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' (9) Here indices r and u are respectively associated with the ray and the atomic velocity directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Once γ has been computed for every couple of polar angle and azimuth, the specific intensity is interpolated in x in order to estimate the I(x = γu) quantity that will contribute to the integral leading to J12(⃗u, τ), after a first integration in frequency along the atomic absorption pro- file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' A mere linear interpolation was used together with our identical x and u grids, hereafter spanning 6 Doppler width in x, with ∆u = ∆x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' To achieve this very task, we use a dedicated function which is called in from the usual SC formal solver, once the specific intensity is available at all frequencies, at a given depth and direc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Practically, we used a 10-point regularly spaced quadrature for φ in the [0, 2π] domain, together with Gauss–Legendre nodes for the direction cosines usually 4 defined in radiation transfer as: µ = cos(θr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Most com- putations used 6 nodes for µ, leading therefore to 60 dis- tinct couples (θ, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Note also that our new formal solver was designed for considering full frequency and angular dependence of the source function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' The numerical calculation of J12 is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Here for the integration over dΩu we use the same angular quadrature as mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' The integration over the modulus of the atomic velocity u can be done either using a simple trapezoidal rule or a Gauss–Hermite quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Then follows another similar numerical integration over atomic velocities, leading to the self-consistent emis- sion profile, according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' It is repeatedly done further using a basic trapezoidal rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' It was obviously being tested setting f2 ≡ f M, for which case we easily recover the usual thermal, or Doppler profile: ϕ(x) = 1 √π e−x2 , (10) accurately i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=', with a relative error better than 1% over the spectral domain we used for the radiation transfer problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Moreover, every additional numerical calculations pre- viously listed have been tested for the recovery of the well-known solution of the two-level atom with complete redistribution in frequency (hereafter CRD) problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' A SIMPLE ITERATIVE SCHEME A first validation was indeed to run the whole pro- cess with the modified formal solver now also comput- ing ¯J12(⃗u, τ) setting f2 ≡ f M, so that we could recover the usual CRD solution, for a Doppler absorption profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Our angular quadrature with 10 azimuths and 6 Gauss– Legendre nodes for the µ’s guarantees a CRD–like solu- tion within 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='5% maximum relative error throughout a semi-infinite atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Second, for benchmarking with the standard PRD case using RI−A redistribution, we used a very simple iterative scheme consisting, once the CRD solution has been com- puted using standard ALI, in a mere computation and successive updates of J12 and J12, then f2 and ψ and update the source function S before moving to the next iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' The latter process is somewhat comparable to Λ-iteration in the sense that it consists in the simplest possible iterative process one could implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' A simi- lar process was for instance successfully used by Paletou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' (1999) for polarized radiative transfer in 2D geom- etry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' The same numerical strategy allows us to consider also cases for which ζ ̸= 0 (see §7 hereafter) without any difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Variation with frequency of the normalized source function, for different values of the optical depth at line center across the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Dashed lines indicate the (constant with frequency) CRD values at τ = 0, 1, 10, 100, 103, 104 for comparison, where both SCRD and S(x = 0) increase with τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' It satisfactorily reproduces the standard PRD results using Hummer’s RI−A (see also Hummer 1969, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' BENCHMARKING AGAINST HUMMER’S RI−A Our first task then has been to reproduce the S(x, τ) results of the original Hummer (1969;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' his Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' 1c) pub- lication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' It is an obvious comparison to be made, which was not conducted in earlier studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' They were obtained for a 1D, semi-infinite, plane parallel atmosphere of to- tal optical thickness at line center τ = 106, ε = 10−4 and ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' In our computations, we used 5 points per decade in order to cover this range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Our solutions for S(x, τ) are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' (1) where increasing values around line core (|x| < 2) correspond to successive optical depths of the order of: τ = 0, 1, 10, 100, 103, 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Suc- cessive dashed lines mark the (frequency independent) CRD values at the same optical depths, for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Even using our simple iterative scheme, we recover eas- ily the Hummer (1969) “historic” solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' We could 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='75 log( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='00 0 1 2 3 4 5 x5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Dependence of the emission profiles ψ(x, τ) on various optical depths across the atmosphere (see the top-right frame giving the correspondence between τ at line center and the color of the relevant profile).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' This distribution is computed self-consistently with the radiation, without the need of any a priori given redistribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' also check that the relative error on S(x, τ) between our new scheme vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' similar results obtained with a “fre- quency by frequency” (hereafter FBF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' see Paletou & Auer, 1995) for RI−A never exceeds 4% with a more typ- ical mean value around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='3% (for this validation, we used FBF with a 6-node Gauss-Legendre angular quadrature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' To achieve this level of accuracy, we compute the CRD solution using the ALI-iteration until the maximum rel- ative error on the source function was better than 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' The new cycle was then iterated up to a maximum rela- tive error on the frequency dependent source function of 3 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' (2), we display the emission profile ψ(x, τ) for different optical depths τ across the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' These are “naturally” computed using our approach, and with- out the need of any a priori given redistribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' The only assumptions we rely on are: (1) that the VDF f1 of the atoms in their ground state is Maxwellian, and (2) that the atomic absorption profile is a priori given, in the present case of coherent scattering by a Dirac func- tion centered at the frequency ν0 of the model-spectral line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Then all relevant quantities, down to the velocity distribution function of the excited atoms at every depth into the atmosphere, are computed consistently with the successive evaluation of the radiation field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Deviations from Maxwellian illustrated by ratios f2(u, τ)/f M(u) at various optical depths across the atmo- sphere using the same convention as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' It shows the “overpopulation” of f2 at large u’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' This very information cannot be accessed to, using standard non–LTE approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Indeed, we can display, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' (3) a more original sample of the ratios f2(u, τ)/f M(u) for differ- ent optical depths, as considered in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' In the present paper, we consider the emission profile ψ(x, τ) and thereby the source function S(x, τ) to be indepen- dent of the polar angles of the radiation field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Therefore, we prefer to illustrate the normalized VDF of the excited atom that depends only on the modulus of velocity u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' This is obtained by integrating f(⃗u, τ) given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' (3) over atomic velocity directions, namely dΩu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Important deviations from Maxwellian can be identified, typically for u > 2, and close to the non-illuminated surface of the semi-infinite atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' We recover easily, using our new numerical procedures, the “overpopulation” of f2 at large u’s already put in evidence by Borsenberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Such a material is only accessible using the Oxenius- like formalism that we adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Note also that the de facto neglected potential effects of velocity-changing elas- tic collisions can only be addressed, and studied in that theoretical frame (see details in PP21 and §7 hereafter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' More generally, such additional information could be very valuable for the detailed coupling between the radiation transfer problem and any other physical processes that would take place within an atmosphere, and for which the very knowledge of the various VDF’s of contributing ele- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='5 1 4 17 68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='4 270 e profile Emission 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='3 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='0 0 1 2 m 4 5 x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='25 25 1 4 17 68 20 270 normalized f2 15 10 5 0 0 1 2 m 4 5 u6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Variation with frequency of the normalized source function at τ ≈ 1, for different values of the velocity-changing elastic collision parameter ζ (indicated in the top-left frame).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' The model-atmosphere is the same as the one considered in §6 and Hummer (1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' As expected, successive solutions range between standard PRD-RI−A values (blue) and the limit of the CRD solution (dashed line) for increasing values of ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' ments, at different excitation and ionization stages would be critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' NON-ZERO VELOCITY-CHANGING ELASTIC COLLISIONS To the best of our knowledge, such preliminary compu- tations were only addressed by Atanackoviˇc et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' (1987) so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' We now go a bit further by illustrating in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' (4) the dependence of the source function at τ ≈ 1 on the velocity-changing elastic collisions parameterized as ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' The model-atmosphere used for this computation is identical to the one adopted in §6, but we have made vary ζ from 0 to 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' As expected, the source function S(x, τ) ranges between the standard PRD-RI−A solutions when ζ = 0, and the frequency independent CRD values (shown as black dashed line) for increasing values of ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' As also expected, the numerical problem becomes eas- ier for increasing values of ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Effects of velocity-changing elastic collisions will be discussed in more details in an- other devoted study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Deviations from Maxwellian illustrated by the changes of the normalized f2 at various optical depths (men- tioned in the top-left frame) across an asymmetrically and strongly illuminated finite slab of total depth τ = 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Largest amplitudes are for these two values around τ = 1, while smaller but significant deviations are still noticeable at midslab (τ = 500).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' A FINITE SLAB CASE Among numerous applications, we are particularly in- terested in the radiative modelling of isolated and illumi- nated finite slabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' As an example of expected effects, we simulated a 1D plane parallel horizontal slab strongly ir- radiated asymmetrically, only from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' This mimics the radiative modelling of a “cold filament” suspended above a stellar disk (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=', Paletou 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' We used a 33 points optical depth grid, logarithmically spaced away from both open surfaces (using an initial δτ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='01), and symmetric around a midslab depth set at τ1/2 = 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' For this example we also set ε = 10−4 and ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' External illumination is applied only at the bot- tom surface, using a flat profile of normalized intensity Iext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Figure (5) displays values of the normalized VDF f2(u, τ)/f M(u) for optical depth values, counted from the top surface, of τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='58 and τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='15 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=', around the critical value of 1, and at midslab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' The quite strong external illumination that we applied generates very sig- nificant departures of f2 from Maxwellian, especially at u > 2 again, but larger than what we already identified for the semi-infinite atmosphere case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='8 1 3 5 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='0 30 50 log(S) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='8 0 1 2 3 4 5 X70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='15 500 60 50 normalized 40 30 20 10 0 0 1 2 m 4 5 u7 IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' DISCUSSION AND CONCLUSION A first critical evaluation of new numerical procedures in radiative transfer for the solution of the “full non– LTE” problem have been conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' They were vali- dated by reproducing both CRD and standard PRD with Hummer’s RI−A (Hummer 1962, 1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' This will allow us to now move forward in several directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' After the very first computations shown here, we shall also be able to evaluate further, in more details, the ad- ditional effects of potential velocity-changing elastic col- lisions with different set of “classic” parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Also finite slab models, with different conditions of external illumination will be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Such cases lead to more significant departures from Maxwellian than what hap- pens for semi-infinite slabs, as indicated by our prelim- inary filament-like computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Relevant astrophysical “objects” should range from solar prominences to circum- stellar environements for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' The next obvious step is to modify the atomic ab- sorption description for the more realistic case of natural broadening of the upper level of the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' It would go beyond the previous studies of Borsenberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' (1987) and Atanackoviˇc et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' (1987) which were limited to “pure” Doppler broadening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Therefore, a Lorentzian profile for atomic absorption will be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' This will also require to implement modifications to the original formal- ism of Oxenius for that case, following rather Bommier’s (1997) approach, as discussed in PP21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' This will be suit- able for a preliminary study of resonance lines such as Ly- man α of H i for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' The successive computations of Voigt-like profiles at every iterative step, according to the departures of f2 to Maxwellian, and throughout the whole atmosphere will certainly benefit from the numer- ical scheme proposed by Paletou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' This may also require us to implement and validate a more robust iterative scheme, more likely inspired by the so-called FBF scheme proposed by Paletou & Auer (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Then we shall proceed with the consideration of at least an additional distribution, either for another excited state or for free electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' This should lead to an alter- native, less heuristic, approach to the so-called “cross- redistribution” model for the multi-level atoms case (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=', Milkey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' 1975, Hubeny & Lites 1995, Sam- poorna & Nagendra 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' ACKNOWLEDGMENTS M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' acknowledges the support from the Science and Engineering Research Board (SERB), Depart- ment of Science and Technology, Government of India via a SERB-Women Excellence Award research grant WEA/2020/000012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' [1] Atanackoviˇc, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=', Borsenberger, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=', Oxenius, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=', Simon- neau, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' 1987, JQSRT, 38, 427 [2] Bommier, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' 1997, A&A, 328, 706 [3] Borsenberger, J.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' Stenflo eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=', Kluwer [15] Paletou, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=', Peymirat, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' 2021, A&A, 649, A165 [16] Paletou, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=', Peymirat, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=', Anterrieu, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=', B¨ohm, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' 2020, A&A, 633, A111 [17] Sampoorna, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=', Nagendra, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' 2017, ApJ, 838, 95 [18] Sampoorna, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=', Nagendra, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=', Frisch, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} +page_content=' 2011, A&A, 527, A89' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE4T4oBgHgl3EQfJwyq/content/2301.04924v1.pdf'} diff --git a/MdFLT4oBgHgl3EQfNC8e/content/2301.12018v1.pdf b/MdFLT4oBgHgl3EQfNC8e/content/2301.12018v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..27557753b1b4e6adb8b6b72e3aeb267ba61e462d --- /dev/null +++ b/MdFLT4oBgHgl3EQfNC8e/content/2301.12018v1.pdf @@ -0,0 +1,3 @@ +version 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Effects of Polymer Network Dynamics and Mesh Confinement on +the Diffusion and Structural Relaxation of Molecular Penetrants +Tsai-Wei Lin1,3, Baicheng Mei2,3, Kenneth S. Schweizer1,2,3, and Charles E. Sing1,3 +1Department of Chemical and Biomolecular Engineering, University of Illinois at Urbana- +Champaign +2Department of Materials Science and Engineering, University of Illinois at Urbana-Champaign +3Materials Research Lboratory, University of Illinois at Urbana-Champaign +Abstract +The diffusion of small molecular penetrants through polymeric materials represents an important +fundamental problem, relevant to the design of materials for applications such as coatings and +membranes. Polymer networks hold promise in these applications, because dramatic differences +in molecular diffusion can result from subtle changes in the network structure. In this paper, we +use molecular simulation to understand the role that crosslinked network polymers have in +governing the molecular motion of penetrants. By considering the local, activated alpha relaxation +time of the penetrant and its long-time diffusive dynamics, we can determine the relative +importance of glassy dynamics versus mesh confinement on penetrant diffusion. We vary several +parameters, such as the crosslinking density, temperature, and penetrant size, to show that +crosslinks primarily affect molecular diffusion through modification of the matrix glass transition, +with local penetrant hopping at least partially coupled to the segmental relaxation of the polymer +network. This coupling is very sensitive to the local glassy dynamics of the surrounding matrix, +and we also show that penetrant transport is affected by dynamic heterogeneity at low +temperatures. To contrast, only at high temperatures and large penetrants does the effect of mesh + + +2 +confinement become significant, even though penetrant diffusion empirically follows similar +trends as established models of mesh confinement-based transport. +Introduction +The diffusive transport of molecular species through polymers is an important fundamental +problem, with the motion of small ‘penetrants’ being integral to the design of polymer materials +for a variety of applications. For example, barrier coatings,1–3 drug delivery vehicles,4,5 and self- +healing microcapsules1,2,6 may be engineered to impede the transport of penetrants, while +membranes are often designed to selectively separate or filter specific small penetrants (e.g. gas,7– +10 water,11–13 or organic molecules14,15). Membrane separations represent an especially important +materials design challenge,7,15 due to significant interest in finding promising alternatives to +distillation for chemical separations. Distillation is a costly and inefficient process that is the source +of an estimated 10-15% of the world’s energy consumption,15 prompting a search for materials +that can selectively transport small molecules based on their physicochemical features (e.g. size, +shape, or interactions). +The predominant strategy for separating molecules is to impose control over pore size in +materials such as zeolites,16 metal organic frameworks,17,18 and covalent organic frameworks.19 +While these form precise structures that are selective to molecular size and chemistry, they are +typically very brittle and difficult to synthesize at scale.16–19 If the molecular penetrants are +especially small, such as in gas separations, then glassy amorphous polymers can be another option +for membranes.8–10 These materials are more mechanically robust, but are limited to small gas +molecules and do not have the same size-selectivity as more precise matrix structures. +Alternatively, rubbery polymer membranes can instead be tailored to use solubility to affect +penetrant transport while retaining high rates of transport even for larger molecules,14 though this + + +3 +again suffers from limited selectivity. Despite this abundance of promising strategies for designing +separation membranes, the design principles for selectivity relies on a still-incomplete +understanding how chemical structure and dynamics contributes to the diffusive transport of +molecular penetrants. Highly-crosslinked networks have recently emerged as a promising option +for the selective transport of small molecules,20–22 exploiting the sensitivity of penetrant diffusion +to the near-𝑇𝑔 coupling between the structural relaxation of the dense molecular mesh and activated +hopping processes. This is motivated by experimental support for the premise that, despite +significant molecular disorder, dense polymer networks can discern small changes in penentrant +size and interactions.21 +Design of crosslinked networks for selective transport requires an understanding of how +the molecular-scale dynamics of penetrant and matrix relaxation contribute to large-scale diffusive +motion. Aspects of this problem have been studied in the literature. For example, there has been +extensive work on studying particle transport through polymer solutions or high-temperature +networks,23–29 which is understood to be governed by a confinement parameter 𝐶 = 𝑑/𝑎𝑥 that +relates the penetrant size 𝑑 to the size of the network mesh 𝑎𝑥. This parameter quantifies the extent +to which the surrounding polymer matrix sterically ‘traps’ the particle, such that the particle +diffusion is an activated process where the network strands must entropically stretch to allow the +particle to hop out of its location within the mesh.23 Several expressions for the diffusion constant +have been considered in this literature, such as the expression proposed by Cai, et al.:23 + +𝐷𝑝 ∼ 𝑑2 +𝜏 𝐶−1 exp(−𝑏𝐶2) = 𝑑2 +𝜏 𝑋 +(1) +This relates the diffusion constant 𝐷𝑝 to the confinement parameter 𝐶, a length scale characterized +by the penetrant size 𝑑, a time scale 𝜏 that Cai, et al. identify as the Rouse time of the network +strand, and a constant 𝑏 of order unity.23 While there are other candidate models,26 these still + + +4 +exhibit a general form given in the second equivalence of Equation 1 as the product of two factors; +the elementary particle diffusion 𝑑2 𝜏 +⁄ and a factor 𝑋 dictated by the network mesh and related to +𝐶 (in the case above, 𝑋 = exp(−𝑏𝐶2) /𝐶). Simulations of large penetrants in rubbery (i.e. high- +temperature) networks are consistent with this physical picture;24,25,30 however, the choice of 𝜏 can +become non-trivial as temperature is lowered for polymer melts and/or when the glass transition +is approached.31 For example, in uncrosslinked systems activated transport of penetrants and +particles becomes dominated by molecular caging near the glass transition.31–34 + +For tight and near-𝑇𝑔 networks, the physical relationship in Equation 1 is complicated by +recent experimental, simulation, and theoretical investigations by the authors and collaborators +demonstrating that the structural relaxation in polymer networks is strongly dependent on the +extent of crosslinking, manifesting as a monotonic increase in the material 𝑇𝑔 with crosslink +density.35 In this case, the time scale 𝜏 in Equation 1 is expected to depend on molecular penetrant +caging, which itself will be sensitive to the confinement parameter 𝐶 due to its relationship with +the crosslink density of the polymer network. Crosslinks could thus impact penetrant motion both +by affecting the structural relaxation of the surrounding network as well as providing a confining +mesh that obstructs penetrant motion. +In this article, we use computer simulation to study the diffusion of penetrants in dense, +highly-crosslinked polymer networks in the weakly-supercooled regime. Simulation will allow us +to separate out the different ways that crosslinks contribute to penetrant diffusion, by quantifying +both the overall diffusion constant 𝐷𝑝 and the characteristic time scale of penetrant hopping 𝜏 = +𝜏𝛼,𝑝 that we identify with a penetrant alpha relaxation time. By relating particle hopping events to +long-time diffusive motion, we can study how these properties are affected by both crosslinking +fraction and temperature. We determine that (1) supercooled networks exhibit strong coupling + + +5 +between the network segmental relaxation and penetrant hopping, with sensitivity to the +crosslinking-dependent 𝑇𝑔, (2) increasing temperatures can lead to slower diffusion at a given +𝑇𝑔/𝑇 due to the dependence of 𝑇𝑔 on crosslink density, but this is mostly attributed to a non-trivial +𝑇 dependence of how local polymer relaxations couple to penetrant motion, (3) the coupling +between penetrant motion and polymer relaxation processes is dependent on penetrant size 𝑑, and +(4) molecular penetrant hopping is correlated with long-time diffusion at several values of 𝑇𝑔/𝑇 +near 𝑇 = 𝑇𝑔, but the mesh confinement only becomes apparent at high 𝑇 and for large penetrants. +This all leads to the overall insight that, under typical conditions for polymer networks and +molecular penetrants, crosslinking primarily affects penetrant transport by changing the effective +𝑇𝑔 of the surrounding matrix and only secondarily through mesh confinement. Serendipitously, +this still manifests in a similar relationship between the diffusion constant 𝐷𝑝 and 𝐶 that is +consistent with the form in Equation 1.23,26 These molecular-level insights into penetrant diffusion +and hopping clarify the fundamental mechanisms governing penetrant transport and help identify +ways in which both temperature and network architecture can be used to engineer selectivity into +a promising class of membrane materials. +Simulation Methods +We use coarse-grained molecular dynamics simulations (MD) to model the diffusion of +spherical penetrants in crosslinked polymer networks,36 using well-established methods for +studying polymers near the glass transition.37–40 Our system is initially composed of 𝑁c = 20 +linear chains with 𝑁m = 30 beads each with diameter 𝜎, modeled as standard semiflexible chains. +These beads are placed in a cubic box with periodic boundary conditions in three dimensions, +along with 𝑁p spherical penetrants of diameter 𝑑̃. The number of penetrants 𝑁p depends on 𝑑̃, and +is chosen so that the volume fraction of the penetrant 𝜙𝑝 = +𝜋𝑑3𝑁𝑝 +6𝑉 is kept below 𝜙𝑝 = 0.01 to + + +6 +ensure that the addition of penetrants will not affect network dynamics and to prevent significant +interactions between penetrant molecules. +Standard dimensionless simulation quantities are employed,36,41 and denoted with tildes +(e.g. 𝑇̃ = 𝑇/𝑇∗). Quantities are rendered dimensionless by several characteristic values: lengths +are related to the bead diameter 𝜎, energies are related to the thermal energy 𝑘𝐵𝑇∗, and times are +related to a time scale 𝜏∗ = √𝑚𝜎2/𝑘𝐵𝑇∗, where 𝑚 is the monomer mass. The characteristic +temperatures 𝑇∗ corresponds to 485 K, which is determined from parametrization against +experiments,35 using crosslinked poly(n-butyl acrylate) (PnBA) as a representative polymer that +has been used by our experimental collaborators (see details in ref. 35). +Our MD simulations consider standard potentials for coarse-grained bead-spring +polymers,42 including a Lennard-Jones potential 𝑈̃LJ between all non-bonded beads, and both +bonding 𝑈̃B and bending 𝑈̃𝜃 potentials to model each semiflexible bead-spring chain. The overall +energy is composed of a sum of these contributions: + +𝑈̃ = 𝑈̃B + 𝑈̃𝜃 + 𝑈̃LJ += ∑ 𝑢̃B,𝑖𝑗 +𝑁𝑡𝑜𝑡,𝑛 +𝑖,𝑗>𝑖 ++ +∑ +𝑢̃𝜃,𝑖𝑗𝑘 +𝑁𝑡𝑜𝑡,𝑛 +𝑖,𝑗>𝑖,𝑘>𝑖,𝑗 ++ ∑ +∑ +∑ +∑ 𝑢̃LJ,𝛼𝛽,𝑖𝑗 +𝑁𝑡𝑜𝑡,𝛽 +𝑗>𝑖 +𝑁𝑡𝑜𝑡,𝛼 +𝑖 +𝛽=𝑛,𝑝 +𝛼=𝑛,𝑝 + +(2) +Here the total system energy is written in terms of independent pairwise contributions 𝑢̃B,α,𝑖, 𝑢̃𝜃,𝛼,𝑖, +and 𝑢̃LJ,𝛼𝛽,𝑖𝑗. Bonded monomers interact through harmonic bonding potential: + +𝑢̃B,𝑖𝑗 = 𝑘̃ +2 (𝑟̃𝑖𝑗 − 1) +2Θ𝑖𝑗 +(3) +A large spring constant 𝑘̃ = 2000 is adopted to enforce the equilibrium distance 𝑟̃𝑖𝑗 = 1 between +bonded beads 𝑖 and 𝑗. The factor Θ𝑖𝑗 determines the connectivity between monomer pairs, with + + +7 +Θ𝑖𝑗 = 1 being connected and Θ𝑖𝑗 = 0 being disconnected; these values depend on the specific +network that is formed. A bending energy is similarly introduced to account for chain stiffness: + +𝑢̃𝜃,𝑖𝑗𝑘 = 𝑘̃𝜃[1 − cos𝜃𝑖𝑗𝑘]Θ𝑖𝑗𝑘 +(4) +Here, the bond angle is between three adjacent beads 𝑖, 𝑗, and 𝑘 whose connectivity is determined +by a similar factor Θ𝑖𝑗𝑘 to the one used for the bonding potential. The bending constant 𝑘̃𝜃 = 1.52 +is selected to reflect the experimental Kuhn length of PnBA.43–46 Further details of the +parametrization are given in the literature.35 A Lennard-Jones (LJ) potential is used to describe all +non-bonded interactions between particles 𝑖 and 𝑗 on species 𝛼 and 𝛽: + +𝑢̃LJ,𝛼𝛽,𝑖𝑗 = {4𝜖̃𝛼𝛽 [( +𝑑̃𝛼 + 𝑑̃𝛽 +2𝑟̃𝛼𝛽 +) +12 +− ( +𝑑̃𝛼 + 𝑑̃𝛽 +2𝑟̃𝛼𝛽 +) +6 +] +0, +otherwise +, 𝑟̃𝛼𝛽 < 𝑟̃cut = 2.5 × +𝑑̃𝛼 + 𝑑̃𝛽 +2 + +(5) +Here, 𝛼, 𝛽 ∈ {n, p}, denotes the species as either network (n) or penetrant (p), with n denoting the +monomer bead in the network and p denotes the penetrant. In the overall summation, 𝑁𝑡𝑜𝑡,𝑛 = +𝑁𝐶𝑁𝑚 is the total number of network beads and 𝑁𝑡𝑜𝑡,𝑝 = 𝑁𝑝 is the total number of penetrant beads. +The monomer size 𝑑̃n = 1 is always unity (i.e., 𝑑n = 𝜎) and the penetrant size 𝑑̃p ≡ 𝑑̃ is an +important parameter for this study. In this study, we do not consider specific penetrant-polymer +attractions, so keep 𝜖̃nn = 𝜖̃np = 𝜖̃pp = 1. + + +8 + +Figure 1. (a) Schematic illustrating the setup of our simulation. Monomer beads (orange) and +crosslinker beads (blue) interact via bonding (𝑢̃B,𝑖𝑗) and angle (𝑢̃𝜃,𝑖𝑗𝑘) potentials, and can interact +with each other and with the penetrant through Lennard-Jones potentials (𝑢̃LJ,𝑖𝑗). Monomer and +crosslinker beads have a diameter 𝑑̃𝑛 = 𝜎 and the penetrant has a diameter 𝑑̃𝑝. Crosslinking +occurs during an initial simulation step where ‘reactive’ beads (light blue) will form bonds with +regular monomers within a cutoff distance 𝑅̃min. (b) Snapshot of a typical simulation, for 𝑑 𝜎 +⁄ += +2.0 and 𝑓𝑐𝑟𝑜𝑠𝑠 = 0.11. + +The system is first equilibrated at 𝑇̃ = 1, 𝑃̃ = 0, and then networks are prepared by +crosslinking the linear chains with reactive beads randomly distributed along the chain.35,47,48 The +total number of reactive beads is 𝑁r = 𝑓r𝑁m𝑁c, where 𝑓r is the fraction of reactive beads which +tunes the crosslink density of networks. If the distance between a reactive bead and a free bead +(orange beads in Figure 1) is within 𝑅̃min = 1.1, a new permanent bond will be formed given an +assigned probability. Once a reactive bead forms a bond with a free bead, both the reactive bead +and the free bead are labeled as crosslink beads (dark blue beads in Figure 1) and these two beads +represent one crosslink ‘molecule’, in a way that reflects the specific chemistry used to synthesize +PnBA networks,21 where the crosslinker molecular weight is roughly twice of the molecular weight +of nBA. Crosslinking reactions were turned off once every reactive bead has formed a new bond +with another free monomer (maximum number of possible bonds has been reached). In the end, + + +9 +𝑁r reactive beads have reacted with the 𝑁r free beads and they turned into 2𝑁r crosslink beads and +belong to the 𝑁r crosslink molecules. The crosslink density, 𝑓cross, of networks is defined as:35 + +𝑓cross = +𝑛crosslink +𝑛crosslink + 𝑛monomer += +𝑁r +𝑁m𝑁c − 𝑁r + +(6) +Here, 𝑛crosslink and 𝑛monomer are the number of crosslink molecules and nBA monomers, +respectively. Four values of crosslink fraction 𝑓cross are considered: 0.11, 025, 0.36, and 0.5. +After crosslinking, the system is cooled to a target temperature at a cooling rate 𝛤̃ = +8.3 × 10−6 (corresponding to 𝛤 = 1.25 × 109 K/s in experimental units) and further equilibrated +at constant 𝑃̃ = 0. Another short NPT run was performed and the mean volume 𝑉 is measured. +We then switch to a NVT ensemble by setting the system volume to the mean volume 𝑉 and +equilibrate the system before final production run at NVT. All simulations are performed in +LAMMPS49 with a standard Nosé-Hoover thermostat and barostat.36,50,51 +The polymer network is characterized by its mesh size (𝑎̃𝑥), Kuhn segment alpha relaxation +time (𝜏̃𝛼,K), and glass transition temperature Tg. The mesh size of networks at different crosslink +densities is defined as the averaged distance between two adjacent crosslink beads on the same +chain, and is tabulated in Table S1. The value of 𝜏̃𝛼,K at each temperature and crosslink density +was defined as the time where the temporal autocorrelation function of a Kuhn monomer vector, +𝐶𝜆(𝑡̃) = 〈𝑃2(𝒓̃3(𝑡̃) ∙ 𝒓̃3(0))〉 decays to 𝐶𝜆(𝑡̃max) + (1 − 𝐶𝜆(𝑡̃max)) 𝑒 +⁄ , where 𝐶𝜆(𝑡̃max) represents +the plateau value at the long-time limit 𝑡̃max.52–56 Here, 𝑃2 is the second Legendre polynomial and +𝒓̃3 is a vector between two beads that are 3 bonds apart which reflects the choice of coarse-grained +bead relate to Kuhn segment of PnBA. 𝑇g is then defined as when the Kuhn segment alpha +relaxation time is 𝜏̃𝛼,K(𝑇̃g) = 105. More information can be found in our previous work.35 + + +10 +We characterize the dynamics of the penetrant motion by its alpha relaxation time, 𝜏̃𝛼,p, +and its diffusion coefficient 𝐷𝑝. To calculate 𝜏̃𝛼,p, we use the self-intermediate scattering function +(ISF) of the penetrant, which is given by:53 + +𝐹𝑠(𝒒̃, 𝑡̃) = 1 +𝑁𝑝 +∑ ⟨exp [−𝑖𝒒̃ ∙ (𝒓̃𝑝𝑗(𝑡̃) − 𝒓̃𝑝𝑗(0))]⟩ +𝑁𝑝 +𝑗 + +(7) +where 𝒓̃𝑝𝑗 is the position of the 𝑗th penetrant at time 𝑡̃, and |𝒒̃| is set to the position of the first peak +of the static structure factor for the networks, corresponding to |𝒒̃| = 7.2. The value of 𝜏̃𝛼,p, is +defined as the time required for 𝐹𝑠(𝒒̃, 𝑡̃) to decay to either 1/𝑒 or 0.1 from its initial (𝑡̃ = 0) value +of unity. We characterize the mean-squared displacement (MSD) as a function of time: MSD(𝑡̃) = +〈𝑟̃𝑝 +2(𝑡̃)〉 = +1 +𝑁𝑝 ∑ +⟨(𝒓̃𝑝𝑗(𝑡̃) − 𝒓̃𝑝𝑗(0)) +2 +⟩ +𝑁𝑝 +𝑗 +. The diffusion coefficient of penetrant is then obtained via +the Einstein relation in the diffusive regime, 𝐷𝑝 = lim +𝑡̃→∞ +1 +6𝑡 〈𝑟̃𝑝 +2(𝑡̃)〉.57 +Results and Discussion +Penetrant Hopping and Diffusion in Dense Networks +We first characterize the diffusive motion of penetrants in polymer networks by studying +the mean square displacement, ⟨𝑟̃𝑝 +2(𝑡̃)⟩, of both small (𝑑/𝜎 = 1.0) and large ( 𝑑/𝜎 = 2.0) +penetrants. The choice of these two particle sizes is motivated by considering particle sizes +characteristic of model experimental systems in previous work by some of the authors;22 using +PnBA as our representative network polymer (coarse-grained so that 𝜎 = 0.573nm), the span +𝑑 𝜎 +⁄ +∼ 1 − 2 is typical for the average size of molecular penetrants.22 This choice also corresponds +to the general length scale of the mesh size in the literature, which here ranges size from 𝑎̃𝑥 ∼ +1.3 − 2.5 (see Table S1). The MSD versus time 𝑡̃ is plotted for both particle sizes in Figures 2a +and 2b for 𝑑/𝜎 = 1.0 and 𝑑/𝜎 = 2.0 respectively, varying both the crosslink density 𝑓𝑐𝑟𝑜𝑠𝑠 and + + +11 +temperature 𝑇. These plots show typical features characteristic of molecular diffusion. At short +times, the particle exhibits ballistic motion such that the MSD scales as ⟨𝑟̃𝑝 +2(𝑡̃)⟩ ∼ 𝑡̃2; however, +this regime gives way to a subdiffusive regime that we attribute to molecular caging by +neighboring network monomers. At long times, the particle can overcome the barrier imposed by +these cages, along with any constraints due to the network mesh, and undergo Fickian diffusion +where ⟨𝑟̃𝑝 +2(𝑡̃)⟩ ∼ 𝑡̃1. These general features of diffusive motion are observed for both penetrant +sizes and all crosslink densities and temperatures, however the specifics of penetrant caging and +diffusive motion will be affected by these parameters. +The subdiffusive regime following the initial, ballistic penetrant motion provides insight +into the time scales of caging, and large differences in the persistence of this regime are observed +as 𝑓𝑐𝑟𝑜𝑠𝑠, 𝑇, and 𝑑/𝜎 are varied. To characterize this regime, we plot in Figure S1 the slope of the +MSD versus 𝑡̃ curve on a log-log plot (𝛽 = 𝑑 log⟨𝑟̃𝑝 +2(𝑡)⟩ 𝑑 log 𝑡̃ +⁄ +) as a function of time 𝑡̃. The +minimum value of 𝛽 is an estimate for the time 𝑡̃𝛽 at which the penetrant is maximally subdifusive +or caged, and the value of ⟨𝑟̃𝑝 +2(𝑡̃𝛽)⟩ +1/2 correspondingly quantifies a transient localization length. +A representative quantity is indicated in Figures 2a and 2b with an open square, showing the +caging onset for 𝑇̃ = 0.62 and 𝑓𝑐𝑟𝑜𝑠𝑠 = 0.25. We only indicate a single representative value for +each particle size, because the displacement ⟨𝑟̃𝑝 +2(𝑡̃𝛽)⟩ +1/2 is only weakly dependent on temperature +and crosslink densities, and is far smaller than the length scale of the penetrant diameter +(⟨𝑟̃𝑝 +2(𝑡̃𝛽)⟩ +1/2 ∼ 0.22 for 𝑑 𝜎 +⁄ += 1 and ⟨𝑟̃𝑝 +2(𝑡̃𝛽)⟩ +1/2 ∼ 0.32 for 𝑑 𝜎 +⁄ += 2). This insensitivity leads +us to interpret this onset of caging as tied to the structural correlations of the surrounding melt +beads. + + +12 +For both small (𝑑 𝜎 +⁄ += 1) and large (𝑑 𝜎 +⁄ += 2) penetrants, the value of 𝛽 → 1 at long times +limits to Fickian behavior, however the time that it takes to reach this limit increases with +decreasing temperature (Figure 2 and Figure S1). This is the expected result of having less thermal +energy to overcome the dynamic caging barrier,20,58 especially as our simulations approach the 𝑇𝑔. +Similarly, an increase crosslink density 𝑓𝑐𝑟𝑜𝑠𝑠 monotonically increases the length of this +subdiffusive regime and the minimum value of 𝛽 concomitantly decreases (Figure S2) so that the +MSD nearly exhibits a transient plateau. In principle, this is due to some combination of (1) an +increase in the caging barrier because of the concomitant increase in the segmental relaxation time +due to crosslinks35 and (2) the impact of mesh confinement impeding penetrant motion.23,24,26 In +this paper, a major result will be to show that the first interpretation is the dominant determinant +of penetrant mobility for the models studied, especially as 𝑇𝑔 is approached. This is apparent +already by comparing the features in the MSD versus 𝑡̃ to the mesh size 𝑎̃𝑥, which were tabulated +in Table S1 for several different values of 𝑓𝑐𝑟𝑜𝑠𝑠 and indicated in Figures 2 and S1 as open circles. +This length scale is often only accessed in or near the Fickian regime, suggesting that any effect +on the subdiffusive transport of the penetrant in these simulations is subtle and not readily apparent +from the MSD versus 𝑡̃ plot alone. Despite this observation, it is apparent that both the temperature +and crosslink density effects still depend on the penetrant size, with more pronounced changes in +the subdiffusive regime for the larger (𝑑 𝜎 +⁄ += 2) penetrants than the smaller (𝑑 𝜎 +⁄ += 1) penetrants +as 𝑇 and 𝑓𝑐𝑟𝑜𝑠𝑠 are varied. + + +13 + +Figure 2. Mean square displacement ⟨𝑟̃𝑝 +2(𝑡̃)⟩ versus time 𝑡̃ of a penetrant particle with sizes (a) +𝑑/𝜎 = 1.0 and (b) 𝑑/𝜎 = 2.0, for several values of 𝑇 and 𝑓𝑐𝑟𝑜𝑠𝑠 . Open squares denote a +representative location for the onset of caging or transient localization, defined as the minimum +slope 𝛽 on this plot (see Figure S1) and chosen for 𝑇 = 300K and 𝑓𝑐𝑟𝑜𝑠𝑠 = 0.11. The analogous +point for other values of 𝑇 and 𝑓𝑐𝑟𝑜𝑠𝑠 would be almost identical. The open circles denote the length +scale of the polymer network mesh size, as described in Table S1. The long-time data Fickian +region, where the slope of this plot 𝛽 = 1, is used to determine the penetrant diffusion constant +𝐷̃𝑝. + +The MSD provides a molecular measure of penetrant transport, with the long-time Fickian +regime providing an estimate of the overall diffusion coefficient 𝐷̃𝑝 of the penetrant. However, +this quantity is informed by the cumulative effect of both local particle motions as well as any +larger length-scale transport effects of the network mesh confinement. To deconvolute these +contributions, we also consider the time evolution of the self-ISF 𝐹s(𝒒̃, 𝑡̃) in Figure 3. Here we +select a value 𝒒̃ = 7.2 that is chosen to reflect the local structural cage of the polymer network and +not the length scale of the network mesh. The curves shown in Figure 3 thus only account for the +dynamics of local relaxation and hopping, and quantify the extent to which the penetrant remains +within its original cage. The first part of this decay is independent of temperature and crosslinking, +and corresponds to the initial ballistic motion of the penetrant observed in Figure 2. There are + +(a) +(b) +105 +105 +T/K: 300 +339 +T/K: 300 +339 +cross +1.0 +cross +0.11 +0.11 +103 +103 +0.25 +0.25 +1.0 +0.36 +0.36 +0.50 +0.50 +t +101 +101 +1 +2.p +2p +-710-1 +10-3 +10-3 +2.0 +d/ = 1.0 +2.0 +d/ = 2.0 +10-5 +10-5 +10-1 +101 +~t +103 +105 +10-1 +101 +~t +103 +105 +14 +some subtle differences between the different penetrant sizes (𝑑 𝜎 +⁄ += 1.0 vs. 2.0) that we attribute +to the different masses of the penetrants and is also apparent in Figure 2. +At longer times, the decay of 𝐹s(𝒒, 𝑡̃) becomes strongly dependent on temperature, +crosslinking, and penetrant size in a manner consistent with the diffusive motion in Figure 2. At +high temperatures, the smaller penetrant (Figure 3a) exhibits a non-exponential, long tail of +𝐹s(𝒒, 𝑡̃) that we attribute to an activated but relatively rapid hopping process for penetrant +relaxation. As crosslinking increases, this long-time tail extends further, and its non-exponential +character becomes more pronounced. This trend is exacerbated by going to lower temperatures. +We find there is a significant slowing down of penetrant motion even at intermediate times; this +occurs around 𝑡̃ ∼ 1, which is roughly when the minimum in the apparent exponent 𝛽 occurs for +the MSD curves (in Figure 2 and S1) and is consistent with the interpretation of this as the onset +of penetrant caging. For larger penetrants (𝑑 𝜎 +⁄ += 2.0, Figure 3b) this slowing down leads to the +near-arrest of particle motion, again at 𝑡̃ ∼ 1, as evidenced by a plateau in 𝐹s(𝐪, 𝑡̃) that only begins +to relax at significantly longer times as the temperature is decreased or crosslinking fraction is +increased. In both cases, we can attribute this arrest to the slow segmental relaxation dynamics of +the surrounding network, as 𝐹s(𝐪, 𝑡̃) is a local measure of hopping that does not probe the length +scales relevant for mesh confinement. +The trends seen in the intermediate scattering function plots in Figure 3 are largely +consistent with those of the MSD plots in Figure 2, however we note several important differences +that we will use to understand the mechanisms of penetrant motion in polymer networks. First, we +note the similarity of the MSD versus time plots in Figure 2a for low temperatures and crosslink +density (𝑇 = 300K and 𝑓𝑐𝑟𝑜𝑠𝑠 = 0.11) and the high temperatures and crosslink density (𝑇 = +339K and 𝑓𝑐𝑟𝑜𝑠𝑠 = 0.36); these curves almost overlap, and exhibit essentially the same diffusive + + +15 +properties. Yet, the shapes of these curves in Figure 3a are distinctly different, with the +characteristics of local caging more apparent in the lower temperature curve. In addition, there is +a distinct difference in the subdiffusive regimes seen in the MSD plots versus 𝐹s(𝐪, 𝑡̃), which are +much more pronounced. This is contrast to normal glass-forming liquids, for which the same +supercooling degree exhibits a weaker or shorter caging regime than for the MSD when compared +to 𝐹s(𝐪, 𝑡̃).59–66 These disparities between the MSD versus time and 𝐹s(𝐪, 𝑡̃) suggest that mesh +confinement may contribute meaningfully to the transport of penetrants in tight networks, and we +seek to refine our physical understanding of this complicated interplay of local and long-time +penetrant and network dynamics. + +Figure 3. Intermediate scattering function 𝐹s(q, 𝑡̃) of the penetrant particle with (a) 𝑑/𝜎 = 1.0 +and (b) 𝑑/𝜎 = 2.0 with |q| = 7.2 at 𝑇 = 300K (solid, blue) and 339K (dashed, gray) for a variety +of crosslink densities 𝑓𝑐𝑟𝑜𝑠𝑠 as a function of time 𝑡̃. The orange and purple horizontal lines define +the penetrant alpha time, with the criteria 𝐹s(q, 𝜏𝛼,𝑝) = 0.1 and 𝐹s(q, 𝜏𝛼,𝑝′) = 1/𝑒. The former +criteria (𝜏𝛼,𝑝) is more often used in this paper, because it includes the long tails observed in these +functions that reflect a subpopulation of penetrants in long-lived cages. The latter criteria (𝜏𝛼,𝑝′) +may more closely relate to the average hopping time that is predicted in the theory developed in +our companion paper, which does not account for dynamic heterogeneity and a distribution of +penetrant hopping times. + + +(a) +(b) +1.0 +1.0 +T/K: 300 +339 +T/K: 300 +339 +0.11 +0.11 +0.8 +0.8 +0.25 +0.25 +0.36 +0.36 +0.50 +0.6 +#0.6 +0.50 +d/ = 1.0 +q9 +d/ = 2.0 +E° 0.4 +1/e +F°0.4 +1/e +0.2 +0.2 +0.1 +0.1 +0.0 +0.0 +10-2 +10-1 +100 +101 +102 +103 +10-2 +10-1 +100 +101~ 102 103 104 105 +t +t +16 +Comparing the Role of Particle Size, Temperature, and Crosslinking on the Hopping and Diffusive +Processes in Dense Networks +To quantify the relationship between local penetrant relaxation and diffusive motion, we +extract the alpha time of the penetrant 𝜏̃𝛼,𝑝 from the ISF and the diffusion constant 𝐷̃𝑝 from the +MSD versus 𝑡̃. As described in the methods section, the diffusion constant is obtained from the +long-time MSD curve via the expression 𝐷𝑝 = lim +𝑡→∞ +1 +6𝑡 〈𝑟̃𝑝 +2(𝑡̃)〉. The alpha time of the penetrant 𝜏̃𝛼,𝑝 +is set from the decay of 𝐹s(𝒒, 𝑡̃), and quantifies the time it takes to reach one of two possible +criteria: 𝐹s(𝒒, 𝜏̃𝛼,𝑝) = 0.1 and 𝐹s(𝒒, 𝜏̃𝛼,𝑝′) = 1/𝑒. While the specific criterion is arbitrary, these +two criteria will be used to illustrate the effect of dynamic heterogeneity;65–68 the quantity 𝜏̃𝛼,𝑝 will +account for the effect of the long-time tails in Figure 3, which do not contribute to the value 𝜏̃𝛼,𝑝′ +that we deduce is closer to the average hopping time predicted in the accompanying theoretical +work based on comparisons of the theory and simulation extracted relaxation times.69 Differences +between how 𝜏̃𝛼,𝑝 and 𝜏̃𝛼,𝑝′ depend on 𝑓𝑐𝑟𝑜𝑠𝑠 do depend on both 𝑇 and 𝑑/𝜎, and as we will show +can be very small or substantial. We will primarily report 𝜏̃𝛼,𝑝, considering 𝜏̃𝛼,𝑝′ only where the +comparison is instructive. We can thus extract the key quantities in Equation 1 directly from +molecular simulations, the long-time diffusion 𝐷̃𝑝 and the hopping time 𝜏̃𝛼,𝑝, and examine how +they are related and affected by the state of the network and penetrant. +We start by considering 𝜏̃𝛼,𝑝, which we indicated in Figure 3 and is typically between +𝜏̃𝛼,𝑝 = 101 − 103 for 𝑑 𝜎 +⁄ += 1.0 and 𝜏̃𝛼,𝑝 = 101 − 104 for 𝑑 𝜎 +⁄ += 2.0 . Our goal is not to +determine any precise dependence on 𝑑 𝜎 +⁄ , but rather to explore the differences between relatively +‘small’ and ‘large’ molecular penetrants in the context of experimental systems. We plot the +inverse penetrant mean alpha times 1/𝜏̃𝛼,𝑝 or 1/𝜏̃𝛼,𝑝′ as a function of 𝑇𝑔(𝑓𝑐𝑟𝑜𝑠𝑠)/𝑇 for several + + +17 +different temperatures 𝑇 and crosslink fractions 𝑓𝑐𝑟𝑜𝑠𝑠 in Figure 4. We emphasize that the abscissa +combines two parameters we consider in our model; the temperature 𝑇 is plotted in different colors +in all panels of Figure 4a and b for 𝑑 𝜎 +⁄ += 1.0 and 𝑑 𝜎 +⁄ += 2.0, respectively, and is directly set in +our simulations. However, we note that 𝑇𝑔/𝑇 has multiple values for a given 𝑇 due to the +dependence of 𝑇𝑔 on 𝑓𝑐𝑟𝑜𝑠𝑠.35 For the Figure 4a inset, we replot the data to show the corresponding +plot of 1/𝜏̃𝛼,𝑝 versus 𝑇𝑔/𝑇, but grouping data by 𝑓𝑐𝑟𝑜𝑠𝑠 instead of 𝑇. + +Figure 4. Inverse penetrant alpha time 1/𝜏̃𝛼,𝑝 calculated from simulations as a function of +𝑇𝑔(𝑓𝑐𝑟𝑜𝑠𝑠)/𝑇 for several temperatures 𝑇 and penetrant sizes (a) 𝑑/𝜎 = 1.0 and (b) 𝑑/𝜎 = 2.0. In +(a) there is a noticeable collapse to a single curve, which we illustrate more clearly in the inset by +grouping with respect to 𝑓𝑐𝑟𝑜𝑠𝑠. This collapse is no longer observed in (b). We also plot the shorter- +time criteria for the penetrant alpha time 1/𝜏̃𝛼,𝑝′ versus 𝑇𝑔(𝑓𝑐𝑟𝑜𝑠𝑠)/𝑇 in (c) for /𝜎 = 1.0 to +demonstrate that the coupling of penetrant hopping to crosslink density is noticeably weaker when +the long-time tail in the intermediate scattering function is no longer included. The slopes of these +plots are calculated and included in Table 1. + +Figures 4a and b show that at a fixed 𝑇, the penetrant alpha relaxation time 1/𝜏̃𝛼,𝑝 behaves +exponentially with 𝑇𝑔(𝑓𝑐𝑟𝑜𝑠𝑠), exhibiting Arrhenius-like behavior for both penetrant sizes. As the +temperature increases, the absolute value of the slope of this curve decreases, demonstrating a +stronger contribution from crosslink-induced 𝑇𝑔 enhancement at lower temperatures. This +underscores the importance of the network segmental dynamics on penetrant diffusion, and can be +quantified by a change in the apparent activation energy that is proportional to the slope log 𝜏̃𝛼,𝑝 + +(a) +(b) +(c) +10-1 +T/K +T/K +100 +T/K +300 +300 +口 +口 +300 +10-2 +320 +320 +320 +339 +339 +339 +T,(fero +10-2 +0/T +0.8 +0.9 +1.0 +TY +10-1 +10-3 +口 +0.11 +4 +0.25 +10-3 +10-4 +0.36 +Q +0.50 +d/a = 1.0 +d/ = 2.0 +d/ = 1.0 +10-4 +10° +10-2 +0.80 +0.85 +0.90 +0.95 +0.80 +0.85 +0.90 +0.95 +0.80 +0.85 +0.90 +0.95 +18 +versus 𝑇𝑔/𝑇. To highlight how the choice of criterion 𝐹s(𝒒, 𝜏̃𝛼,𝑝) = 0.1 affects our results, we plot +in Figure 4c how Figure 4a would be changed upon choosing the alternative criterion, +𝐹s(𝒒, 𝜏̃𝛼,𝑝′) = 1/𝑒. In this case, similar trends are observed (i.e. an increasing apparent activation +energy with decreasing temperature), however the magnitudes of the slopes are significantly +smaller. We speculate that this is due to the neglect of longer time-scale processes that are apparent +in the extended tails seen in Figure 3, meaning that the 1/e criterion does not fully capture the +hopping processes thereby justifying our use of the 0.1 criterion. For both criteria and penetrant +sizes 𝑑/𝜎, the values for the slopes of log 𝜏̃𝛼,𝑝 versus 𝑇𝑔/𝑇 are calculated and tabulated in Table +1 for different values of 𝑇. We note that for both penetrant sizes, these slopes exhibit similar trends +to the predictions from theory in our companion paper, showing a modest decrease in the +magnitude of the slope with increasing temperature for both 𝜏̃𝛼,𝑝 and 𝜏̃𝛼,𝑝′ . However, the +magnitudes of the slopes are significantly larger for 𝜏̃𝛼,𝑝 than the slopes for 𝜏̃𝛼,𝑝′, with the latter +values in parenthesis in Table 1. We attribute this to the inclusion of the long-time tails of the ISF +in 𝜏̃𝛼,𝑝, which indicate that penetrant hopping is influenced by dynamic heterogeneity near 𝑇𝑔 due +to the slower relaxation of a significant fraction (~5-10%) of particles.66 To contrast, the long-time +tails (dynamic heterogeneity) do not (or weakly/negligibly) affect 𝜏̃𝛼,𝑝′, leading to a weaker +apparent coupling between the near-𝑇𝑔 dynamics of the network and the penetrant hopping. +Table 1. Slope for log 𝜏̃𝛼,𝑝 or 𝐷̃𝑝 +−1 versus 𝑇g/𝑇 from simulations at several temperatures (𝑇 = +300, 320, and 339K) and penetrant sizes (𝑑/𝜎 = 1 and 2). Both criteria 𝜏̃𝛼,𝑝 and 𝜏̃𝛼,𝑝′ (in +parentheses) are considered. +T/K +𝒅/𝝈 +300 +320 +339 +𝐥𝐨𝐠 (𝝉̃𝜶,𝐩) vs 𝑻𝐠/𝑻 +1.0 +15.57 (7.79) +8.59 (3.98) +8.01 (1.97) +𝐥𝐨𝐠 (𝟏/𝑫𝐩) vs 𝑻𝐠/𝑻 +1.0 +9.34 +7.48 +5.79 +𝐥𝐨𝐠 (𝝉̃𝜶,𝐩) vs 𝑻𝐠/𝑻 +2.0 +18.13 (16.10) +12.86 (8.78) +14.80 (9.60) + + +19 +𝐥𝐨𝐠 (𝟏/𝑫𝐩) vs 𝑻𝐠/𝑻 +2.0 +17.47 +19.27 +18.12 + +The two different penetrant sizes in Figure 4a and 4b exhibit similar trends, including +nearly the same overall decrease of relaxation rate by a little over two decades (over all values of +𝑇 and 𝑓𝑐𝑟𝑜𝑠𝑠 ). However, there are subtle but important differences. At a fixed value of +𝑇𝑔(𝑓𝑐𝑟𝑜𝑠𝑠)/𝑇 (i.e. at a fixed supercooling degree), we observe that the penetrant alpha time is +roughly constant with decreasing temperature for 𝑑 𝜎 +⁄ += 1.0 . This lower temperature also +corresponds to a lower degree of crosslinking 𝑓𝑐𝑟𝑜𝑠𝑠. This means that the degree of supercooling +𝑇𝑔(𝑓𝑐𝑟𝑜𝑠𝑠)/𝑇 largely dictates penetrant hopping, with the absolute temperature 𝑇 playing a similar +role in affecting transport as the effective 𝑇𝑔. In this case, several effects – changing 𝑇𝑔, absolute 𝑇, +and crosslinking (𝑓𝑐𝑟𝑜𝑠𝑠) – tend to cancel resulting in a near collapse of the 1/𝜏̃𝛼,𝑝 versus 𝑇𝑔/𝑇 +data. This is highlighted in the inset of Figure 4a, which is the same data as in the main frame but +grouped by 𝑓𝑐𝑟𝑜𝑠𝑠 instead of 𝑇, and more clearly shows how well this data collapses to a single +curve. Figure 4b shows that this balance no longer holds for larger penetrants 𝑑 𝜎 +⁄ += 2.0, with +lower 𝑇 values exhibiting faster dynamics (i.e. higher 1/𝜏̃𝛼,𝑝) at a given degree of supercooling +𝑇𝑔(𝑓𝑐𝑟𝑜𝑠𝑠)/𝑇 due to the corresponding decrease in the 𝑓𝑐𝑟𝑜𝑠𝑠. +This same competition is apparent in the diffusion constant 𝐷̃𝑝, which we plot in Figure +5a and b for 𝑑 𝜎 +⁄ += 1.0 and 𝑑 𝜎 +⁄ += 2.0 respectively, and at the same conditions plotted in Figure +4a and b. Like 1/𝜏̃𝛼,𝑝, 𝐷̃𝑝 exhibits Arrhenius-like behavior as defined by a straight line on a log 𝐷̃𝑝 +versus 𝑇𝑔/𝑇 plot. The primary difference is in the relative magnitude change in 𝐷𝑝 at different +degrees of supercooling 𝑇𝑔/𝑇. 𝐷̃𝑝 still exhibits a near-collapse for 𝑑 𝜎 +⁄ += 1.0, however generally +decreases more slowly than 1/𝜏̃𝛼,𝑝 in Figure 4a. For the larger penetrant size 𝑑 𝜎 +⁄ += 2.0, the +situation is more complicated. In this case, high temperatures 𝑇 in Figure 5b show a more + + +20 +dramatic decrease in 𝐷̃𝑝 with 𝑇𝑔/𝑇 than that of 1/𝜏̃𝛼,𝑝 in Figure 4b, but a similar decrease at lower +temperatures. These effects are quantified by calculating the slope of the − log 𝐷̃𝑝 versus 𝑇𝑔/𝑇 +relationship, which we include in Table 1 for both penetrant sizes. Here, the 𝐷̃𝑝 slopes are +significantly smaller than the 𝜏̃𝛼,𝑝 slopes for 𝑑 𝜎 +⁄ += 1.0, but are similar or larger than the 𝜏̃𝛼,𝑝 for +𝑑 𝜎 +⁄ += 2.0. The latter behavior for 𝑑 𝜎 +⁄ += 2.0 is qualitatively consistent with the results in our +companion theory paper,69 which finds that the diffusion constant 𝐷̃𝑝 changes with supercooling +𝑇𝑔/𝑇 similarly or more pronounced than the corresponding inverse 𝜏̃𝛼,𝑝. For 𝑑 𝜎 +⁄ += 1.0, however, +the theory predicts that the 𝐷̃𝑝 slopes will be similar to inverse 𝜏̃𝛼,𝑝 slopes, in contrast to the +simulation observation that the 𝐷̃𝑝 slopes are much smaller. We attribute this disparity between +simulation and theory to the larger importance of dynamic heterogeneity for smaller penetrants in +our simulations,66 because the theory only makes predictions for a mean penetrant alpha time that +is closer to our value 𝜏̃𝛼,𝑝′. Indeed, for 𝑑 𝜎 +⁄ += 1.0 the relative slopes for 𝐷̃𝑝 and 1/𝜏̃𝛼,𝑝′ calculated +using this criterion (in parentheses in Table 1) are much more consistent with the theoretical +predictions.69 + +Figure 5. (a) Penetrant diffusion constant 𝐷̃𝑝 as a function of 𝑇𝑔(𝑓𝑐𝑟𝑜𝑠𝑠)/𝑇 for several +temperatures 𝑇 and penetrant sizes (a) 𝑑/𝜎 = 1.0 and (b) 𝑑/𝜎 = 2.0. These are the same + +(a) +(b) +10- +T/K +T/K +300 +口 +口 +300 +320 +KH +320 +10-4 +339 +339 +10-3 +KOH +~D +六 +10-4 +10-6 +d/g = 1.0 +d/ = 2.0 +0.80 +0.85 +0.90 +0.95 +0.80 +0.85 +0.90 +0.95 +21 +conditions as in Figure 4a and b, and similarly there is a collapse in (a) that is no longer present +in (b). The slopes of these plots are calculated and included in Table 1. + + +To contextualize the relationship between the diffusion constant 𝐷̃𝑝, inverse penetrant +alpha time 1/𝜏̃𝛼,𝑝, temperature 𝑇, and particle size 𝑑/𝜎, we plot both 𝐷̃𝑝 and 1/𝜏̃𝛼,𝑝 as a function +of 𝑇𝑔/𝑇 in Figure 6 for two temperatures each. We have used a multiplier 𝐴 to align these data +sets at the leftmost points, to compare how the effect of changing the crosslinking fraction 𝑓𝑐𝑟𝑜𝑠𝑠 +(and concomitantly the 𝑇𝑔) on 𝐷̃𝑝 versus 1/𝜏̃𝛼,𝑝. For small particles (𝑑 𝜎 +⁄ += 1.0), the value of 𝐷̃𝑝 +exhibits a weaker decrease with the degree of supercooling 𝑇𝑔/𝑇 than 1/𝜏̃𝛼,𝑝 for both temperatures +𝑇. We believe that this is again a dynamic heterogeneity effect, which manifests as the well-known +breakdown of the Stokes-Einstein relation in glassy liquids,59–66 where the diffusion constant is +understood to be dominated by the subpopulation of particles undergoing rapid transport and is +thus less dependent on 𝑇𝑔/𝑇 than the particle hopping time that characterizes a relaxation +process.66 To contrast, for larger particles (𝑑 𝜎 +⁄ += 2.0) the change in the diffusion constant 𝐷̃𝑝 +tracks the change in 1/𝜏̃𝛼,𝑝 with increasing 𝑓𝑐𝑟𝑜𝑠𝑠 at low temperatures (𝑇 = 300K) and even +decreases more than 1/𝜏̃𝛼,𝑝 with increasing 𝑓𝑐𝑟𝑜𝑠𝑠 at higher temperatures (𝑇 = 339K). This +indicates that there is an additional mechanism slowing down diffusive penetrant motion at length +scales longer than what is probed in the intermediate scattering function (the local molecular cage, +1/𝜏̃𝛼,𝑝), which we attribute to the effect of mesh confinement. + + +22 + +Figure 6. Comparison of the penetrant diffusion constant 𝐷̃𝑝 and inverse penetrant alpha time +1/𝜏̃𝛼,p as a function of 𝑇𝑔(𝑓𝑐𝑟𝑜𝑠𝑠)/𝑇 at both high (339K) and a low (339K) temperatures for (a) +𝑑/𝜎 = 1.0 and (b) 𝑑/𝜎 = 2.0. 1/𝜏̃𝛼,p is vertically shifted down by a factor (a) 𝐴 = 3.40 × 10−3 +and (b) 𝐴 = 2.55 × 10−3 such that the leftmost point of alpha time 1/𝜏̃𝛼,p at a given temperature +coincides with the corresponding 𝐷̃𝑝 data point for better comparison. In (a), this comparison +shows that 𝐷̃𝑝 exhibits a weaker dependence than 1/𝜏̃𝛼,p with 𝑇𝑔(𝑓𝑐𝑟𝑜𝑠𝑠)/𝑇 for all temperatures, +while in (b) 𝐷̃𝑝 exhibits a stronger dependence than 1/𝜏̃𝛼,p at high temperatures. + + +To isolate the role of mesh confinement, we refer to the form of Equation 1 that writes the +diffusion constant as the product of 1/𝜏̃𝛼,𝑝 and a mesh confinement factor 𝑋. The form predicted +for 𝑋 is model-dependent,23,24,26 but we can directly evaluate this quantity from simulation by +taking the product 𝑋 = 𝐷̃𝑝𝜏̃𝛼,𝑝. We plot 𝐷̃𝑝𝜏̃𝛼,𝑝 in Figure 7 for both penetrant sizes as a function +of 𝑇𝑔/𝑇 and 𝑇, in analogy with Figures 4 and 5. Figure 7a plots this product for small penetrants +(𝑑 𝜎 +⁄ += 1.0), and as expected from Figures 4a and 5a the data roughly collapses to a single curve +that increases monotonically with the degree of supercooling 𝑇𝑔/𝑇. This trend is generically +expected for one-component glass-forming liquids due to alpha process dynamic heterogeneity,66 +so in this case the network primarily affects penetrant transport through the dependence of 𝑇𝑔 on +𝑓𝑐𝑟𝑜𝑠𝑠. Notably, this monotonic increase in 𝐷̃𝑝𝜏̃𝛼,𝑝 is no longer seen when the 1/e criterion is + +(a) +b +d/ = 1.0 +d/α = 2.0 +10-3 +10-4 +口 +KOHH +空 +10-5 +TY +Y +Y +300, Dp +300, D. +300, A/tα.p +TOI +300, A/tα.p +10-6 +口 +339, D. +339, Dp +I +339, A/t +339, A/t. +α,p +10-5 +10-7 +0.80 +0.85 +0.90 +0.95 +0.80 +0.85 +0.90 +0.95 +23 +considered (see inset to Figure 7a), demonstrating that the long-time tail in the ISF should be +responsible for the behavior in the main panel of Figure 7a. To contrast, Figure 7b plots the +product 𝐷̃𝑝𝜏̃𝛼,𝑝 for larger penetrants (𝑑 𝜎 +⁄ += 2.0), and exhibits a significant decrease with 𝑇𝑔/𝑇 +for each of the higher temperatures 𝑇 = 339 and 320. For a constant temperature, the increase +with 𝑇𝑔/𝑇 corresponds to an increase in both 𝑓𝑐𝑟𝑜𝑠𝑠 and 𝐶. We can thus attribute this dependence +to the effect of mesh confinement (and also the reduced importance of glassy dynamic +heterogeneity since larger particles tend to average over it), as this decrease of 𝐷̃𝑝𝜏̃𝛼,𝑝 = 𝑋 with 𝐶 +is predicted by both Cai, et al.23 and Dell and Schweizer.26 However, this trend is less noticeable +for the lowest temperature 𝑇 = 300K, where within error 𝐷̃𝑝𝜏̃𝛼,𝑝 remains roughly constant. The +expected increase in 𝐷̃𝑝𝜏̃𝛼,𝑝 for glassy melts appears to cancel out any mesh confinement effects, +indicating that the local caging and mesh confinement both contribute to penetrant dynamics in +this limit. + +Figure 7. The product 𝐷̃𝑝𝜏̃𝛼,p as a function of 𝑇𝑔(𝑓𝑐𝑟𝑜𝑠𝑠)/𝑇 for several temperatures 𝑇 for (a) +𝑑/𝜎 = 1.0 and (b) 𝑑/𝜎 = 2.0 with the 0.1 criterion. For (a) there is a collapse to an increasing +curve that reflects the breakdown of Stokes-Einstein that is characteristic of dynamic heterogeneity +in glass forming liquids. The inset shows the same plot, but using the 1/e criterion. For (b) at high +temperatures, each temperature exhibits a monotonic decrease in the product 𝐷̃𝑝𝜏̃𝛼,p that we +attribute to the effect of mesh confinement. At low temperatures (𝑇 = 300K) and 𝑑/𝜎 = 2.0, the + +a +Tg(feross)/T +d/g = 1.0 +d/g = 2.0 +0.800.850.90 +0.95 +10-2 +TOI +α,p +TOH +10 +~D10 +10-3 +T/K +T/K +口 +300 +口 +300 +320 +320 +339 +339 +10-3 +0.80 +0.85 +0.90 +0.95 +0.80 +0.85 +0.90 +0.95 +24 +increase due to dynamic heterogeneity appears to counteract the effect of mesh confinement, +leading to a roughly constant value of 𝐷̃𝑝𝜏̃𝛼,p. + +Network and Penetrant Dynamic Decoupling and Particle Size + +We quantified penetrant dynamics, determining both the local hopping via 1/𝜏̃𝛼,𝑝 and the +long-time diffusion via 𝐷̃𝑝. These results demonstrated that (1) both processes were primarily +governed by local network segmental motion (i.e. glass physics) at low temperatures, yet (2) +crosslinks nevertheless had a more pronounced effect on 𝐷̃𝑝 for larger penetrants (𝑑/𝜎 = 2.0 +versus 𝑑 𝜎 +⁄ += 1.0 in Table 1). To understand these two observations, we quantify the coupling +between the penetrant alpha time 𝜏̃𝛼,𝑝′ and the network alpha relaxation time 𝜏̃𝛼,𝐾, where the latter +is obtained from the autocorrelation function of the Kuhn monomer vector per our prior work.35 In +this case, we use the short-time criterion 𝜏̃𝛼,𝑝′ due to its better consistency with both the theoretical +predictions and to match the 1/𝑒 criterion used in calculating 𝜏̃𝛼,𝐾 previously. The ratio between +these two properties 𝜏̃𝛼,𝑝′/𝜏̃𝛼,𝐾 is plotted as a function of 𝑇𝑔/𝑇 in Figure 8 for both penetrant sizes +and all temperatures 𝑇 studied; this comparison has previously demonstrated to be useful in the +self0consistent cooperative hopping (SCCH) theory of hard sphere mixtures and dilute penetrants +in polymer melts,20,58,70 and quantifies the degree of ‘dynamic coupling/decoupling’. In these +previous theories, the ratio 𝜏̃𝛼,𝑝′/𝜏̃𝛼,𝐾 was measured in the context of ‘packing fraction’, which is +qualitatively analogous to inverse temperature.58 The theory predicted that the ratio 𝜏̃𝛼,𝑝′/𝜏̃𝛼,𝐾 (1) +initially increases slightly as the temperature is lowered from high temperature, and then (2) +sharply decreases in a penetrant size-dependent manner (stronger decrease for smaller penetrants) +as temperature is further lowered towards the glassy state.58,69 This indicates a strong decoupling +between penetrant and matrix dynamics. + + +25 + +In Figure 8, both penetrants show only a modest dependence of 𝜏̃𝛼,𝑝′/𝜏̃𝛼,𝐾 on 𝑇𝑔/𝑇, +spanning only roughly a decade in the simulation-accessible weakly supercooled regime. We do +not observe the characteristic decrease in 𝜏̃𝛼,𝑝′/𝜏̃𝛼,𝐾 predicted for the deeply supercooled regime, +which is expected to be computationally inaccessible, though our trends are consistent with the +weak changes in 𝜏̃𝛼,𝑝′/𝜏̃𝛼,𝐾 predicted by the theory in the polymer alpha relaxation time range +probed in our simulation.69 Importantly, the relative magnitude of these ratios is instructive. +𝜏̃𝛼,𝑝/𝜏̃𝛼,𝐾 for the 𝑑/𝜎 = 2.0 penetrant is at least an order of magnitude larger than that for the +𝑑/𝜎 = 1.0 penetrant, indicating that it is more strongly coupled to the surrounding matrix as +generically expected and as predicted by SCCH theory. This helps explain our results for 1/𝜏̃𝛼,𝑝 +and 𝐷̃𝑝 in the previous section; in those results, there was a stronger dependence of 𝐷̃𝑝 for the +larger penetrants on 𝑓𝑐𝑟𝑜𝑠𝑠even at lower temperatures where mesh confinement is not dominant +(see Table 1). We can now attribute this to the stronger matrix-penetrant coupling for larger +penetrants, rather than enhanced mesh confinement. + +Figure 8. Degree of decoupling between penetrant and Kuhn segment dynamics, as quantified by +the ratio 𝜏̃𝛼,p'/𝜏̃𝛼,K, as a function of 𝑇𝑔(𝑓𝑐𝑟𝑜𝑠𝑠)/𝑇 for several temperatures and both 𝑑/𝜎 = 1.0 and +𝑑/𝜎 = 2.0. While the trends with 𝑇𝑔(𝑓𝑐𝑟𝑜𝑠𝑠)/𝑇 are quite weak, the larger penetrant 𝑑/𝜎 = 2.0 + +100 +T/K +300 +310 +10-1 +320 +K +口 +2.10-2 +d/ = 2.0 +d/g = 1.0 +10-3 +0.80 +0.85 +0.90 +0.95 +Tg(feross)/T +26 +couples significantly more strongly to the segmental dynamics of the surrounding network, as +evident in the order-of-magnitude larger value of 𝜏̃𝛼,p'/𝜏̃𝛼,K when compared to the smaller +penetrant 𝑑/𝜎 = 1.0. + +Degeneracy of Mesh Size Effects Due to Confinement and Glassy Dynamics + +From a molecular viewpoint, we only see significant mesh confinement effects on +penetrant diffusion in the limit of large particles and high temperatures, and for small molecules +or low temperatures show that crosslinking instead impacts penetrant diffusion primarily through +its effect on the local segmental relaxation of the polymer matrix. However, even in this limit we +can demonstrate results consistent with aspects of Equation 1 if we plot our data versus the +confinement ratio 𝐶 rather than the extent of supercooling 𝑇𝑔/𝑇. We consider two candidate +models for 𝑋, by plotting 𝐶𝐷𝑝 versus 𝐶2 in Figure 9a based on the predictions of Cai, et al.23 and +𝐷̃𝑝 versus 𝐶 in Figure 9b based on the predictions of Dell and Schweizer.26 Despite already +demonstrating that mesh confinement does not play a significant role for small penetrants, these +plots exhibit remarkably good agreement with the functional forms predicted by both models as +indicated by the nearly linear trends in these semi-log plots. However, there is a non-negligible +change in slope in both plots of Figure 9 as the temperature 𝑇 is changed, which is not a prediction +for either entropic mesh confinement model.23,25,26 As per Equation 1, these models that only have +a temperature dependence in the prefactor 1/𝜏𝛼,𝑝 (plotted also versus 𝐶 from simulation in Figure +S2) and not in the exponent of the entropic mesh confinement factor 𝑋. This key observation +allows the effects of crosslinking on the penetrant relaxation time to be distinguished from the +effects of crosslinking due to mesh confinement, in the absence of simulation determination of +1/𝜏𝛼,𝑝, though this distinguishing feature may become weak as in the case of larger penetrants as +is shown in the supporting information (𝑑/𝜎 = 2.0, Figure S3). + + +27 + +Figure 9. The penetrant diffusion constant 𝐷̃𝑝 plotted with respect to the confinement ratio 𝐶 as +suggested by the forms of various literature predictions for mesh confinement effects. (a) Cai, et +al.23 predicts that 𝐷̃𝑝 ∝ 𝐶−1 exp(−𝐶2) and (b) Dell and Schweizer26 predict that 𝐷̃𝑝 ∝ exp(−𝑏𝐶). +Both ways of plotting this data appear linear, even though mesh confinement does not play a major +role in penetrant transport for the 𝑑/𝜎 = 1.0 case plotted here. The main indication that the +simulations do not agree with the idea that entropic mesh confinement is dominant lie in the +temperature-dependent slopes, which for mesh confinement should not change with temperature. +Conclusion +In this paper, we used simulation to understand the role of glassy dynamics versus mesh +confinement on the diffusion and alpha relaxation of molecular penetrants in polymer networks. +We consider a specific form for the diffusion constant, 𝐷𝑝 ∼ (𝑑2 𝜏𝛼,𝑝 +⁄ +)𝑋, that allows us to isolate +the effect of network mesh confinement 𝑋 through calculating and studying both the diffusion +constant 𝐷̃𝑝 and the penetrant hopping or alpha relaxation time 𝜏̃𝛼,𝑝. Both quantities exhibit +Arrhenius-like trends with the degree of supercooling 𝑇𝑔/𝑇 at a fixed temperature, due to the +dependence of 𝑇𝑔 on the extent of crosslinking. This strong coupling between penetrant hopping +and crosslinking-dependent glassy dynamics is consistent with theoretical predictions in the +companion paper,69 but also exhibits significant breakdown in the Stokes-Einstein relationship if +a strong dynamic heterogeneity effect65–68 is apparent in the hopping dynamics of the smaller +molecular penetrants as the system is supercooled to larger values of 𝑇𝑔/𝑇.66 For small penetrants, + +(a) +(b) +d/g = 1.0 +d/a = 1.0 +10-3 +鱼 +10-3 +D +p +C +~D +T/K +T/K +口 +300 +口 +300 +10-4 +△ +320 +10-4 +△ +320 +HH +口 +339 +D +339 +0.2 +0.3 +0.4 +0.5 +0.6 +0.4 +0.5 +0.6 +0.7 +0.8 +C +C2 +28 +the effect of crosslinking on 𝑇𝑔 is demonstrated to dominate diffusive motion, with a collapse to a +single dependence of both 𝜏̃𝛼,𝑝 and 𝐷̃𝑝 on 𝑇𝑔/𝑇. This collapse is no longer apparent for larger +penetrants, where we show by plotting the product 𝐷̃𝑝𝜏̃𝛼,𝑝 with 𝑇𝑔/𝑇 that we see the signatures of +mesh confinement at low temperatures. While we show that crosslinking affects molecular +penetrant transport through both its effect on 𝑇𝑔 and through mesh confinement, the former appears +to be the dominant mechanism for our simulations. We can also show, however, that the diffusion +constant 𝐷̃𝑝 still follows trends predicted in the literature for mesh confinement,23,24,26 due to a +degeneracy of how crosslink fraction changes with 𝑇𝑔 and mesh size as predicted by SCCH +theory.69 This underscores the difficulty of drawing conclusions on transport mechanisms from +diffusion data alone. +Acknowledgement +This research was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, +Division of Materials Sciences and Engineering (Award No. DE-SC0020858), through the +Materials Research Laboratory at the University of Illinois at Urbana-Champaign. Helpful +discussions with Christopher Evans are gratefully acknowledged. +References: +(1) Blaiszik, B. J.; Kramer, S. L. B.; Olugebefola, S. C.; Moore, J. S.; Sottos, N. R.; White, S. R. +Self-Healing Polymers and Composites. Annu. Rev. Mater. Res. 2010, 40 (1), 179–211. +https://doi.org/10.1146/annurev-matsci-070909-104532. +(2) Patrick, J. F.; Robb, M. 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Representative plots of the effective exponent 𝛽 for the mean square displacement +versus time ⟨𝑟̃𝑝 +2(𝑡̃)⟩ ∼ 𝑡̃𝛽 of a penetrant particle with sizes (a) 𝑑/𝜎 = 1.0 and (b) 𝑑/𝜎 = 2.0, for +the same values of 𝑇 and 𝑓𝑐𝑟𝑜𝑠𝑠 as shown in Figure 2. 𝛽 is the slope of Figure 2, and both plots +show that penetrant diffusion reaches the Fickian diffusion limit of 𝛽 = 1 in the long-time region. +The open circles denote the length scale of the polymer network mesh size, as described in Table +S1. Intermediate regions with more noise reflect the need to perform several distinct computational +runs with different data output frequency. + + +Figure S2. The inverse penetrant alpha time 1/𝜏̃𝛼,𝑝 plotted with respect to the confinement ratio +𝐶 for penetrant particles with sizes (a) 𝑑/𝜎 = 1.0 and (b) 𝑑/𝜎 = 2.0. For both particle sizes, we +find that they are consistent with the trend 𝜏̃𝛼,𝑝 ∝ exp(−𝑏𝐶). This primarily reflects the local +glassy dynamics and not mesh confinement effects, and so this trend contributes to the apparent +degeneracy between the two mechanisms for crosslink effects on penetrant transport in networks. + +(a) +2.0 +T/K:300 +339 +0.11 +1.5 +0.25 +0.36 +0.50 +B1.0 +d/ = 1.0 +0.5 +d/ = 1.0 +0.0 +10-2 +100 +~102 +104 +106(b) +2.0 +T/K:300 +339 +0.11 +1.5 +0.25 +0.36 +0.50 +3 1.0 +0.5 +d/ = 2.0 +0.0 +10-2 +100 +~102 +104 +106 +t(a) +10-1 +d/a = 1.0 +空 +10-2 +克 +10-3 +T/K +口 +300 +320 +339 +10-4 +0.4 +0.5 +0.6 +0.7 +0.8 +C(b) +d/a = 2.0 +10-2 +空 +P +OT +T/K +口 +300 +10-4 +320 +339 +0.8 +0.9 +1.0 +1.11.2 +1.3 +1.4 1.5 +C +35 + +Figure S3: + +Figure S3. The penetrant diffusion constant 𝐷̃𝑝 plotted with respect to the confinement ratio 𝐶 as +suggested by the forms of various literature predictions for mesh confinement effects. (a) Cai, et +al.23 predicts that 𝐷̃𝑝 ∝ 𝐶−1 exp(−𝐶2) and (b) Dell and Schweizer26 predict that 𝐷̃𝑝 ∝ exp(−𝑏𝐶). +Both ways of plotting this data appear linear, even though mesh confinement does not play the +sole role in penetrant transport for the 𝑑/𝜎 = 2.0 case plotted here. + +Table S1. Mesh size (in unit of 𝜎) of networks at various fcross. +fcross +0.11 +0.25 +0.36 +0.50 +𝒅/𝝈 = 𝟏. 𝟎 +2.43 +1.75 +1.48 +1.28 +𝒅/𝝈 = 𝟐. 𝟎 +2.45 +1.76 +1.49 +1.29 + + +(a)10-3 +d/a = 2.0 +10-4 +10-5 +T/K +口 +300 +320 +10-6 +339 +0.5 +1.0 +1.5 +2.0 +2.5(b)10-3 +d/a = 2.0 +10-4 +KH +210-5 +T/K +口 +300 +10-6 +320 +339 +0.8 +0.9 +1.0 +1.11.2 +1.3 +1.4 1.5 +C \ No newline at end of file diff --git a/N9E4T4oBgHgl3EQf9g6Z/content/tmp_files/load_file.txt b/N9E4T4oBgHgl3EQf9g6Z/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9954ddc3497166ce9da0694ca66bee7ee180feda --- /dev/null +++ b/N9E4T4oBgHgl3EQf9g6Z/content/tmp_files/load_file.txt @@ -0,0 +1,1784 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf,len=1783 +page_content='1 Simulation Study of the Effects of Polymer Network Dynamics and Mesh Confinement on the Diffusion and Structural Relaxation of Molecular Penetrants Tsai-Wei Lin1,3, Baicheng Mei2,3, Kenneth S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Schweizer1,2,3, and Charles E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Sing1,3 1Department of Chemical and Biomolecular Engineering, University of Illinois at Urbana- Champaign 2Department of Materials Science and Engineering, University of Illinois at Urbana-Champaign 3Materials Research Lboratory, University of Illinois at Urbana-Champaign Abstract The diffusion of small molecular penetrants through polymeric materials represents an important fundamental problem, relevant to the design of materials for applications such as coatings and membranes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Polymer networks hold promise in these applications, because dramatic differences in molecular diffusion can result from subtle changes in the network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' In this paper, we use molecular simulation to understand the role that crosslinked network polymers have in governing the molecular motion of penetrants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' By considering the local, activated alpha relaxation time of the penetrant and its long-time diffusive dynamics, we can determine the relative importance of glassy dynamics versus mesh confinement on penetrant diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' We vary several parameters, such as the crosslinking density, temperature, and penetrant size, to show that crosslinks primarily affect molecular diffusion through modification of the matrix glass transition, with local penetrant hopping at least partially coupled to the segmental relaxation of the polymer network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' This coupling is very sensitive to the local glassy dynamics of the surrounding matrix, and we also show that penetrant transport is affected by dynamic heterogeneity at low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' To contrast, only at high temperatures and large penetrants does the effect of mesh 2 confinement become significant, even though penetrant diffusion empirically follows similar trends as established models of mesh confinement-based transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Introduction The diffusive transport of molecular species through polymers is an important fundamental problem, with the motion of small ‘penetrants’ being integral to the design of polymer materials for a variety of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' For example, barrier coatings,1–3 drug delivery vehicles,4,5 and self- healing microcapsules1,2,6 may be engineered to impede the transport of penetrants, while membranes are often designed to selectively separate or filter specific small penetrants (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' gas,7– 10 water,11–13 or organic molecules14,15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Membrane separations represent an especially important materials design challenge,7,15 due to significant interest in finding promising alternatives to distillation for chemical separations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Distillation is a costly and inefficient process that is the source of an estimated 10-15% of the world’s energy consumption,15 prompting a search for materials that can selectively transport small molecules based on their physicochemical features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' size, shape, or interactions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The predominant strategy for separating molecules is to impose control over pore size in materials such as zeolites,16 metal organic frameworks,17,18 and covalent organic frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='19 While these form precise structures that are selective to molecular size and chemistry, they are typically very brittle and difficult to synthesize at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='16–19 If the molecular penetrants are especially small, such as in gas separations, then glassy amorphous polymers can be another option for membranes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='8–10 These materials are more mechanically robust, but are limited to small gas molecules and do not have the same size-selectivity as more precise matrix structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Alternatively, rubbery polymer membranes can instead be tailored to use solubility to affect penetrant transport while retaining high rates of transport even for larger molecules,14 though this 3 again suffers from limited selectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Despite this abundance of promising strategies for designing separation membranes, the design principles for selectivity relies on a still-incomplete understanding how chemical structure and dynamics contributes to the diffusive transport of molecular penetrants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Highly-crosslinked networks have recently emerged as a promising option for the selective transport of small molecules,20–22 exploiting the sensitivity of penetrant diffusion to the near-𝑇𝑔 coupling between the structural relaxation of the dense molecular mesh and activated hopping processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' This is motivated by experimental support for the premise that, despite significant molecular disorder, dense polymer networks can discern small changes in penentrant size and interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='21 Design of crosslinked networks for selective transport requires an understanding of how the molecular-scale dynamics of penetrant and matrix relaxation contribute to large-scale diffusive motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Aspects of this problem have been studied in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' For example, there has been extensive work on studying particle transport through polymer solutions or high-temperature networks,23–29 which is understood to be governed by a confinement parameter 𝐶 = 𝑑/𝑎𝑥 that relates the penetrant size 𝑑 to the size of the network mesh 𝑎𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' This parameter quantifies the extent to which the surrounding polymer matrix sterically ‘traps’ the particle, such that the particle diffusion is an activated process where the network strands must entropically stretch to allow the particle to hop out of its location within the mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='23 Several expressions for the diffusion constant have been considered in this literature, such as the expression proposed by Cai, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' :23 𝐷𝑝 ∼ 𝑑2 𝜏 𝐶−1 exp(−𝑏𝐶2) = 𝑑2 𝜏 𝑋 (1) This relates the diffusion constant 𝐷𝑝 to the confinement parameter 𝐶, a length scale characterized by the penetrant size 𝑑, a time scale 𝜏 that Cai, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' identify as the Rouse time of the network strand, and a constant 𝑏 of order unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='23 While there are other candidate models,26 these still 4 exhibit a general form given in the second equivalence of Equation 1 as the product of two factors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' the elementary particle diffusion 𝑑2 𝜏 ⁄ and a factor 𝑋 dictated by the network mesh and related to 𝐶 (in the case above, 𝑋 = exp(−𝑏𝐶2) /𝐶).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Simulations of large penetrants in rubbery (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' high- temperature) networks are consistent with this physical picture;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='24,25,30 however, the choice of 𝜏 can become non-trivial as temperature is lowered for polymer melts and/or when the glass transition is approached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='31 For example, in uncrosslinked systems activated transport of penetrants and particles becomes dominated by molecular caging near the glass transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='31–34 For tight and near-𝑇𝑔 networks, the physical relationship in Equation 1 is complicated by recent experimental, simulation, and theoretical investigations by the authors and collaborators demonstrating that the structural relaxation in polymer networks is strongly dependent on the extent of crosslinking, manifesting as a monotonic increase in the material 𝑇𝑔 with crosslink density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='35 In this case, the time scale 𝜏 in Equation 1 is expected to depend on molecular penetrant caging, which itself will be sensitive to the confinement parameter 𝐶 due to its relationship with the crosslink density of the polymer network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Crosslinks could thus impact penetrant motion both by affecting the structural relaxation of the surrounding network as well as providing a confining mesh that obstructs penetrant motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' In this article, we use computer simulation to study the diffusion of penetrants in dense, highly-crosslinked polymer networks in the weakly-supercooled regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Simulation will allow us to separate out the different ways that crosslinks contribute to penetrant diffusion, by quantifying both the overall diffusion constant 𝐷𝑝 and the characteristic time scale of penetrant hopping 𝜏 = 𝜏𝛼,𝑝 that we identify with a penetrant alpha relaxation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' By relating particle hopping events to long-time diffusive motion, we can study how these properties are affected by both crosslinking fraction and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' We determine that (1) supercooled networks exhibit strong coupling 5 between the network segmental relaxation and penetrant hopping,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' with sensitivity to the crosslinking-dependent 𝑇𝑔,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' (2) increasing temperatures can lead to slower diffusion at a given 𝑇𝑔/𝑇 due to the dependence of 𝑇𝑔 on crosslink density,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' but this is mostly attributed to a non-trivial 𝑇 dependence of how local polymer relaxations couple to penetrant motion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' (3) the coupling between penetrant motion and polymer relaxation processes is dependent on penetrant size 𝑑,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' and (4) molecular penetrant hopping is correlated with long-time diffusion at several values of 𝑇𝑔/𝑇 near 𝑇 = 𝑇𝑔,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' but the mesh confinement only becomes apparent at high 𝑇 and for large penetrants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' This all leads to the overall insight that, under typical conditions for polymer networks and molecular penetrants, crosslinking primarily affects penetrant transport by changing the effective 𝑇𝑔 of the surrounding matrix and only secondarily through mesh confinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Serendipitously, this still manifests in a similar relationship between the diffusion constant 𝐷𝑝 and 𝐶 that is consistent with the form in Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='23,26 These molecular-level insights into penetrant diffusion and hopping clarify the fundamental mechanisms governing penetrant transport and help identify ways in which both temperature and network architecture can be used to engineer selectivity into a promising class of membrane materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Simulation Methods We use coarse-grained molecular dynamics simulations (MD) to model the diffusion of spherical penetrants in crosslinked polymer networks,36 using well-established methods for studying polymers near the glass transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='37–40 Our system is initially composed of 𝑁c = 20 linear chains with 𝑁m = 30 beads each with diameter 𝜎, modeled as standard semiflexible chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' These beads are placed in a cubic box with periodic boundary conditions in three dimensions, along with 𝑁p spherical penetrants of diameter 𝑑̃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The number of penetrants 𝑁p depends on 𝑑̃, and is chosen so that the volume fraction of the penetrant 𝜙𝑝 = 𝜋𝑑3𝑁𝑝 6𝑉 is kept below 𝜙𝑝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='01 to 6 ensure that the addition of penetrants will not affect network dynamics and to prevent significant interactions between penetrant molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Standard dimensionless simulation quantities are employed,36,41 and denoted with tildes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' 𝑇̃ = 𝑇/𝑇∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Quantities are rendered dimensionless by several characteristic values: lengths are related to the bead diameter 𝜎, energies are related to the thermal energy 𝑘𝐵𝑇∗, and times are related to a time scale 𝜏∗ = √𝑚𝜎2/𝑘𝐵𝑇∗, where 𝑚 is the monomer mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The characteristic temperatures 𝑇∗ corresponds to 485 K, which is determined from parametrization against experiments,35 using crosslinked poly(n-butyl acrylate) (PnBA) as a representative polymer that has been used by our experimental collaborators (see details in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' 35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Our MD simulations consider standard potentials for coarse-grained bead-spring polymers,42 including a Lennard-Jones potential 𝑈̃LJ between all non-bonded beads, and both bonding 𝑈̃B and bending 𝑈̃𝜃 potentials to model each semiflexible bead-spring chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The overall energy is composed of a sum of these contributions: 𝑈̃ = 𝑈̃B + 𝑈̃𝜃 + 𝑈̃LJ = ∑ 𝑢̃B,𝑖𝑗 𝑁𝑡𝑜𝑡,𝑛 𝑖,𝑗>𝑖 + ∑ 𝑢̃𝜃,𝑖𝑗𝑘 𝑁𝑡𝑜𝑡,𝑛 𝑖,𝑗>𝑖,𝑘>𝑖,𝑗 + ∑ ∑ ∑ ∑ 𝑢̃LJ,𝛼𝛽,𝑖𝑗 𝑁𝑡𝑜𝑡,𝛽 𝑗>𝑖 𝑁𝑡𝑜𝑡,𝛼 𝑖 𝛽=𝑛,𝑝 𝛼=𝑛,𝑝 (2) Here the total system energy is written in terms of independent pairwise contributions 𝑢̃B,α,𝑖, 𝑢̃𝜃,𝛼,𝑖, and 𝑢̃LJ,𝛼𝛽,𝑖𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Bonded monomers interact through harmonic bonding potential: 𝑢̃B,𝑖𝑗 = 𝑘̃ 2 (𝑟̃𝑖𝑗 − 1) 2Θ𝑖𝑗 (3) A large spring constant 𝑘̃ = 2000 is adopted to enforce the equilibrium distance 𝑟̃𝑖𝑗 = 1 between bonded beads 𝑖 and 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The factor Θ𝑖𝑗 determines the connectivity between monomer pairs, with 7 Θ𝑖𝑗 = 1 being connected and Θ𝑖𝑗 = 0 being disconnected;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' these values depend on the specific network that is formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' A bending energy is similarly introduced to account for chain stiffness: 𝑢̃𝜃,𝑖𝑗𝑘 = 𝑘̃𝜃[1 − cos𝜃𝑖𝑗𝑘]Θ𝑖𝑗𝑘 (4) Here, the bond angle is between three adjacent beads 𝑖, 𝑗, and 𝑘 whose connectivity is determined by a similar factor Θ𝑖𝑗𝑘 to the one used for the bonding potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The bending constant 𝑘̃𝜃 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='52 is selected to reflect the experimental Kuhn length of PnBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='43–46 Further details of the parametrization are given in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='35 A Lennard-Jones (LJ) potential is used to describe all non-bonded interactions between particles 𝑖 and 𝑗 on species 𝛼 and 𝛽: 𝑢̃LJ,𝛼𝛽,𝑖𝑗 = {4𝜖̃𝛼𝛽 [( 𝑑̃𝛼 + 𝑑̃𝛽 2𝑟̃𝛼𝛽 ) 12 − ( 𝑑̃𝛼 + 𝑑̃𝛽 2𝑟̃𝛼𝛽 ) 6 ] 0, otherwise , 𝑟̃𝛼𝛽 < 𝑟̃cut = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='5 × 𝑑̃𝛼 + 𝑑̃𝛽 2 (5) Here, 𝛼, 𝛽 ∈ {n, p}, denotes the species as either network (n) or penetrant (p), with n denoting the monomer bead in the network and p denotes the penetrant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' In the overall summation, 𝑁𝑡𝑜𝑡,𝑛 = 𝑁𝐶𝑁𝑚 is the total number of network beads and 𝑁𝑡𝑜𝑡,𝑝 = 𝑁𝑝 is the total number of penetrant beads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The monomer size 𝑑̃n = 1 is always unity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=', 𝑑n = 𝜎) and the penetrant size 𝑑̃p ≡ 𝑑̃ is an important parameter for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' In this study, we do not consider specific penetrant-polymer attractions, so keep 𝜖̃nn = 𝜖̃np = 𝜖̃pp = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' 8 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' (a) Schematic illustrating the setup of our simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Monomer beads (orange) and crosslinker beads (blue) interact via bonding (𝑢̃B,𝑖𝑗) and angle (𝑢̃𝜃,𝑖𝑗𝑘) potentials, and can interact with each other and with the penetrant through Lennard-Jones potentials (𝑢̃LJ,𝑖𝑗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Monomer and crosslinker beads have a diameter 𝑑̃𝑛 = 𝜎 and the penetrant has a diameter 𝑑̃𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Crosslinking occurs during an initial simulation step where ‘reactive’ beads (light blue) will form bonds with regular monomers within a cutoff distance 𝑅̃min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' (b) Snapshot of a typical simulation, for 𝑑 𝜎 ⁄ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 and 𝑓𝑐𝑟𝑜𝑠𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The system is first equilibrated at 𝑇̃ = 1, 𝑃̃ = 0, and then networks are prepared by crosslinking the linear chains with reactive beads randomly distributed along the chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='35,47,48 The total number of reactive beads is 𝑁r = 𝑓r𝑁m𝑁c, where 𝑓r is the fraction of reactive beads which tunes the crosslink density of networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' If the distance between a reactive bead and a free bead (orange beads in Figure 1) is within 𝑅̃min = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='1, a new permanent bond will be formed given an assigned probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Once a reactive bead forms a bond with a free bead, both the reactive bead and the free bead are labeled as crosslink beads (dark blue beads in Figure 1) and these two beads represent one crosslink ‘molecule’, in a way that reflects the specific chemistry used to synthesize PnBA networks,21 where the crosslinker molecular weight is roughly twice of the molecular weight of nBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Crosslinking reactions were turned off once every reactive bead has formed a new bond with another free monomer (maximum number of possible bonds has been reached).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' In the end, 9 𝑁r reactive beads have reacted with the 𝑁r free beads and they turned into 2𝑁r crosslink beads and belong to the 𝑁r crosslink molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The crosslink density, 𝑓cross, of networks is defined as:35 𝑓cross = 𝑛crosslink 𝑛crosslink + 𝑛monomer = 𝑁r 𝑁m𝑁c − 𝑁r (6) Here, 𝑛crosslink and 𝑛monomer are the number of crosslink molecules and nBA monomers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Four values of crosslink fraction 𝑓cross are considered: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='11, 025, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='36, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' After crosslinking, the system is cooled to a target temperature at a cooling rate 𝛤̃ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='3 × 10−6 (corresponding to 𝛤 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='25 × 109 K/s in experimental units) and further equilibrated at constant 𝑃̃ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Another short NPT run was performed and the mean volume 𝑉 is measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' We then switch to a NVT ensemble by setting the system volume to the mean volume 𝑉 and equilibrate the system before final production run at NVT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' All simulations are performed in LAMMPS49 with a standard Nosé-Hoover thermostat and barostat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='36,50,51 The polymer network is characterized by its mesh size (𝑎̃𝑥), Kuhn segment alpha relaxation time (𝜏̃𝛼,K), and glass transition temperature Tg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The mesh size of networks at different crosslink densities is defined as the averaged distance between two adjacent crosslink beads on the same chain, and is tabulated in Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The value of 𝜏̃𝛼,K at each temperature and crosslink density was defined as the time where the temporal autocorrelation function of a Kuhn monomer vector, 𝐶𝜆(𝑡̃) = 〈𝑃2(𝒓̃3(𝑡̃) ∙ 𝒓̃3(0))〉 decays to 𝐶𝜆(𝑡̃max) + (1 − 𝐶𝜆(𝑡̃max)) 𝑒 ⁄ , where 𝐶𝜆(𝑡̃max) represents the plateau value at the long-time limit 𝑡̃max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='52–56 Here, 𝑃2 is the second Legendre polynomial and 𝒓̃3 is a vector between two beads that are 3 bonds apart which reflects the choice of coarse-grained bead relate to Kuhn segment of PnBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' 𝑇g is then defined as when the Kuhn segment alpha relaxation time is 𝜏̃𝛼,K(𝑇̃g) = 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' More information can be found in our previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='35 10 We characterize the dynamics of the penetrant motion by its alpha relaxation time, 𝜏̃𝛼,p, and its diffusion coefficient 𝐷𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' To calculate 𝜏̃𝛼,p, we use the self-intermediate scattering function (ISF) of the penetrant, which is given by:53 𝐹𝑠(𝒒̃, 𝑡̃) = 1 𝑁𝑝 ∑ ⟨exp [−𝑖𝒒̃ ∙ (𝒓̃𝑝𝑗(𝑡̃) − 𝒓̃𝑝𝑗(0))]⟩ 𝑁𝑝 𝑗 (7) where 𝒓̃𝑝𝑗 is the position of the 𝑗th penetrant at time 𝑡̃, and |𝒒̃| is set to the position of the first peak of the static structure factor for the networks, corresponding to |𝒒̃| = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The value of 𝜏̃𝛼,p, is defined as the time required for 𝐹𝑠(𝒒̃, 𝑡̃) to decay to either 1/𝑒 or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='1 from its initial (𝑡̃ = 0) value of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' We characterize the mean-squared displacement (MSD) as a function of time: MSD(𝑡̃) = 〈𝑟̃𝑝 2(𝑡̃)〉 = 1 𝑁𝑝 ∑ ⟨(𝒓̃𝑝𝑗(𝑡̃) − 𝒓̃𝑝𝑗(0)) 2 ⟩ 𝑁𝑝 𝑗 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The diffusion coefficient of penetrant is then obtained via the Einstein relation in the diffusive regime, 𝐷𝑝 = lim 𝑡̃→∞ 1 6𝑡 〈𝑟̃𝑝 2(𝑡̃)〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='57 Results and Discussion Penetrant Hopping and Diffusion in Dense Networks We first characterize the diffusive motion of penetrants in polymer networks by studying the mean square displacement, ⟨𝑟̃𝑝 2(𝑡̃)⟩, of both small (𝑑/𝜎 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0) and large ( 𝑑/𝜎 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0) penetrants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The choice of these two particle sizes is motivated by considering particle sizes characteristic of model experimental systems in previous work by some of the authors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='22 using PnBA as our representative network polymer (coarse-grained so that 𝜎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='573nm), the span 𝑑 𝜎 ⁄ ∼ 1 − 2 is typical for the average size of molecular penetrants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='22 This choice also corresponds to the general length scale of the mesh size in the literature, which here ranges size from 𝑎̃𝑥 ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='3 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='5 (see Table S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The MSD versus time 𝑡̃ is plotted for both particle sizes in Figures 2a and 2b for 𝑑/𝜎 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 and 𝑑/𝜎 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 respectively, varying both the crosslink density 𝑓𝑐𝑟𝑜𝑠𝑠 and 11 temperature 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' These plots show typical features characteristic of molecular diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' At short times, the particle exhibits ballistic motion such that the MSD scales as ⟨𝑟̃𝑝 2(𝑡̃)⟩ ∼ 𝑡̃2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' however, this regime gives way to a subdiffusive regime that we attribute to molecular caging by neighboring network monomers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' At long times, the particle can overcome the barrier imposed by these cages, along with any constraints due to the network mesh, and undergo Fickian diffusion where ⟨𝑟̃𝑝 2(𝑡̃)⟩ ∼ 𝑡̃1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' These general features of diffusive motion are observed for both penetrant sizes and all crosslink densities and temperatures, however the specifics of penetrant caging and diffusive motion will be affected by these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The subdiffusive regime following the initial, ballistic penetrant motion provides insight into the time scales of caging, and large differences in the persistence of this regime are observed as 𝑓𝑐𝑟𝑜𝑠𝑠, 𝑇, and 𝑑/𝜎 are varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' To characterize this regime, we plot in Figure S1 the slope of the MSD versus 𝑡̃ curve on a log-log plot (𝛽 = 𝑑 log⟨𝑟̃𝑝 2(𝑡)⟩ 𝑑 log 𝑡̃ ⁄ ) as a function of time 𝑡̃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The minimum value of 𝛽 is an estimate for the time 𝑡̃𝛽 at which the penetrant is maximally subdifusive or caged, and the value of ⟨𝑟̃𝑝 2(𝑡̃𝛽)⟩ 1/2 correspondingly quantifies a transient localization length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' A representative quantity is indicated in Figures 2a and 2b with an open square, showing the caging onset for 𝑇̃ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='62 and 𝑓𝑐𝑟𝑜𝑠𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' We only indicate a single representative value for each particle size, because the displacement ⟨𝑟̃𝑝 2(𝑡̃𝛽)⟩ 1/2 is only weakly dependent on temperature and crosslink densities, and is far smaller than the length scale of the penetrant diameter (⟨𝑟̃𝑝 2(𝑡̃𝛽)⟩ 1/2 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='22 for 𝑑 𝜎 ⁄ = 1 and ⟨𝑟̃𝑝 2(𝑡̃𝛽)⟩ 1/2 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='32 for 𝑑 𝜎 ⁄ = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' This insensitivity leads us to interpret this onset of caging as tied to the structural correlations of the surrounding melt beads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' 12 For both small (𝑑 𝜎 ⁄ = 1) and large (𝑑 𝜎 ⁄ = 2) penetrants, the value of 𝛽 → 1 at long times limits to Fickian behavior, however the time that it takes to reach this limit increases with decreasing temperature (Figure 2 and Figure S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' This is the expected result of having less thermal energy to overcome the dynamic caging barrier,20,58 especially as our simulations approach the 𝑇𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Similarly, an increase crosslink density 𝑓𝑐𝑟𝑜𝑠𝑠 monotonically increases the length of this subdiffusive regime and the minimum value of 𝛽 concomitantly decreases (Figure S2) so that the MSD nearly exhibits a transient plateau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' In principle, this is due to some combination of (1) an increase in the caging barrier because of the concomitant increase in the segmental relaxation time due to crosslinks35 and (2) the impact of mesh confinement impeding penetrant motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='23,24,26 In this paper, a major result will be to show that the first interpretation is the dominant determinant of penetrant mobility for the models studied, especially as 𝑇𝑔 is approached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' This is apparent already by comparing the features in the MSD versus 𝑡̃ to the mesh size 𝑎̃𝑥, which were tabulated in Table S1 for several different values of 𝑓𝑐𝑟𝑜𝑠𝑠 and indicated in Figures 2 and S1 as open circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' This length scale is often only accessed in or near the Fickian regime, suggesting that any effect on the subdiffusive transport of the penetrant in these simulations is subtle and not readily apparent from the MSD versus 𝑡̃ plot alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Despite this observation, it is apparent that both the temperature and crosslink density effects still depend on the penetrant size, with more pronounced changes in the subdiffusive regime for the larger (𝑑 𝜎 ⁄ = 2) penetrants than the smaller (𝑑 𝜎 ⁄ = 1) penetrants as 𝑇 and 𝑓𝑐𝑟𝑜𝑠𝑠 are varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' 13 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Mean square displacement ⟨𝑟̃𝑝 2(𝑡̃)⟩ versus time 𝑡̃ of a penetrant particle with sizes (a) 𝑑/𝜎 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 and (b) 𝑑/𝜎 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0, for several values of 𝑇 and 𝑓𝑐𝑟𝑜𝑠𝑠 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Open squares denote a representative location for the onset of caging or transient localization, defined as the minimum slope 𝛽 on this plot (see Figure S1) and chosen for 𝑇 = 300K and 𝑓𝑐𝑟𝑜𝑠𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The analogous point for other values of 𝑇 and 𝑓𝑐𝑟𝑜𝑠𝑠 would be almost identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The open circles denote the length scale of the polymer network mesh size, as described in Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The long-time data Fickian region, where the slope of this plot 𝛽 = 1, is used to determine the penetrant diffusion constant 𝐷̃𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The MSD provides a molecular measure of penetrant transport, with the long-time Fickian regime providing an estimate of the overall diffusion coefficient 𝐷̃𝑝 of the penetrant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' However, this quantity is informed by the cumulative effect of both local particle motions as well as any larger length-scale transport effects of the network mesh confinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' To deconvolute these contributions, we also consider the time evolution of the self-ISF 𝐹s(𝒒̃, 𝑡̃) in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Here we select a value 𝒒̃ = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='2 that is chosen to reflect the local structural cage of the polymer network and not the length scale of the network mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The curves shown in Figure 3 thus only account for the dynamics of local relaxation and hopping, and quantify the extent to which the penetrant remains within its original cage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The first part of this decay is independent of temperature and crosslinking, and corresponds to the initial ballistic motion of the penetrant observed in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' There are (a) (b) 105 105 T/K: 300 339 T/K: 300 339 cross 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 cross 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='11 103 103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='50 t 101 101 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='p 2p 710-1 10-3 10-3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 d/ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 d/ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 10-5 10-5 10-1 101 ~t 103 105 10-1 101 ~t 103 105 14 some subtle differences between the different penetrant sizes (𝑑 𝜎 ⁄ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0) that we attribute to the different masses of the penetrants and is also apparent in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' At longer times, the decay of 𝐹s(𝒒, 𝑡̃) becomes strongly dependent on temperature, crosslinking, and penetrant size in a manner consistent with the diffusive motion in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' At high temperatures, the smaller penetrant (Figure 3a) exhibits a non-exponential, long tail of 𝐹s(𝒒, 𝑡̃) that we attribute to an activated but relatively rapid hopping process for penetrant relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' As crosslinking increases, this long-time tail extends further, and its non-exponential character becomes more pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' This trend is exacerbated by going to lower temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' We find there is a significant slowing down of penetrant motion even at intermediate times;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' this occurs around 𝑡̃ ∼ 1, which is roughly when the minimum in the apparent exponent 𝛽 occurs for the MSD curves (in Figure 2 and S1) and is consistent with the interpretation of this as the onset of penetrant caging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' For larger penetrants (𝑑 𝜎 ⁄ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0, Figure 3b) this slowing down leads to the near-arrest of particle motion, again at 𝑡̃ ∼ 1, as evidenced by a plateau in 𝐹s(𝐪, 𝑡̃) that only begins to relax at significantly longer times as the temperature is decreased or crosslinking fraction is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' In both cases, we can attribute this arrest to the slow segmental relaxation dynamics of the surrounding network, as 𝐹s(𝐪, 𝑡̃) is a local measure of hopping that does not probe the length scales relevant for mesh confinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The trends seen in the intermediate scattering function plots in Figure 3 are largely consistent with those of the MSD plots in Figure 2, however we note several important differences that we will use to understand the mechanisms of penetrant motion in polymer networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' First, we note the similarity of the MSD versus time plots in Figure 2a for low temperatures and crosslink density (𝑇 = 300K and 𝑓𝑐𝑟𝑜𝑠𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='11) and the high temperatures and crosslink density (𝑇 = 339K and 𝑓𝑐𝑟𝑜𝑠𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='36);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' these curves almost overlap, and exhibit essentially the same diffusive 15 properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Yet, the shapes of these curves in Figure 3a are distinctly different, with the characteristics of local caging more apparent in the lower temperature curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' In addition, there is a distinct difference in the subdiffusive regimes seen in the MSD plots versus 𝐹s(𝐪, 𝑡̃), which are much more pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' This is contrast to normal glass-forming liquids, for which the same supercooling degree exhibits a weaker or shorter caging regime than for the MSD when compared to 𝐹s(𝐪, 𝑡̃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='59–66 These disparities between the MSD versus time and 𝐹s(𝐪, 𝑡̃) suggest that mesh confinement may contribute meaningfully to the transport of penetrants in tight networks, and we seek to refine our physical understanding of this complicated interplay of local and long-time penetrant and network dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Intermediate scattering function 𝐹s(q, 𝑡̃) of the penetrant particle with (a) 𝑑/𝜎 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 and (b) 𝑑/𝜎 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 with |q| = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='2 at 𝑇 = 300K (solid, blue) and 339K (dashed, gray) for a variety of crosslink densities 𝑓𝑐𝑟𝑜𝑠𝑠 as a function of time 𝑡̃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The orange and purple horizontal lines define the penetrant alpha time, with the criteria 𝐹s(q, 𝜏𝛼,𝑝) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='1 and 𝐹s(q, 𝜏𝛼,𝑝′) = 1/𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The former criteria (𝜏𝛼,𝑝) is more often used in this paper, because it includes the long tails observed in these functions that reflect a subpopulation of penetrants in long-lived cages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The latter criteria (𝜏𝛼,𝑝′) may more closely relate to the average hopping time that is predicted in the theory developed in our companion paper, which does not account for dynamic heterogeneity and a distribution of penetrant hopping times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' (a) (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 T/K: 300 339 T/K: 300 339 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='6 #0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='50 d/ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 q9 d/ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 E° 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='4 1/e F°0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='4 1/e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 10-2 10-1 100 101 102 103 10-2 10-1 100 101~ 102 103 104 105 t t 16 Comparing the Role of Particle Size, Temperature, and Crosslinking on the Hopping and Diffusive Processes in Dense Networks To quantify the relationship between local penetrant relaxation and diffusive motion, we extract the alpha time of the penetrant 𝜏̃𝛼,𝑝 from the ISF and the diffusion constant 𝐷̃𝑝 from the MSD versus 𝑡̃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' As described in the methods section, the diffusion constant is obtained from the long-time MSD curve via the expression 𝐷𝑝 = lim 𝑡→∞ 1 6𝑡 〈𝑟̃𝑝 2(𝑡̃)〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The alpha time of the penetrant 𝜏̃𝛼,𝑝 is set from the decay of 𝐹s(𝒒, 𝑡̃), and quantifies the time it takes to reach one of two possible criteria: 𝐹s(𝒒, 𝜏̃𝛼,𝑝) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='1 and 𝐹s(𝒒, 𝜏̃𝛼,𝑝′) = 1/𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' While the specific criterion is arbitrary, these two criteria will be used to illustrate the effect of dynamic heterogeneity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='65–68 the quantity 𝜏̃𝛼,𝑝 will account for the effect of the long-time tails in Figure 3, which do not contribute to the value 𝜏̃𝛼,𝑝′ that we deduce is closer to the average hopping time predicted in the accompanying theoretical work based on comparisons of the theory and simulation extracted relaxation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='69 Differences between how 𝜏̃𝛼,𝑝 and 𝜏̃𝛼,𝑝′ depend on 𝑓𝑐𝑟𝑜𝑠𝑠 do depend on both 𝑇 and 𝑑/𝜎, and as we will show can be very small or substantial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' We will primarily report 𝜏̃𝛼,𝑝, considering 𝜏̃𝛼,𝑝′ only where the comparison is instructive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' We can thus extract the key quantities in Equation 1 directly from molecular simulations, the long-time diffusion 𝐷̃𝑝 and the hopping time 𝜏̃𝛼,𝑝, and examine how they are related and affected by the state of the network and penetrant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' We start by considering 𝜏̃𝛼,𝑝, which we indicated in Figure 3 and is typically between 𝜏̃𝛼,𝑝 = 101 − 103 for 𝑑 𝜎 ⁄ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 and 𝜏̃𝛼,𝑝 = 101 − 104 for 𝑑 𝜎 ⁄ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Our goal is not to determine any precise dependence on 𝑑 𝜎 ⁄ , but rather to explore the differences between relatively ‘small’ and ‘large’ molecular penetrants in the context of experimental systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' We plot the inverse penetrant mean alpha times 1/𝜏̃𝛼,𝑝 or 1/𝜏̃𝛼,𝑝′ as a function of 𝑇𝑔(𝑓𝑐𝑟𝑜𝑠𝑠)/𝑇 for several 17 different temperatures 𝑇 and crosslink fractions 𝑓𝑐𝑟𝑜𝑠𝑠 in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' We emphasize that the abscissa combines two parameters we consider in our model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' the temperature 𝑇 is plotted in different colors in all panels of Figure 4a and b for 𝑑 𝜎 ⁄ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 and 𝑑 𝜎 ⁄ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0, respectively, and is directly set in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' However, we note that 𝑇𝑔/𝑇 has multiple values for a given 𝑇 due to the dependence of 𝑇𝑔 on 𝑓𝑐𝑟𝑜𝑠𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='35 For the Figure 4a inset, we replot the data to show the corresponding plot of 1/𝜏̃𝛼,𝑝 versus 𝑇𝑔/𝑇, but grouping data by 𝑓𝑐𝑟𝑜𝑠𝑠 instead of 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Inverse penetrant alpha time 1/𝜏̃𝛼,𝑝 calculated from simulations as a function of 𝑇𝑔(𝑓𝑐𝑟𝑜𝑠𝑠)/𝑇 for several temperatures 𝑇 and penetrant sizes (a) 𝑑/𝜎 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 and (b) 𝑑/𝜎 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' In (a) there is a noticeable collapse to a single curve, which we illustrate more clearly in the inset by grouping with respect to 𝑓𝑐𝑟𝑜𝑠𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' This collapse is no longer observed in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' We also plot the shorter- time criteria for the penetrant alpha time 1/𝜏̃𝛼,𝑝′ versus 𝑇𝑔(𝑓𝑐𝑟𝑜𝑠𝑠)/𝑇 in (c) for /𝜎 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 to demonstrate that the coupling of penetrant hopping to crosslink density is noticeably weaker when the long-time tail in the intermediate scattering function is no longer included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The slopes of these plots are calculated and included in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Figures 4a and b show that at a fixed 𝑇, the penetrant alpha relaxation time 1/𝜏̃𝛼,𝑝 behaves exponentially with 𝑇𝑔(𝑓𝑐𝑟𝑜𝑠𝑠), exhibiting Arrhenius-like behavior for both penetrant sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' As the temperature increases, the absolute value of the slope of this curve decreases, demonstrating a stronger contribution from crosslink-induced 𝑇𝑔 enhancement at lower temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' This underscores the importance of the network segmental dynamics on penetrant diffusion, and can be quantified by a change in the apparent activation energy that is proportional to the slope log 𝜏̃𝛼,𝑝 (a) (b) (c) 10-1 T/K T/K 100 T/K 300 300 口 口 300 10-2 320 320 320 339 339 339 T,(fero 10-2 0/T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 TY 10-1 10-3 口 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='11 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='25 10-3 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='36 Q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='50 d/a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 d/ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 d/ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 10-4 10° 10-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='95 18 versus 𝑇𝑔/𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' To highlight how the choice of criterion 𝐹s(𝒒, 𝜏̃𝛼,𝑝) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='1 affects our results, we plot in Figure 4c how Figure 4a would be changed upon choosing the alternative criterion, 𝐹s(𝒒, 𝜏̃𝛼,𝑝′) = 1/𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' In this case, similar trends are observed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' an increasing apparent activation energy with decreasing temperature), however the magnitudes of the slopes are significantly smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' We speculate that this is due to the neglect of longer time-scale processes that are apparent in the extended tails seen in Figure 3, meaning that the 1/e criterion does not fully capture the hopping processes thereby justifying our use of the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='1 criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' For both criteria and penetrant sizes 𝑑/𝜎, the values for the slopes of log 𝜏̃𝛼,𝑝 versus 𝑇𝑔/𝑇 are calculated and tabulated in Table 1 for different values of 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' We note that for both penetrant sizes, these slopes exhibit similar trends to the predictions from theory in our companion paper, showing a modest decrease in the magnitude of the slope with increasing temperature for both 𝜏̃𝛼,𝑝 and 𝜏̃𝛼,𝑝′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' However, the magnitudes of the slopes are significantly larger for 𝜏̃𝛼,𝑝 than the slopes for 𝜏̃𝛼,𝑝′, with the latter values in parenthesis in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' We attribute this to the inclusion of the long-time tails of the ISF in 𝜏̃𝛼,𝑝, which indicate that penetrant hopping is influenced by dynamic heterogeneity near 𝑇𝑔 due to the slower relaxation of a significant fraction (~5-10%) of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='66 To contrast, the long-time tails (dynamic heterogeneity) do not (or weakly/negligibly) affect 𝜏̃𝛼,𝑝′, leading to a weaker apparent coupling between the near-𝑇𝑔 dynamics of the network and the penetrant hopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Slope for log 𝜏̃𝛼,𝑝 or 𝐷̃𝑝 −1 versus 𝑇g/𝑇 from simulations at several temperatures (𝑇 = 300, 320, and 339K) and penetrant sizes (𝑑/𝜎 = 1 and 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Both criteria 𝜏̃𝛼,𝑝 and 𝜏̃𝛼,𝑝′ (in parentheses) are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' T/K 𝒅/𝝈 300 320 339 𝐥𝐨𝐠 (𝝉̃𝜶,𝐩) vs 𝑻𝐠/𝑻 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='57 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='79) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='59 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='98) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='01 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='97) 𝐥𝐨𝐠 (𝟏/𝑫𝐩) vs 𝑻𝐠/𝑻 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='34 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='48 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='79 𝐥𝐨𝐠 (𝝉̃𝜶,𝐩) vs 𝑻𝐠/𝑻 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='13 (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='10) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='86 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='78) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='80 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='60) 19 𝐥𝐨𝐠 (𝟏/𝑫𝐩) vs 𝑻𝐠/𝑻 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='47 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='27 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='12 The two different penetrant sizes in Figure 4a and 4b exhibit similar trends, including nearly the same overall decrease of relaxation rate by a little over two decades (over all values of 𝑇 and 𝑓𝑐𝑟𝑜𝑠𝑠 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' However, there are subtle but important differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' At a fixed value of 𝑇𝑔(𝑓𝑐𝑟𝑜𝑠𝑠)/𝑇 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' at a fixed supercooling degree), we observe that the penetrant alpha time is roughly constant with decreasing temperature for 𝑑 𝜎 ⁄ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' This lower temperature also corresponds to a lower degree of crosslinking 𝑓𝑐𝑟𝑜𝑠𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' This means that the degree of supercooling 𝑇𝑔(𝑓𝑐𝑟𝑜𝑠𝑠)/𝑇 largely dictates penetrant hopping, with the absolute temperature 𝑇 playing a similar role in affecting transport as the effective 𝑇𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' In this case, several effects – changing 𝑇𝑔, absolute 𝑇, and crosslinking (𝑓𝑐𝑟𝑜𝑠𝑠) – tend to cancel resulting in a near collapse of the 1/𝜏̃𝛼,𝑝 versus 𝑇𝑔/𝑇 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' This is highlighted in the inset of Figure 4a, which is the same data as in the main frame but grouped by 𝑓𝑐𝑟𝑜𝑠𝑠 instead of 𝑇, and more clearly shows how well this data collapses to a single curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Figure 4b shows that this balance no longer holds for larger penetrants 𝑑 𝜎 ⁄ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0, with lower 𝑇 values exhibiting faster dynamics (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' higher 1/𝜏̃𝛼,𝑝) at a given degree of supercooling 𝑇𝑔(𝑓𝑐𝑟𝑜𝑠𝑠)/𝑇 due to the corresponding decrease in the 𝑓𝑐𝑟𝑜𝑠𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' This same competition is apparent in the diffusion constant 𝐷̃𝑝, which we plot in Figure 5a and b for 𝑑 𝜎 ⁄ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 and 𝑑 𝜎 ⁄ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 respectively, and at the same conditions plotted in Figure 4a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Like 1/𝜏̃𝛼,𝑝, 𝐷̃𝑝 exhibits Arrhenius-like behavior as defined by a straight line on a log 𝐷̃𝑝 versus 𝑇𝑔/𝑇 plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The primary difference is in the relative magnitude change in 𝐷𝑝 at different degrees of supercooling 𝑇𝑔/𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' 𝐷̃𝑝 still exhibits a near-collapse for 𝑑 𝜎 ⁄ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0, however generally decreases more slowly than 1/𝜏̃𝛼,𝑝 in Figure 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' For the larger penetrant size 𝑑 𝜎 ⁄ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0, the situation is more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' In this case, high temperatures 𝑇 in Figure 5b show a more 20 dramatic decrease in 𝐷̃𝑝 with 𝑇𝑔/𝑇 than that of 1/𝜏̃𝛼,𝑝 in Figure 4b, but a similar decrease at lower temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' These effects are quantified by calculating the slope of the − log 𝐷̃𝑝 versus 𝑇𝑔/𝑇 relationship, which we include in Table 1 for both penetrant sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Here, the 𝐷̃𝑝 slopes are significantly smaller than the 𝜏̃𝛼,𝑝 slopes for 𝑑 𝜎 ⁄ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0, but are similar or larger than the 𝜏̃𝛼,𝑝 for 𝑑 𝜎 ⁄ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The latter behavior for 𝑑 𝜎 ⁄ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 is qualitatively consistent with the results in our companion theory paper,69 which finds that the diffusion constant 𝐷̃𝑝 changes with supercooling 𝑇𝑔/𝑇 similarly or more pronounced than the corresponding inverse 𝜏̃𝛼,𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' For 𝑑 𝜎 ⁄ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0, however, the theory predicts that the 𝐷̃𝑝 slopes will be similar to inverse 𝜏̃𝛼,𝑝 slopes, in contrast to the simulation observation that the 𝐷̃𝑝 slopes are much smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' We attribute this disparity between simulation and theory to the larger importance of dynamic heterogeneity for smaller penetrants in our simulations,66 because the theory only makes predictions for a mean penetrant alpha time that is closer to our value 𝜏̃𝛼,𝑝′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Indeed, for 𝑑 𝜎 ⁄ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 the relative slopes for 𝐷̃𝑝 and 1/𝜏̃𝛼,𝑝′ calculated using this criterion (in parentheses in Table 1) are much more consistent with the theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='69 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' (a) Penetrant diffusion constant 𝐷̃𝑝 as a function of 𝑇𝑔(𝑓𝑐𝑟𝑜𝑠𝑠)/𝑇 for several temperatures 𝑇 and penetrant sizes (a) 𝑑/𝜎 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 and (b) 𝑑/𝜎 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' These are the same (a) (b) 10- T/K T/K 300 口 口 300 320 KH 320 10-4 339 339 10-3 KOH ~D 六 10-4 10-6 d/g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 d/ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='95 21 conditions as in Figure 4a and b, and similarly there is a collapse in (a) that is no longer present in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The slopes of these plots are calculated and included in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' To contextualize the relationship between the diffusion constant 𝐷̃𝑝, inverse penetrant alpha time 1/𝜏̃𝛼,𝑝, temperature 𝑇, and particle size 𝑑/𝜎, we plot both 𝐷̃𝑝 and 1/𝜏̃𝛼,𝑝 as a function of 𝑇𝑔/𝑇 in Figure 6 for two temperatures each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' We have used a multiplier 𝐴 to align these data sets at the leftmost points, to compare how the effect of changing the crosslinking fraction 𝑓𝑐𝑟𝑜𝑠𝑠 (and concomitantly the 𝑇𝑔) on 𝐷̃𝑝 versus 1/𝜏̃𝛼,𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' For small particles (𝑑 𝜎 ⁄ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0), the value of 𝐷̃𝑝 exhibits a weaker decrease with the degree of supercooling 𝑇𝑔/𝑇 than 1/𝜏̃𝛼,𝑝 for both temperatures 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' We believe that this is again a dynamic heterogeneity effect, which manifests as the well-known breakdown of the Stokes-Einstein relation in glassy liquids,59–66 where the diffusion constant is understood to be dominated by the subpopulation of particles undergoing rapid transport and is thus less dependent on 𝑇𝑔/𝑇 than the particle hopping time that characterizes a relaxation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='66 To contrast, for larger particles (𝑑 𝜎 ⁄ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0) the change in the diffusion constant 𝐷̃𝑝 tracks the change in 1/𝜏̃𝛼,𝑝 with increasing 𝑓𝑐𝑟𝑜𝑠𝑠 at low temperatures (𝑇 = 300K) and even decreases more than 1/𝜏̃𝛼,𝑝 with increasing 𝑓𝑐𝑟𝑜𝑠𝑠 at higher temperatures (𝑇 = 339K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' This indicates that there is an additional mechanism slowing down diffusive penetrant motion at length scales longer than what is probed in the intermediate scattering function (the local molecular cage, 1/𝜏̃𝛼,𝑝), which we attribute to the effect of mesh confinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' 22 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Comparison of the penetrant diffusion constant 𝐷̃𝑝 and inverse penetrant alpha time 1/𝜏̃𝛼,p as a function of 𝑇𝑔(𝑓𝑐𝑟𝑜𝑠𝑠)/𝑇 at both high (339K) and a low (339K) temperatures for (a) 𝑑/𝜎 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 and (b) 𝑑/𝜎 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' 1/𝜏̃𝛼,p is vertically shifted down by a factor (a) 𝐴 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='40 × 10−3 and (b) 𝐴 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='55 × 10−3 such that the leftmost point of alpha time 1/𝜏̃𝛼,p at a given temperature coincides with the corresponding 𝐷̃𝑝 data point for better comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' In (a), this comparison shows that 𝐷̃𝑝 exhibits a weaker dependence than 1/𝜏̃𝛼,p with 𝑇𝑔(𝑓𝑐𝑟𝑜𝑠𝑠)/𝑇 for all temperatures, while in (b) 𝐷̃𝑝 exhibits a stronger dependence than 1/𝜏̃𝛼,p at high temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' To isolate the role of mesh confinement, we refer to the form of Equation 1 that writes the diffusion constant as the product of 1/𝜏̃𝛼,𝑝 and a mesh confinement factor 𝑋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The form predicted for 𝑋 is model-dependent,23,24,26 but we can directly evaluate this quantity from simulation by taking the product 𝑋 = 𝐷̃𝑝𝜏̃𝛼,𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' We plot 𝐷̃𝑝𝜏̃𝛼,𝑝 in Figure 7 for both penetrant sizes as a function of 𝑇𝑔/𝑇 and 𝑇, in analogy with Figures 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Figure 7a plots this product for small penetrants (𝑑 𝜎 ⁄ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0), and as expected from Figures 4a and 5a the data roughly collapses to a single curve that increases monotonically with the degree of supercooling 𝑇𝑔/𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' This trend is generically expected for one-component glass-forming liquids due to alpha process dynamic heterogeneity,66 so in this case the network primarily affects penetrant transport through the dependence of 𝑇𝑔 on 𝑓𝑐𝑟𝑜𝑠𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Notably, this monotonic increase in 𝐷̃𝑝𝜏̃𝛼,𝑝 is no longer seen when the 1/e criterion is (a) b d/ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 d/α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 10-3 10-4 口 KOHH 空 10-5 TY Y Y 300, Dp 300, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' 300, A/tα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='p TOI 300, A/tα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='p 10-6 口 339, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' 339, Dp I 339, A/t 339, A/t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' α,p 10-5 10-7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='95 23 considered (see inset to Figure 7a), demonstrating that the long-time tail in the ISF should be responsible for the behavior in the main panel of Figure 7a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' To contrast, Figure 7b plots the product 𝐷̃𝑝𝜏̃𝛼,𝑝 for larger penetrants (𝑑 𝜎 ⁄ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0), and exhibits a significant decrease with 𝑇𝑔/𝑇 for each of the higher temperatures 𝑇 = 339 and 320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' For a constant temperature, the increase with 𝑇𝑔/𝑇 corresponds to an increase in both 𝑓𝑐𝑟𝑜𝑠𝑠 and 𝐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' We can thus attribute this dependence to the effect of mesh confinement (and also the reduced importance of glassy dynamic heterogeneity since larger particles tend to average over it), as this decrease of 𝐷̃𝑝𝜏̃𝛼,𝑝 = 𝑋 with 𝐶 is predicted by both Cai, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='23 and Dell and Schweizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='26 However, this trend is less noticeable for the lowest temperature 𝑇 = 300K, where within error 𝐷̃𝑝𝜏̃𝛼,𝑝 remains roughly constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The expected increase in 𝐷̃𝑝𝜏̃𝛼,𝑝 for glassy melts appears to cancel out any mesh confinement effects, indicating that the local caging and mesh confinement both contribute to penetrant dynamics in this limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The product 𝐷̃𝑝𝜏̃𝛼,p as a function of 𝑇𝑔(𝑓𝑐𝑟𝑜𝑠𝑠)/𝑇 for several temperatures 𝑇 for (a) 𝑑/𝜎 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 and (b) 𝑑/𝜎 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 with the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='1 criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' For (a) there is a collapse to an increasing curve that reflects the breakdown of Stokes-Einstein that is characteristic of dynamic heterogeneity in glass forming liquids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The inset shows the same plot, but using the 1/e criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' For (b) at high temperatures, each temperature exhibits a monotonic decrease in the product 𝐷̃𝑝𝜏̃𝛼,p that we attribute to the effect of mesh confinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' At low temperatures (𝑇 = 300K) and 𝑑/𝜎 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0, the a Tg(feross)/T d/g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 d/g = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='850.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='95 10-2 TOI α,p TOH 10 ~D10 10-3 T/K T/K 口 300 口 300 320 320 339 339 10-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='95 24 increase due to dynamic heterogeneity appears to counteract the effect of mesh confinement, leading to a roughly constant value of 𝐷̃𝑝𝜏̃𝛼,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Network and Penetrant Dynamic Decoupling and Particle Size We quantified penetrant dynamics, determining both the local hopping via 1/𝜏̃𝛼,𝑝 and the long-time diffusion via 𝐷̃𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' These results demonstrated that (1) both processes were primarily governed by local network segmental motion (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' glass physics) at low temperatures, yet (2) crosslinks nevertheless had a more pronounced effect on 𝐷̃𝑝 for larger penetrants (𝑑/𝜎 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 versus 𝑑 𝜎 ⁄ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 in Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' To understand these two observations, we quantify the coupling between the penetrant alpha time 𝜏̃𝛼,𝑝′ and the network alpha relaxation time 𝜏̃𝛼,𝐾, where the latter is obtained from the autocorrelation function of the Kuhn monomer vector per our prior work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='35 In this case, we use the short-time criterion 𝜏̃𝛼,𝑝′ due to its better consistency with both the theoretical predictions and to match the 1/𝑒 criterion used in calculating 𝜏̃𝛼,𝐾 previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The ratio between these two properties 𝜏̃𝛼,𝑝′/𝜏̃𝛼,𝐾 is plotted as a function of 𝑇𝑔/𝑇 in Figure 8 for both penetrant sizes and all temperatures 𝑇 studied;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' this comparison has previously demonstrated to be useful in the self0consistent cooperative hopping (SCCH) theory of hard sphere mixtures and dilute penetrants in polymer melts,20,58,70 and quantifies the degree of ‘dynamic coupling/decoupling’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' In these previous theories, the ratio 𝜏̃𝛼,𝑝′/𝜏̃𝛼,𝐾 was measured in the context of ‘packing fraction’, which is qualitatively analogous to inverse temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='58 The theory predicted that the ratio 𝜏̃𝛼,𝑝′/𝜏̃𝛼,𝐾 (1) initially increases slightly as the temperature is lowered from high temperature, and then (2) sharply decreases in a penetrant size-dependent manner (stronger decrease for smaller penetrants) as temperature is further lowered towards the glassy state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='58,69 This indicates a strong decoupling between penetrant and matrix dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' 25 In Figure 8, both penetrants show only a modest dependence of 𝜏̃𝛼,𝑝′/𝜏̃𝛼,𝐾 on 𝑇𝑔/𝑇, spanning only roughly a decade in the simulation-accessible weakly supercooled regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' We do not observe the characteristic decrease in 𝜏̃𝛼,𝑝′/𝜏̃𝛼,𝐾 predicted for the deeply supercooled regime, which is expected to be computationally inaccessible, though our trends are consistent with the weak changes in 𝜏̃𝛼,𝑝′/𝜏̃𝛼,𝐾 predicted by the theory in the polymer alpha relaxation time range probed in our simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='69 Importantly, the relative magnitude of these ratios is instructive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' 𝜏̃𝛼,𝑝/𝜏̃𝛼,𝐾 for the 𝑑/𝜎 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 penetrant is at least an order of magnitude larger than that for the 𝑑/𝜎 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 penetrant, indicating that it is more strongly coupled to the surrounding matrix as generically expected and as predicted by SCCH theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' This helps explain our results for 1/𝜏̃𝛼,𝑝 and 𝐷̃𝑝 in the previous section;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' in those results, there was a stronger dependence of 𝐷̃𝑝 for the larger penetrants on 𝑓𝑐𝑟𝑜𝑠𝑠even at lower temperatures where mesh confinement is not dominant (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' We can now attribute this to the stronger matrix-penetrant coupling for larger penetrants, rather than enhanced mesh confinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=" Degree of decoupling between penetrant and Kuhn segment dynamics, as quantified by the ratio 𝜏̃𝛼,p'/𝜏̃𝛼,K, as a function of 𝑇𝑔(𝑓𝑐𝑟𝑜𝑠𝑠)/𝑇 for several temperatures and both 𝑑/𝜎 = 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 and 𝑑/𝜎 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' While the trends with 𝑇𝑔(𝑓𝑐𝑟𝑜𝑠𝑠)/𝑇 are quite weak, the larger penetrant 𝑑/𝜎 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 100 T/K 300 310 10-1 320 K 口 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='10-2 d/ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 d/g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 10-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content="95 Tg(feross)/T 26 couples significantly more strongly to the segmental dynamics of the surrounding network, as evident in the order-of-magnitude larger value of 𝜏̃𝛼,p'/𝜏̃𝛼,K when compared to the smaller penetrant 𝑑/𝜎 = 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Degeneracy of Mesh Size Effects Due to Confinement and Glassy Dynamics From a molecular viewpoint, we only see significant mesh confinement effects on penetrant diffusion in the limit of large particles and high temperatures, and for small molecules or low temperatures show that crosslinking instead impacts penetrant diffusion primarily through its effect on the local segmental relaxation of the polymer matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' However, even in this limit we can demonstrate results consistent with aspects of Equation 1 if we plot our data versus the confinement ratio 𝐶 rather than the extent of supercooling 𝑇𝑔/𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' We consider two candidate models for 𝑋, by plotting 𝐶𝐷𝑝 versus 𝐶2 in Figure 9a based on the predictions of Cai, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='23 and 𝐷̃𝑝 versus 𝐶 in Figure 9b based on the predictions of Dell and Schweizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='26 Despite already demonstrating that mesh confinement does not play a significant role for small penetrants, these plots exhibit remarkably good agreement with the functional forms predicted by both models as indicated by the nearly linear trends in these semi-log plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' However, there is a non-negligible change in slope in both plots of Figure 9 as the temperature 𝑇 is changed, which is not a prediction for either entropic mesh confinement model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='23,25,26 As per Equation 1, these models that only have a temperature dependence in the prefactor 1/𝜏𝛼,𝑝 (plotted also versus 𝐶 from simulation in Figure S2) and not in the exponent of the entropic mesh confinement factor 𝑋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' This key observation allows the effects of crosslinking on the penetrant relaxation time to be distinguished from the effects of crosslinking due to mesh confinement, in the absence of simulation determination of 1/𝜏𝛼,𝑝, though this distinguishing feature may become weak as in the case of larger penetrants as is shown in the supporting information (𝑑/𝜎 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0, Figure S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' 27 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The penetrant diffusion constant 𝐷̃𝑝 plotted with respect to the confinement ratio 𝐶 as suggested by the forms of various literature predictions for mesh confinement effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' (a) Cai, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='23 predicts that 𝐷̃𝑝 ∝ 𝐶−1 exp(−𝐶2) and (b) Dell and Schweizer26 predict that 𝐷̃𝑝 ∝ exp(−𝑏𝐶).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Both ways of plotting this data appear linear, even though mesh confinement does not play a major role in penetrant transport for the 𝑑/𝜎 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 case plotted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The main indication that the simulations do not agree with the idea that entropic mesh confinement is dominant lie in the temperature-dependent slopes, which for mesh confinement should not change with temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Conclusion In this paper, we used simulation to understand the role of glassy dynamics versus mesh confinement on the diffusion and alpha relaxation of molecular penetrants in polymer networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' We consider a specific form for the diffusion constant, 𝐷𝑝 ∼ (𝑑2 𝜏𝛼,𝑝 ⁄ )𝑋, that allows us to isolate the effect of network mesh confinement 𝑋 through calculating and studying both the diffusion constant 𝐷̃𝑝 and the penetrant hopping or alpha relaxation time 𝜏̃𝛼,𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Both quantities exhibit Arrhenius-like trends with the degree of supercooling 𝑇𝑔/𝑇 at a fixed temperature, due to the dependence of 𝑇𝑔 on the extent of crosslinking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' This strong coupling between penetrant hopping and crosslinking-dependent glassy dynamics is consistent with theoretical predictions in the companion paper,69 but also exhibits significant breakdown in the Stokes-Einstein relationship if a strong dynamic heterogeneity effect65–68 is apparent in the hopping dynamics of the smaller molecular penetrants as the system is supercooled to larger values of 𝑇𝑔/𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='66 For small penetrants, (a) (b) d/g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 d/a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 10-3 鱼 10-3 D p C ~D T/K T/K 口 300 口 300 10-4 △ 320 10-4 △ 320 HH 口 339 D 339 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='8 C C2 28 the effect of crosslinking on 𝑇𝑔 is demonstrated to dominate diffusive motion, with a collapse to a single dependence of both 𝜏̃𝛼,𝑝 and 𝐷̃𝑝 on 𝑇𝑔/𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' This collapse is no longer apparent for larger penetrants, where we show by plotting the product 𝐷̃𝑝𝜏̃𝛼,𝑝 with 𝑇𝑔/𝑇 that we see the signatures of mesh confinement at low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' While we show that crosslinking affects molecular penetrant transport through both its effect on 𝑇𝑔 and through mesh confinement, the former appears to be the dominant mechanism for our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' We can also show, however, that the diffusion constant 𝐷̃𝑝 still follows trends predicted in the literature for mesh confinement,23,24,26 due to a degeneracy of how crosslink fraction changes with 𝑇𝑔 and mesh size as predicted by SCCH theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='69 This underscores the difficulty of drawing conclusions on transport mechanisms from diffusion data alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Acknowledgement This research was supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering (Award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' DE-SC0020858), through the Materials Research Laboratory at the University of Illinois at Urbana-Champaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Helpful discussions with Christopher Evans are gratefully acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' References: (1) Blaiszik, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' 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Matrix-Fluctuation-Mediated Activated Transport of Dilute Penetrants in Glass-Forming Liquids and Suspensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' 2017, 146 (19), 194906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='4983224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' 34 Supporting Information Figures Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Representative plots of the effective exponent 𝛽 for the mean square displacement versus time ⟨𝑟̃𝑝 2(𝑡̃)⟩ ∼ 𝑡̃𝛽 of a penetrant particle with sizes (a) 𝑑/𝜎 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 and (b) 𝑑/𝜎 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0, for the same values of 𝑇 and 𝑓𝑐𝑟𝑜𝑠𝑠 as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' 𝛽 is the slope of Figure 2, and both plots show that penetrant diffusion reaches the Fickian diffusion limit of 𝛽 = 1 in the long-time region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The open circles denote the length scale of the polymer network mesh size, as described in Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Intermediate regions with more noise reflect the need to perform several distinct computational runs with different data output frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The inverse penetrant alpha time 1/𝜏̃𝛼,𝑝 plotted with respect to the confinement ratio 𝐶 for penetrant particles with sizes (a) 𝑑/𝜎 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 and (b) 𝑑/𝜎 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' For both particle sizes, we find that they are consistent with the trend 𝜏̃𝛼,𝑝 ∝ exp(−𝑏𝐶).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' This primarily reflects the local glassy dynamics and not mesh confinement effects, and so this trend contributes to the apparent degeneracy between the two mechanisms for crosslink effects on penetrant transport in networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' (a) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 T/K:300 339 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='50 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 d/ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='5 d/ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 10-2 100 ~102 104 106(b) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 T/K:300 339 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='50 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='5 d/ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 10-2 100 ~102 104 106 t(a) 10-1 d/a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 空 10-2 克 10-3 T/K 口 300 320 339 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='8 C(b) d/a = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 10-2 空 P OT T/K 口 300 10-4 320 339 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='5 C 35 Figure S3: Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' The penetrant diffusion constant 𝐷̃𝑝 plotted with respect to the confinement ratio 𝐶 as suggested by the forms of various literature predictions for mesh confinement effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' (a) Cai, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='23 predicts that 𝐷̃𝑝 ∝ 𝐶−1 exp(−𝐶2) and (b) Dell and Schweizer26 predict that 𝐷̃𝑝 ∝ exp(−𝑏𝐶).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Both ways of plotting this data appear linear, even though mesh confinement does not play the sole role in penetrant transport for the 𝑑/𝜎 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 case plotted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' Mesh size (in unit of 𝜎) of networks at various fcross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' fcross 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='50 𝒅/𝝈 = 𝟏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' 𝟎 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='43 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='48 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='28 𝒅/𝝈 = 𝟐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content=' 𝟎 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='76 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='49 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='29 (a)10-3 d/a = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 10-4 10-5 T/K 口 300 320 10-6 339 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='5(b)10-3 d/a = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 10-4 KH 210-5 T/K 口 300 10-6 320 339 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} +page_content='5 C' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E4T4oBgHgl3EQf9g6Z/content/2301.05356v1.pdf'} diff --git a/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf b/NdAyT4oBgHgl3EQfs_n5/content/2301.00589v1.pdf new file mode 100644 index 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b/R9E0T4oBgHgl3EQfkwEq/content/tmp_files/2301.02476v1.pdf.txt @@ -0,0 +1,3068 @@ +manuscript submitted to JGR: Planets +Is there a semi-molten layer at the base of the lunar +mantle? +Michaela Walterov´a1, Marie Bˇehounkov´a2, Michael Efroimsky3 +1Institute of Planetary Research, German Aerospace Center (DLR), Berlin, Germany +2Department of Geophysics, Faculty of Mathematics and Physics, Charles University, Prague, Czech +Republic +3US Naval Observatory, Washington DC 20392 USA +Key Points: +• A lunar mantle governed by the Andrade model fits selenodetic constraints only +with a very weak frequency dependence of tidal dissipation +• We seek the parameters of the Sundberg-Cooper model that would explain the anoma- +lous frequency dependence of tidal Q measured by LLR +• Both a dissipative basal layer and elastically-accommodated grain-boundary slid- +ing in the deep mantle result in the same tidal response +Corresponding author: Michaela Walterov´a, kanovami@gmail.com +–1– +arXiv:2301.02476v1 [astro-ph.EP] 6 Jan 2023 + +manuscript submitted to JGR: Planets +Abstract +Parameterised by the Love number k2 and the tidal quality factor Q, and inferred from +lunar laser ranging (LLR), tidal dissipation in the Moon follows an unexpected frequency +dependence often interpreted as evidence for a highly dissipative, melt-bearing layer en- +compassing the core-mantle boundary. Within this, more or less standard interpretation, +the basal layer’s viscosity is required to be of order 1015 to 1016 Pa s and its outer ra- +dius is predicted to extend to the zone of deep moonquakes. While the reconciliation of +those predictions with the mechanical properties of rocks might be challenging, alterna- +tive lunar interior models without the basal layer are said to be unable to fit the frequency +dependence of tidal Q. +The purpose of our paper is to illustrate under what conditions the frequency-dependence +of lunar tidal Q can be interpreted without the need for deep-seated partial melt. De- +vising a simplified lunar model, in which the mantle is described by the Sundberg-Cooper +rheology, we predict the relaxation strength and characteristic timescale of elastically- +accommodated grain boundary sliding in the mantle that would give rise to the desired +frequency dependence. Along with developing this alternative model, we test the tra- +ditional model with a basal partial melt; and we show that the two models cannot be +distinguished from each other by the available selenodetic measurements. Additional in- +sight into the nature of lunar tidal dissipation can be gained either by measurements of +higher-degree Love numbers and quality factors or by farside lunar seismology. +Plain Language Summary +As the Moon raises ocean tides on the Earth, the Earth itself gives rise to periodic +deformation of the Moon. Precise measurements of lunar shape and motion can reveal +those deformations and even relate them to our natural satellite’s interior structure. In +this work, we discuss two interpretations of those measurements. According to the first +one, the lunar interior is hot and there is a thick layer of partial melt or other weak ma- +terial buried more than 1000 km deep under the lunar surface. According to the second +one, there is no such layer, and the measured deformation can be explained by the be- +haviour of solid rocks at relatively low temperatures. We show that the two possibili- +ties cannot be distinguished from each other by the existing data. +1 Motivation +Fitting of the lunar laser ranging (LLR) data to the quality-factor power scaling +law Q ∼ χp rendered a small negative value of the exponential: p = −0.19 (Williams +et al., 2001). Further attempts by the JPL team to reprocess the data led to p = −0.07 . +According to Williams and Boggs (2009), +“ Q for rock is expected to have a weak dependence on tidal period, but it is ex- +pected to decrease with period rather than increase. ” +The most recent estimates of the tidal contribution to the lunar physical librations +(Williams & Boggs, 2015) still predict a mild increase of Q with period: from Q = 38± +4 at one month to Q = 41 ± 9 at one year, yielding p = −0.03 ± 0.09. +Efroimsky (2012a, 2012b) suggested that since the frequency-dependence of k2/Q +always has a kink shape, like in Figure 1, the negative slope found by the LLR measure- +ments could be consistent with the peak of the kink residing between the monthly and +annual frequencies. This interpretation entails, for a Maxwell or Andrade moon, very +low values of the mean viscosity, indicating the presence of partial melt. +–2– + +manuscript submitted to JGR: Planets +Our goal now is to devise an interpretation based on the Sundberg-Cooper model. +Within that model, the kink contains not one but two peaks, and we are considering the +possibility that the negative slope of our interest is due to the monthly and annual fre- +quencies bracketing either this peak or the local inter-peak minimum. (It is unlikely that +both of these frequencies are located on the negative-slope side of the peak, because the +slope of that peak is too steep.) +2 Introduction +2.1 Overview of Previous Works +The knowledge of the interior structure of the Moon is essential for understand- +ing its thermal, geochemical, and orbital evolution as well as the coupled evolution of +the Earth-Moon system. The proximity of our natural satellite to the Earth has also made +it a frequent target of geophysical exploration. A large amount of data was collected by +lunar seismic stations, deployed by the Apollo missions, that were functional for several +years between 1972 and 1977 (for a review, see, e.g., Garcia et al., 2019; Nunn et al., 2020). +Other constraints are being placed by selenodetic measurements or by geochemical and +petrological considerations. However, the deepest interior of the Moon still remains some- +what mysterious. Although different models based on the inversion of seismic travel times +generally agree on the lunar mantle structure down to ∼ 1200 km, below these depths +they start to diverge greatly (Garcia et al., 2019). +After the acquisition of the first data by the lunar seismic network, it was pointed +out by Nakamura et al. (1973, 1974) that direct shear-waves from the farside of the Moon +are not being detected by some of the near-side seismometers. Moreover, deep moonquakes, +a class of tidally-triggered seismic events originating at around 1000 km depth, were al- +most absent on the farside. This puzzling phenomenon was interpreted by Nakamura et +al. (1973) as an indication for a shear-wave shadow zone caused by a highly attenuat- +ing region around the core. Later, Nakamura (2005) reported his further efforts to find +farside moonquakes among the discovered nests of deep moonquakes. Having had iden- +tified about 30 nests likely to be on the farside, his updated analysis still demonstrated +that either the region of the Moon’s deep interior within about 40 degrees from the an- +tipode (the centre of the farside) is nearly aseismic or a portion of lunar lower mantle +severely attenuates or deflects seismic waves. Lunar seismic data were also reprocessed +by Weber et al. (2011) and Garcia et al. (2011). However, while Weber et al. (2011) also +found evidence for deep mantle layering and a strongly attenuating zone at the mantle +base, Garcia et al. (2011) did not include such a feature in their lunar interior model. +The discussion about the seismic evidence for a strongly attenuating zone is thus still +ongoing (Garcia et al., 2019). +Several authors argued for the existence of a low-velocity zone (LVZ) at the base +of the mantle also on other than seismological grounds. They linked it to partial melt- +ing in deep lunar interior, which might be triggered either by tidal dissipation (Harada +et al., 2014), or by the presence of incompatible, radiogenic elements buried after an an- +cient mantle overturn. The idea of an overturn has been suggested by numerical mod- +elling of magma ocean solidification with the emplacement of ilmenite-bearing cumulates +above core-mantle boundary. Moreover, it is potentially supported by observations of +near-surface gravity anomalies (Zhang et al., 2013). +Evidence for a low-rigidity/low-viscosity zone has also been sought in the lunar li- +bration signal obtained by LLR (e.g., Williams et al., 2001; Williams & Boggs, 2015), +and in selenodetic measurements (including orbiter tracking) that are sensitive to the +lunar gravity field and tidal deformation (e.g., Konopliv et al., 2013; Lemoine et al., 2013). +One of the most surprising findings resulting from fitting the LLR data was the low value +and unexpected frequency dependence of the tidal quality factor Q, as mentioned in Sec- +tion 1 above. The inferred frequency dependence can be explained by a low effective vis- +–3– + +manuscript submitted to JGR: Planets +cosity of the Moon (Efroimsky, 2012a, 2012b), or by the presence of a secondary peak +in the dissipation spectrum (e.g., Williams & Boggs, 2015), possibly caused by the pu- +tative basal layer (Harada et al., 2014; Matsumoto et al., 2015). Earlier results from LLR +indicated that the lunar core-mantle boundary (CMB) might still be out of equilibrium, +which would imply long relaxation times and high lower-mantle viscosities, in contra- +diction to the presence of a partial melt. However, this hypothesis is not supported by +more recent evaluations of LLR data (Viswanathan et al., 2019), showing a CMB at hy- +drostatic equilibrium. +Despite relative consistency of the evidence for and the theoretical expectation of +a highly dissipative basal layer, alternative models of a “melt-free” Moon have been pro- +posed (Nimmo et al., 2012; Karato, 2013; Matsuyama et al., 2016). One argument for +high values of lower-mantle viscosities comes from the observations of deep moonquakes. +Kawamura et al. (2017) reevaluated an ensemble of moonquakes occurring at depths be- +tween 750 and 1200 km and found a brittle-ductile transition temperature of approxi- +mately 1240 – 1275 K, implying a cold lunar interior with temperatures below solidus of +dry peridotite. Moreover, the employment of a realistic, microphysically substantiated +models of the tidal response (Nimmo et al., 2012) can explain the low tidal Q and the +observed k2 of the Moon without requiring the existence of a weak basal layer, which is +necessitated in some of the other studies by the model settings and the simplified rhe- +ological assumptions. +A feature of the selenodetic measurements that is difficult to explain without the +existence of a highly dissipative basal layer is the aforementioned frequency dependence +of the lunar Q, repeatedly derived from LLR measurements in the series of works by Williams +et al. (2001); Williams and Boggs (2009); Williams et al. (2014), and Williams and Boggs +(2015). Even an independent implementation of the LLR software by Pavlov et al. (2016) +predicts the same value of Q for the monthly period as for the annual period, which is +still not consistent with the expected frequency dependence of tidal dissipation in melt- +free silicates. +In the absence of other than LLR-based data on the lunar Q, the most plausible +explanation for the unexpected frequency dependence might still be an observational un- +certainty, rather than an effect contained in a tidal model. Nevertheless, in this work, +we shall explore two possible implications of the frequency dependence under the explicit +assumption that the fitted values are a result of a natural phenomenon and not of a model’s +limitations or an observation error. +2.2 A Putative Weak Basal Layer: Pros and Contras +The following paragraphs review the last ten years of discussion about the pres- +ence or absence of a low-viscosity basal layer, with the argumentation derived mainly +from the lunar tidal response. +We begin by noting that a negative value of the exponent in Q ∼ χp is impossi- +ble for the seismic quality factor of rocks obeying simple rheologies like the Maxwell or +Andrade models. This can be easily understood if we express the seismic Q via the real +and imaginary parts of the complex compliance (Efroimsky, 2015, eqn 46). By insert- +ing into this expression either the Maxwell model or any other simple model lacking peaks, +we obtain a monotonic function Q(χ). On the other hand, even for simple rheologies the +exponential p can assume negative values if we are fitting to the Q ∼ χp law not a seis- +mic but a tidal quality factor (Efroimsky, 2015, eqn 45). The tidal Q tends to zero at +both very low and very high loading frequencies χ, and has a maximum in between. The +maximum is called into being by interplay of rheology and self-gravity. +This theoretical frequency dependence of the tidal quality factor motivated Efroimsky +(2012a, Section 5.2) to hypothesise that the small negative exponent p reported by Williams +et al. (2001) and Williams and Boggs (2009) may result from a proximity of the major +–4– + +manuscript submitted to JGR: Planets +tidal frequencies in the Moon to the frequency delimiting the peak dissipation. Efroimsky +(2012a, Section 5.7) also noted that this interpretation would imply a low effective vis- +cosity of the Moon (modeled with a homogeneous body governed by the Maxwell or the +combined Maxwell-Andrade rheology), with an estimated value of η = 3 × 1015 Pa s. +Such a low viscosity would support seismic models containing a layer of partial melt (Nakamura +et al., 1974; Weber et al., 2011). +Nimmo et al. (2012) aimed at answering the question whether basal partial melt +is indeed required for reproducing the tidal data, and studied the effect of lunar ther- +mal structure on the seismic and tidal Q. They described the rheology of the lunar in- +terior with the extended Burgers model of Jackson et al. (2010), which contains an ab- +sorption band corresponding to high-temperature background, as well as an additional +low-temperature peak. The peak represents the elastically-accommodated grain bound- +ary sliding, a phenomenon that will be considered also in our work, although within an- +other rheology. Nimmo et al. (2012) further considered a radially heterogeneous elastic +structure of the mantle and accounted for the temperature-, pressure-, and grain-size- +dependence of the characteristic relaxation times. Using this model, they were able to +match the tidal Love numbers k2 and h2 and the monthly quality factor, and they also +deduced that the lower-mantle viscosity should be as high as 1023 Pa s and must be in- +creasing towards the surface. However, the model used did not succeed in fitting the un- +expected slope of Q as a function of frequency. Although the authors showed that a model +case with grain size of 1 mm (instead of their baseline value of 1 cm) would imply a neg- +ative value of the exponential, p = −0.02, they dismissed this model as a poor fit to +both k2 and Q. Moreover, they argued that the smaller grain size would not match the +tentative observation of unrelaxed CMB (Williams et al., 2012). +An original explanation of the high tidal dissipation in the Moon was provided by +Karato (2013), who linked the measurements of electrical conductivity and Q to the wa- +ter content in the lunar mantle. That the water content might not be as low as had been +presumed in earlier models was illustrated by geochemical studies of lunar samples, and +Karato (2013) combined this observation with his own results to propose a new theory +of lunar formation. Using the observational constraints on Q and electrical conductiv- +ity, he further concluded that the temperature at an 800 km depth of the lunar mantle +is ∼ 1200–1500 K for a water content between 10−3 and 10−2 wt.%. Karato (2013) was +sceptical to the idea of partial melting at the base of the lunar mantle. He argued that +the melt-bearing seismic model of Weber et al. (2011) would require more than ∼ 1% +of melt and that retaining such an amount of melt would be difficult due to efficient com- +paction. Regarding the frequency-dependence of Q, Karato (2013) rejected the models +of Efroimsky (2012a) and Nimmo et al. (2012) and suggested that the negative exponent +p might be caused by non-linear anelasticity of the monthly tide and linear anelasticity +of the annual tide. However, this idea was partly based on the incorrect assumption that +the tide at the annual frequency is due to Sun-raised tidal deformation of the Moon. As +explained by Williams et al. (2001), the annual modulation is produced by solar pertur- +bations to lunar orbit only. The annual tide is thus raised by the Earth, just as the monthly +tide. Still, the remark on a possible non-linearity of the lunar tide remains valid. +Adopting the density and rigidity profiles from a 10-layer structural model by Weber +et al. (2011), Harada et al. (2014) explored the possible effects of a low-viscosity layer +at the base of the mantle. To keep the number of unknowns reasonable, the authors set +constant viscosity values for the lithosphere, mantle, low-viscosity layer, outer core, and +inner core, and applied the Maxwell rheological model. They then calculated the tidal +parameters for various thicknesses (outer radii 450–500 km) and viscosities (109–1021 Pa s) +of the basal layer, at both the monthly and annual tidal frequencies, assuming that the +rest of the mantle has a constant viscosity of η = 1021 Pa s. With the highest consid- +ered basal layer thickness (DLVZ = 170 km) and a viscosity of about 2×1016 Pa s, Harada +et al. (2014) were able to reproduce the quality factors given by Williams et al. (2001) +as well as their frequency dependence. Their value for the Love number at the monthly +period falls into the interval k2 = 0.0242±0.0004 suggested by Yan et al. (2012), while +–5– + +manuscript submitted to JGR: Planets +their value of the Love number at the annual period fits into the interval k2 = 0.0255± +0.0016 observed by Goossens et al. (2011). Viscoelastic, the model of Harada et al. (2014) +rendered different values of k2 at the monthly and annual frequencies. This said, neither +Yan et al. (2012) nor Goossens et al. (2011) considered frequency-dependence of their +empirical values of k2. +An updated version of the forward-modelling approach by Harada et al. (2014) was +presented in Harada et al. (2016). Using the improved set of tidal parameters (limits on +Q at four tidal frequencies and the values of k2, k3, and h2 at the monthly frequency), +the estimate of the basal layer’s outer radius was expanded from 500 km to 540−560 km +(i.e., layer thickness DLVZ = 210 − 230 km for a core radius of 330 km) and the corre- +sponding basal viscosity slightly changed to 3×1016 Pa s. In a recent follow-up study, +Tan and Harada (2021) considered full radial profile of the lunar interior (Weber et al., +2011; Garcia et al., 2011) and assumed a temperature-dependent viscosity structure of +the basal layer. The viscosity structure either followed a convective temperature profile +(viscosity almost constant with depth) or a conductive profile (linear decrease of viscos- +ity with depth). Since the former model was shown to match the selenodetic data bet- +ter, the authors argued that the low-viscosity layer should be locally convecting. More- +over, they concluded that the layer’s outer radius reaches 560 or 580 km (that is, to the +depths of ∼ 1160 km) and that the viscosity is the same as found by Harada et al. (2016). +The question whether a basal partial melt is required by the selenodetic data was +also raised by Khan et al. (2014), though with an answer different from Nimmo et al. +(2012). Khan et al. (2014) concentrated on detailed modelling of the lunar mantle petrol- +ogy, and performed a Bayesian inversion of the mean density, the moment of inertia, the +apparent resistivity, and the tidal data (k2 and Q) at the monthly period. To model the +tidal response of the lunar mantle within a purely elastic model, they calculated an anelas- +tic correction to k2 based on a homogeneous spherical model and the power-law depen- +dence of tidal dissipation, which is valid for large seismic quality factors (or weak seis- +mic wave attenuation; Zharkov & Gudkova, 2005). For cases with the Andrade param- +eter α > 0.1, the resulting elastic k2 clearly implied the existence of a partial melt in +a basal layer with the thickness of 150−200 km (i.e., a depth range ∼ 1250−1400 km +or the outer radii between ∼340-490 km). Khan et al. (2014) also found that, in order +to be neutrally buoyant, the partially molten material should be enriched in FeO and +TiO2 with respect to the bulk mantle. In addition to the models with a partially molten +layer, the authors tested a model with a fully solid mantle: this model still fitted all ob- +servations, except for the anelastically-corrected k2. +Similarly, Matsumoto et al. (2015) performed a Bayesian inversion of seismic travel +times and a set of available selenodetic data (mean density, moment of inertia, k2, and +Q at the monthly and annual frequencies), to infer the interior structure of an eight-layered +lunar model. As in Harada et al. (2014), the authors considered the Maxwell rheolog- +ical model, in which the existence of a low-viscosity layer is required not only by the slope +of Q’s frequency dependence but also by the magnitude of k2. The viscosity of the solid +mantle was always set to 1021 Pa s; otherwise, Matsumoto et al. (2015) varied a wide range +of parameters. While their inverted structure of the shallow mantle agrees with the re- +sults of Weber et al. (2011) and Garcia et al. (2011), the lower mantle, mainly constrained +by selenodetic data, slightly differs from the melt-containing model of Weber et al. (2011). +The outer radius of the highly dissipative layer is around 570 km and the predicted vis- +cosity in that region reaches 2.5+1.5 +−0.9×1016 Pa s. The authors noted that with the model +used, k2 and the annual Q are slightly biased from the observed values, although not be- +yond 1σ. Matsumoto et al. (2015) also reported a trade-off between the outer core ra- +dius and the LVZ thickness. The thickness of the LVZ corresponding to the calculated +outer radius is at least 170 km and, for the core size estimate of Weber et al. (2011), it +may reach 240 km. +In a paper presenting their interpretation of LLR data, Williams and Boggs (2015) +compared several rheological models and endeavoured to fit the lunar k2/Q at the monthly +–6– + +manuscript submitted to JGR: Planets +and annual tidal periods, considering physical libration at five periods (1 month, 206 days, +1 year, 3 years, and 6 years). Aware of the complex properties of the lunar interior and +the possible unmodelled effects of its lateral heterogeneity, the authors proposed a model +consisting of an absorption band and a narrow Debye peak: the former characterising +the dissipation in the solid mantle, the latter describing the contribution of the partially +molten layer suggested by Harada et al. (2014). For the thickness of the partially molten +layer, Williams and Boggs (2015) obtained DLVZ ≥ 205 km, placing its outer radius at +≥ 535 km. +The results of Williams and Boggs (2015) are relatively consistent with the pre- +dictions by Harada et al. (2014); Matsumoto et al. (2015), and Harada et al. (2016). As +in the other studies containing a LVZ, they indicate that if partial melt is present, it might +extend to the zone of deep moonquakes. On the one hand, the coexistence of partially +molten material with seismic sources is hard to imagine: while the former requires that +the lower-mantle temperatures exceed solidus, the latter should be concentrated in re- +gions where the mantle rocks undergo brittle deformation, limited to lower temperatures. +On the other hand, the movement of small amounts of melt to the zone of moonquake +nests might be considered one of the mechanisms triggering seismic events. Frohlich and +Nakamura (2009) proposed an explanation for the periodic occurrence of deep moonquakes, +which combines dehydration embrittlement due to partial melting and crack opening by +moving fluids. The authors pointed out the correlation between tidal loading and seis- +mic events associated with magma movements in terrestrial volcanoes and remarked that +a similar process may be active in the lunar interior. Tentative evidence for a link be- +tween deep moonquakes and magma movements might also be seen in the correlation +between the locations of deep moonquake nests and lunar maria (Qin et al., 2012). How- +ever, a definitive answer to the question of whether a rheologically weak layer and seis- +mic sources can exist at comparable depths awaits further modelling efforts. +The specific effect of a partially-molten basal layer on the elastic Love number k2,e +was discussed in the study of Raevskiy et al. (2015), which combined seismic and geode- +tic data with models of lunar mantle composition. Depending on the model used, the +rigidity of the basal layer was required to be 20–50% lower than the rigidity of the over- +lying solid mantle and the outer radius of that zone was determined to reach 530–550 km. +From the petrological perspective, the authors argued that partial melting of a peridotite/harzburgite +mantle above the core-mantle boundary (CMB) would require temperatures in the depth +of 1000 km to be in the range of 1350–1400 ◦C, unless the temperature gradients in the +lower mantle become steeper. Furthermore, they concluded that the seismic velocities +of Weber et al. (2011) are inconsistent with temperature profiles approaching solidus at +the CMB. Although the models of Raevskiy et al. (2015) assume elastic response of the +Moon, the authors also mentioned that anelasticity might explain the observed Love num- +ber without the need for a basal semi-molten layer. +Matsuyama et al. (2016) constrained their lunar interior models by the elastic Love +numbers k2 and h2 (calculated using the same anelastic correction for Q at the monthly +period as in Khan et al., 2014), the mean density of the Moon, and the moment of in- +ertia. After carrying out MCMC-type inversion, the authors concluded that although +the chosen observables do not rule out the existence of a semi-molten layer, there is a +strong preference for higher, solid-mantle-like values of the lower-mantle rigidity. If the +semi-molten layer exists, its thickness calculated by Matsuyama et al. (2016) is DLVZ = +194+66 +−186 km, its rigidity is µLVZ = 43+26 +−9 GPa, and its density may reach exceptionally +high values, ρLVZ = 4676+410 +−1179 kg m−3. +Recently, the combined geochemical, seismic, and selenogetic ensemble of Raevskiy +et al. (2015) was further studied by Kronrod et al. (2022), who extended the former work +by considering explicitly a viscoelastic lunar interior. Regarding the division into inte- +rior layers and the adopted rheological model, the authors followed Matsumoto et al. (2015); +i.e., they assumed the Maxwell model for the mantle and included a semi-molten basal +–7– + +manuscript submitted to JGR: Planets +layer. Besides the main results of their Bayesian analysis, indicating a major difference +in the chemical composition of the bulk silicate Earth and the Moon, Kronrod et al. (2022) +presented probability distributions for the seismic wave velocities, mean density, and the +thickness of the basal layer. The resulting distributions are wide, constraining the basal +layer’s density to 3400–3800 kg m−3 and the thickness to 100–350 km, depending on the +mantle composition. As in Khan et al. (2014), the authors conclude that the layer should +be enriched in TiO2 and FeO, if it is present. +In summary, the literature discussing the unexpected frequency dependence of lu- +nar tidal Q as well as the properties of a hypothetical semi-molten layer atop the lunar +core is rich, and the proposed values of the layer’s thickness range from 0 to 350 km. Mod- +els considering linear viscoelastic Maxwell rheology (both for the basal layer and for the +bulk mantle; Harada et al., 2014, 2016; Matsumoto et al., 2015; Tan & Harada, 2021) +typically arrive at viscosities of order 1016 Pa s. If the semi-molten layer exists, its up- +per radius extends to the depths of ∼ 1150 km, i.e., just below the regions that are rel- +atively well mapped by seismological studies and contain the nests of tidally-triggered +deep moonquakes. Nevertheless, the existence of a low-viscosity layer is not necessarily +required by selenodetic measurements at the best accessible, monthly period (Nimmo +et al., 2012; Matsuyama et al., 2016). The main advantage of melt-bearing models lies +in their ability to explain the possible increase in tidal Q from the monthly to the an- +nual period. +2.3 Lunar k2 and Q +Here, we shall use the potential tidal Love number derived from the GRAIL mis- +sion tracking data. Two independent analyses performed by the JPL group (Konopliv +et al., 2013, the GL0660B solution) and the GSFC group (Lemoine et al., 2013, the GRGM660PRIM +solution) yielded two possible values of the parameter: k2 = 0.02405 ± 0.000176 and +k2 = 0.02427±0.00026, respectively. The unweighted mean of the two alternative val- +ues is k2 = 0.02416 ± 0.000222 for a reference radius of 1738 km, and k2 = 0.02422 ± +0.000222 for the actual mean radius of 1737.151 km (Williams et al., 2014). For compar- +ison, the recent analysis of the data from the Chang’e 5T1 mission gives k2 = 0.02430± +0.0001 (Yan et al., 2020). We note that the value obtained from satellite tracking data +corresponds, in particular, to the real part of the complex Love number introduced later +in Subsection 4.1. The GRAIL data are dominated by data arcs collected throughout +a one-month time interval, and the resulting k2 is thus interpreted as indicative of the +deformation at monthly frequency (A. Konopliv, private communication). +The tidal quality factor Q was obtained by fitting tidal contribution to lunar phys- +ical libration measured by LLR (Williams et al., 2001, 2014; Williams & Boggs, 2015). +Interpreting the measurements of physical libration presents a highly complex problem, +depending on cross interactions of tides raised by the Earth and the Sun, precise mod- +eling of the lunar orbit and of the instantaneous positions of the Earth-based stations +and the Moon-based retroreflectors, and on an adequate incorporation of the lunar core- +mantle friction (Williams et al., 2001). In practice, the tidal time delay at a monthly pe- +riod and the dissipation-related corrections to the periodic latitudinal and longitudinal +variations in the Moon’s orientation are outputted and related analytically to linear com- +binations of k2/Q at a number of loading frequencies. Since many of the loading frequen- +cies are close to each other, the periodic corrections enable approximate estimation of +the leading dissipation terms. Specifically, the strongest correction (compared to its un- +certainty) is related to the annual longitudinal libration. Assuming a fixed k2 at the monthly +frequency, equal to the above-mentioned unweighted average, and using a complex rhe- +ological model best fitting the dissipation-related corrections to libration angles, Williams +and Boggs (2015) derived the following frequency-dependent values of tidal quality fac- +tor: Q = 38 ± 4 at the period of 1 month, Q = 41 ± 9 at 1 year, and lower bounds of +Q ≥ 74 at 3 years and Q ≥ 58 at 6 years. The tidal quality factors at other than the +monthly frequency are model-dependent because the actual quantities extracted from +–8– + +manuscript submitted to JGR: Planets +the dissipation-related corrections to libration angles are the ratios (k2/Q)χ/(k2/Q)monthly, +where χ denotes frequency. +Williams and Boggs (2015) also attempted to find the frequency-dependence of k2; +however, the effect could not be detected by existing measurements. We note that in con- +trast to the unexpected frequency dependence of Q found with the JPL-based software +(Williams et al., 2001, 2014; Williams & Boggs, 2015), an independent implementation +of the fitting tool with different preset solutions for part of the geophysical phenomena +(Pavlov et al., 2016) predicted Q = 45 at both the monthly and the annual frequen- +cies. +As an additional, though a relatively weak constraint on the lunar interior struc- +ture, we consider the degree-3 potential tidal Love number k3 and the degree-2 defor- +mational Love number h2 corresponding to radial deformation. The k3 number has been +derived from GRAIL mission tracking data and, as with k2 above, we adopt the unweighted +average of the two existing independent solutions (Lemoine et al., 2013; Konopliv et al., +2013): k3 = 0.0081±0.0018. The h2 number has been measured by LLR and by laser +altimetry (Mazarico et al., 2014; Pavlov et al., 2016; Viswanathan et al., 2018; Thor et +al., 2021), the most recent value, presented by Thor et al. (2021), being h2 = 0.0387± +0.0025. +We would finally mention the reason why the constraints on the lunar interior from +the measurements of k3 are weak. A degree-l component of the internal tidal potential +is proportional to rl, where r is the distance between the centres of mass of the tidally +perturbed body and the perturber. For this reason, with increasing degree l, the shal- +lower depths contribute more and more to the Love numbers kl. The sensitivity of the +higher-degree Love numbers to the deep interior is, therefore, limited as compared to de- +gree 2. +2.4 Outline of This Work +After an overview of the models and interpretations proposed in recent literature +(with the focus on the last ten years of the discussion), we are ready to continue with +the central part of this project. Our plan is to provide an interpretation of the unexpected +frequency dependence of tidal Q which does not require partial melting (in a way sim- +ilar to Nimmo et al., 2012) and compare it with a model containing a highly dissipative +basal layer (Harada et al., 2014; Matsumoto et al., 2015). Section 3 introduces and gives +a justification for the rheological model employed. Namely, it discusses the Sundberg- +Cooper extension of the Andrade model and the dissipation related to elastically accom- +modated grain-boundary sliding (GBS). The following Section 4 links the non-elastic rhe- +ology to Love numbers and tidal quality factors. In Section 5, we first illustrate the ex- +pected position of a secondary peak in the dissipation spectrum of a homogeneous Moon, +and then attempt to find the parameters of two- or three-layered lunar models that would +produce the values of the monthly tidal Q and annual k2/Q reported by Williams and +Boggs (2015). At the same time, we fit the empirical values of lunar k2, k3, and h2 given +in Subsection 2.3. Section 6 discusses implication of both our models, and the results +are briefly summarised in Section 7. +3 General Facts on Rheologies +3.1 Constitutive Equation +Rheological properties of a material are encoded in a constitutive equation inter- +connecting the present-time deviatoric strain tensor uγν(t) with the values that have +been assumed by the deviatoric stress σγν(t ′) over the time period t ′ ≤ t . Under lin- +–9– + +manuscript submitted to JGR: Planets +ear deformation, the equation has the form of convolution, in the time domain: +2 uγν(t) = ˆJ(t) σγν = +� t +−∞ +� +J (t − t ′) σγν(t ′) dt ′ , +(1) +and the form of product, in the frequency domain: +2 ¯uγν(χ) = ¯J(χ) ¯σγν(χ) . +(2) +Here ¯uγν(χ) and ¯σγν(χ) are the Fourier images of strain and stress, while the complex +compliance ¯J(χ) is a Fourier image of the kernel +˙J(t−t ′) of the integral operator (1), +see, e.g., Efroimsky (2012a, 2012b) for details. +3.2 The Maxwell and Andrade Models +At low frequencies, deformation of most minerals is viscoelastic and obeys the Maxwell +model: +� +U = +1 +2 µ +� +S + 1 +2 η S +(3a) +or, equivalently: +� +S + 1 +τM +S = 2 µ +� +U +, +(3b) +U and S being the deviatoric strain and stress; η and µ denoting the viscosity and +rigidity. (Below, we shall address the question as to whether µ is the unrelaxed or re- +laxed rigidity.) The Maxwell time is introduced as +τM ≡ +η +µ . +(4) +For this rheological model, the kernel of the convolution operator (1) is a time deriva- +tive of the compliance function +(M)J(t − t ′) = +� +Je + (t − t ′) 1 +η +� +Θ(t − t ′) +, +(5) +where Θ(t − t ′) is the Heaviside step function, while the elastic compliance Je is the +inverse of the shear rigidity µ : +Je ≡ +1 +µ . +(6) +In the frequency domain, equation (3) can be cast into form (2), with the complex com- +pliance given by +(M) ¯J (χ) = Je − +i +ηχ = Je +� +1 − +i +χ τM +� +, +(7) +and the terms Je and − i/(ηχ) being the elastic and viscous parts of deformation, cor- +respondingly. So a Maxwell material is elastic at high frequencies, viscous at low. +More general is the combined Maxwell-Andrade rheology, often referred to simply +as the Andrade rheology. It comprises inputs from elasticity, viscosity, and anelastic pro- +cesses: +(A)J(t − t ′) = +� +Je + β (t − t ′)α + t − t ′ +η +� +Θ(t − t ′) , +(8) +the corresponding complex compliance being +(A) ¯J (χ) += +Je + β (iχ)−α Γ (1 + α) − +i +ηχ +(9a) += +Je + β (iχ)−α Γ (1 + α) − i J (χ τM)−1 , +(9b) +–10– + +manuscript submitted to JGR: Planets +where Γ is the Gamma function, while α and β denote the dimensionless and dimen- +sional Andrade parameters. +Expressions (9a - 9b) suffer an inconvenient feature, the fractional dimensions of +the parameter β . It was therefore suggested in Efroimsky (2012a, 2012b) to shape the +compliance into a more suitable form +(A)J(t − t ′) = +� +Je + Je +�t − t ′ +τA +�α ++ Je +t − t ′ +τM +� +Θ(t − t ′) , +(10) +(A) ¯J (χ) = Je +� +1 + (i χ τA)−α Γ (1 + α) − i (χ τM)−1� +, +(11) +with the parameter τA christened as the Andrade time and linked to β through +β = Je τ −α +A +. +(12) +Compliance (11) is identical to (9a) and (9b), but is spared of the parameter β of frac- +tional dimensions. +3.3 Why the Maxwell and Andrade Models Require Refinement +In the literature, it is common to postulate that both the rigidity and compliance +assume their unrelaxed values denoted with µU and JU . +This convention is reasonable for sufficiently high frequencies: +χ is high +=⇒ +µ = µU +and +Je = JU +. +(13) +The convention, however, becomes unjustified for low frequencies. In that situation, the +material has, at each loading cycle, enough time to relax, wherefore both the rigidity mod- +ulus and its inverse assume values different from the unrelaxed ones. In the zero-frequency +limit, they must acquire the relaxed values: +χ → 0 +=⇒ +µ → µR +and +Je → JR . +(14) +This fact must be taken care of, both within the Maxwell and Andrade models. +3.4 Generalisation of the Maxwell and Andrade Models, +according to Sundberg and Cooper (2010) +The simplest expression for the time relaxation of the elastic part of the compli- +ance is +Je(t) += +JU + (JR − JU) +� +1 − e−t/τ � +(15a) += +JU +� +1 + ∆ +� +1 − e− t/τ�� +, +(15b) +where the so-called relaxation strength is introduced as +∆ ≡ JR +JU +− 1 , +(16) +while τ is the characteristic relaxation time. When relaxation of Je is due to elastically +accommodated grain-boundary sliding, this time can be calculated as +τ = τgbs = ηgb d +µU δ , +(17) +where ηgb is the grain-boundary viscosity, d is the grain size, while δ is the structural +width of the grain boundary. +–11– + +manuscript submitted to JGR: Planets +In the frequency domain, this compliance is written as +¯Je(χ) = JU +� +1 + +∆ +1 + χ2 τ 2 + i +χ τ ∆ +1 + χ2 τ 2 +� +, +(18) +its imaginary part demonstrating a Debye peak. Our goal is to trace how this Debye peak +translates into the frequency-dependence of the inverse tidal quality factor 1/Q and of +k2/Q of a near-spherical celestial body. +Substitution of formula (18) into the overall expression (11) for the Andrade com- +plex compliance will produce the Sundberg and Cooper (2010) rheology: +¯J (χ) += +JU +� +1 + +∆ +1 + χ2τ 2 − i +χ τ ∆ +1 + χ2τ 2 + (iχτA)−α Γ(1 + α) − i(χτM)−1 +� +(19a) += +JU +� +1 + +∆ +1 + χ2 τ 2 + Γ(1 + α) ζ−α (χτM)−α cos +�απ +2 +� � +(19b) +− +i JU +� +χ τ ∆ +1 + χ2 τ 2 + Γ(1 + α) ζ−α (χτM)−α sin +�απ +2 +� ++ (χτM)−1 +� +, +where we introduced the dimensionless Andrade time +ζ = τA +τM +. +(20) +Be mindful that in expression (10) it is only the first term, Je, that is changed to func- +tion (15b). Accordingly, in equation (11), it is only the first term, Je, that is substituted +with function (18). In the other terms, both the Maxwell and Andrade times are still +introduced through the unrelaxed value Je = JU : +τM ≡ η JU , +τA ≡ +�JU +β +�1/α +. +(21) +Had we combined the elastic relaxation rule (18) with the Maxwell model (7) in- +stead of Andrade, we would have arrived at the Burgers model — which would be equa- +tion (19) with the Andrade terms omitted, i.e. with τA −→ ∞. Simply speaking, in the +absence of transient processes, Andrade becomes Maxwell, while Sundberg-Cooper be- +comes Burgers. +The presently standard term “Sundberg-Cooper rheology” was coined by Renaud +and Henning (2018) who studied tidal heating in mantles obeying this rheological law. +Soon thereafter, this law was later employed for Mars (Bagheri et al., 2019) and for Pluto +and Charon (Bagheri et al., 2022). +Along with the dimensionless Andrade time ζ, below we shall employ the relative +relaxation time +trel = τ +τM +(22) +relating the relaxation timescale for the compliance Je to the Maxwell time. +3.5 Further Options +The characteristic relaxation time τ can be replaced with a distribution D(τ) of +times spanning an interval from a lower bound τL to an upper bound τH. So the relax- +ation of the elastic part of the compliance will be not +Je(t) = JU +� +1 + ∆ +� +1 − e− t/τ�� +(23) +–12– + +manuscript submitted to JGR: Planets +but +Je(t) = JU +� +1 + ∆ +� τH +τL +D(τ) +� +1 − exp +� +− t +τ +�� +dτ +� +. +(24) +If the relaxation is due to elastically-accommodated GBS, this distribution would be a +consequence of variable grain-boundary viscosity, grain sizes and shapes, and non-uniform +orientation of grain boundaries with respect to the applied stress (see also Lee & Mor- +ris, 2010). +Insertion of expression (24) in the Maxwell model (5) or in the Andrade model (10) +produces the extended Burgers model or the extended Sundberg-Cooper model, correspond- +ingly. For details, see Bagheri et al. (2022) and references therein. +4 Complex Love Numbers and Quality Functions +The perturbing potential wherewith the Earth is acting on the Moon can be de- +composed in series over Fourier modes ωlmpq parameterised with four integers lmpq. If +the tidal response of the Moon is linear, both the produced deformation and the result- +ing additional tidal potential of the Moon are expandable over the same Fourier modes, +as proved in Efroimsky and Makarov (2014, Appendix C). The proof is based on the fact +that a linear integral operator (convolution) in the time domain corresponds to a prod- +uct of Fourier images in the frequency domain. +While the Fourier modes can be of either sign, the physical forcing frequencies in +the body are +χlmpq = |ωlmpq | . +(25) +An extended discussion of this fact can be found in Section 4.3 of Efroimsky and Makarov +(2013). +Wherever this causes no confusion, we omit the subscript to simplify the notation: +ω ≡ ωlmpq , +χ ≡ χlmpq . +(26) +4.1 The Complex Love Number +Writing the degree-l complex Love number as +¯kl(ω) = ℜ +�¯kl(ω) +� ++ i ℑ +�¯kl(ω) +� += |¯kl(ω)| e +−iϵl(ω) , +(27) +we conventionally denote the phase as − ϵl , with a “minus” sign. This convention im- +parts ϵl with the meaning of phase lag. We also introduce the so-called dynamical Love +number +kl(ω) = |¯kl(ω)| . +(28) +A key role in the tidal theory is played by the quality functions +Kl(ω) ≡ − ℑ +� ¯kl(ω) +� += ¯kl(ω) sin ϵl(ω) +(29a) +entering the series expansions for tidal forces, torques, dissipation rate (Efroimsky & Makarov, +2014), and orbital evolution (Bou´e & Efroimsky, 2019) +Since Sign ϵl(ω) = Sign ω (Efroimsky & Makarov, 2013), they can be written as +Kl(ω) ≡ − ℑ +� ¯kl(ω) +� += kl(ω) +Ql(ω) Sign ω , +(29b) +where the tidal quality factor is introduced via +Q−1 +l +(ω) = | sin ϵl(ω)| . +(30) +–13– + +manuscript submitted to JGR: Planets +The dependency sin ϵl(ω) being odd, the function Ql(ω) is even. Also, even is the +function kl(ω). Therefore, for any sign of ω and ϵl, it is always possible to treat both Ql(ω) +and kl(ω) as functions of the forcing frequency χ ≡ |ω| : +Ql(ω) = Ql(χ) , +kl(ω) = kl(χ) +. +(31) +Often attributed to Biot (1954), though known yet to Sir George Darwin (1879), +the so-called correspondence principle, or the elastic-viscoelastic analogy, is a valuable +key to numerous problems of viscoelasticity. It enables one to derive solutions to these +problems from the known solutions to analogous static problems. In application to bod- +ily tides, this principle says that the complex Love number of a uniform spherical vis- +coelastic body, ¯kl(χ) , is linked to the complex compliance ¯J(χ) by the same algebraic +expression through which the static Love number kl of that body is linked to the relaxed +compliance JR : +¯kl(χ) = +3 +2 (l − 1) +1 +1 + Bl/ ¯J(χ) +, +(32) +where +Bl ≡ (2 l 2 + 4 l + 3) +l g ρ R += 3 (2 l 2 + 4 l + 3) +4 l π G ρ2 R2 +, +(33) +ρ, R, and g being the density, radius, and surface gravity of the body, and G being New- +ton’s gravitational constant. +As an aside, we would mention that while −ℑ [kl(ω)] emerges in the tidal torque, +the real part of the complex Love number, ℜ [kl(ω)] = kl(ω) cos ϵl(ω), shows up in the +expansion for the tidal potential. Not considered further in the present study, the gen- +eral expression for this product and its version for the Maxwell and other rheologies can +be found in Efroimsky (2015, Appendix A6). +4.2 kl(χ)/Ql(χ) and 1/Ql(χ) for an Arbitrary Rheology +Expression (32) entails: +Kl(χ) = kl(χ) sin ϵl(χ) = +− +3 +2(l − 1) +Bl ℑ +� ¯J(χ) +� +� +ℜ +� ¯J(χ) +� ++ Bl +�2 + +� +ℑ +� ¯J(χ) +��2 +, +(34) +the coefficients Bl rendered by equation (33). We see that for a homogeneous incom- +pressible sphere, the information needed to calculate the quality function comprises the +radius, the density, and the rheological law ¯J(χ) . +The inverse tidal quality factor of degree l is given by (Efroimsky, 2015) +Ql(χ)−1 ≡ | sin ϵl(χ)| , +(35) +sin ϵl(χ) = − +Bl ℑ +� ¯J(χ) +� +�� +ℜ +� ¯J(χ) +� �2 + +� +ℑ +� ¯J(χ) +� �2 �� +ℜ +� ¯J(χ) +� ++ Bl +�2 + +� +ℑ +� ¯J(χ) +��2 +. +(36) +All new is well-forgotten old. As we were writing this paper, it became known to us that +for the Maxwell rheology the frequency-dependence of sin ϵ2 was studied yet by Gerstenkorn +(1967, Fig. 2) in a work that went virtually unnoticed. Because of different notation and +Gerstenkorn’s terse style, it is not apparent if his values for the peak’s magnitude and +location are the same as ours. However, the overall shape of the frequency-dependence +of sin ϵ2 obtained by Gerstenkorn (1967) seems right. +–14– + +manuscript submitted to JGR: Planets +4.3 Notational Point: Q and Q2 +In publications where both seismic and tidal dissipation are considered, it is nec- +essary to distinguish between the seismic and tidal quality factors. In that situation, the +letter Q without a subscript is preserved for the seismic factor. +In the literature on tides, it is common to employ Q as a shorter notation for the +quadrupole tidal factor Q2. We shall follow the latter convention: +Q ≡ Q2 , +(37) +and shall use the two notations intermittently. +4.4 The frequency-dependencies of kl/Ql and 1/Ql +for the Maxwell and Andrade models +For a homogeneous sphere composed of a Maxwell or Andrade material, the qual- +ity function Kl(ω) has a kink form, as in Figure 1. The function sin ϵl(ω) is shaped sim- +ilarly. +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +0.10 +0.05 +0.00 +0.05 +0.10 +tidal mode ω +kl (ω) sin εl (ω) +Figure 1. +A typical shape of the quality function Kl(ω) += +kl(ω) sin ϵl(ω) , where ω is a +shortened notation for the tidal Fourier mode ωlmpq . (From Noyelles et al., 2014). +Insertion of expression (7) into equation (34) shows that for a spherical Maxwell +body the extrema of the kink Kl(ω) are located at +ωpeakl = ± +τ −1 +M +1 + Bl µ +(38) +the corresponding extrema assuming the values +K(peak) +l += ± +3 +4(l − 1) +Bl µ +1 + Bl µ , +(39) +wherefrom |Kl| < +3 +4(l − 1) . +Inside the interval between peaks, the quality functions are near-linear in ω : +|ω| < |ωpeakl | +=⇒ +Kl(ω) +≃ +3 +2(l − 1) +Bl µ +1 + Bl µ +ω +|ωpeakl | . +(40) +–15– + +manuscript submitted to JGR: Planets +Outside the inter-peak interval, they fall off as about ω−1 : +|ω| > |ωpeakl | +=⇒ +Kl(ω) ≃ +3 +2(l − 1) +Bl µ +1 + Bl µ +|ωpeakl | +ω +. +(41) +While the peak magnitudes (39) are ignorant of the viscosity η, the spread between +the peaks scales as the inverse η, as evident from expression (38). The lower the mean +viscosity, the higher the peak frequency |ωpeakl|. +It can be demonstrated using equation (36) that for a homogeneous Maxwell body +the extrema of sin ϵl(ω) are located at +ωpeak of sin ϵl = ± +τ −1 +M +√1 + Blµ . +(42) +For the Moon, this peak is located within a decade from its counterpart for Kl given +by formula (38). +In many practical situations, only the quadrupole (l = 2) terms matter. The cor- +responding peaks are located at +ωpeak2 = ± +τ −1 +M +1 + B2 µ ≈ ± +1 +B2 η = ± 8 π G ρ2 R2 +57 η +. +(43) +The approximation in this expression relies on the inequality Bl µ ≫ 1, fulfilment whereof +depends on the size of the body. For a Maxwell Moon with µ = 6.4×1010 Pa and G(ρR)2 ≈ +2.24 × 109 Pa, we have B2 µ ≈ 64.5, so the approximation works. +While for the Maxwell and Andrade models each of the functions Kl(ω) and sin ϵl(ω) +possesses only one peak for a positive argument, the situation changes for bodies of a +more complex rheology. For example, the existence of an additional peak is ensured by +the insertion of the Sundberg-Cooper compliance (19) into expressions (34) or (36). +5 Application to the Moon +5.1 The “Wrong” Slope Interpreted with the Maxwell Model +As we explained in Section 1, fitting of the LLR-obtained quadrupole tidal qual- +ity factor Q = Q2 to the power law Q ∼ χp resulted in small negative value of the +exponential p (Williams & Boggs, 2015). An earlier attempt to explain this phenomenon +implied an identification of this slightly negative slope with the incline located to the left +of the maximum of the quality function (k2/Q2)(χ), see Figure 1. Within this interpre- +tation, χpeak ≡ |ωpeak| should be residing somewhere between the monthly and annual +frequencies explored in Williams and Boggs (2015). As was explained in Efroimsky (2012a) +, this sets the mean viscosity of the Moon as low as +η ≈ 3 × 1015 Pa s , +(44) +The extrema of (1/Q2)(χ) are close to those of (k2/Q2)(χ), as can be observed from +equations (19) and (45) Efroimsky (2015). Therefore, had we used instead of the max- +imum of k2/Q2 given by (43) the maximum of 1/Q2 given by (42), the ensuing value would +have been only an order higher: +η ≈ 4 × 1016 Pa s . +(45) +Such values imply a high concentration of the partial melt in the mantle – quite in ac- +cordance with the seismological models by Nakamura et al. (1974) and Weber et al. (2011). +However, employment of a rheology more realistic than Maxwell may entail not so +low a viscosity — in which case the existence of a semi-molten layer may be questioned. +–16– + +manuscript submitted to JGR: Planets +5.2 Frequency Dependence of Tidal Dissipation in the Sundberg-Cooper +Model +The Debye peak emerging in the imaginary part of ¯Je (equation (18)) will, obvi- +ously, show itself also in the shape of the imaginary part of the overall ¯J , the bottom +line of equation (19b). Consequently, substitution of expression (19) in equations (34) +and (36) will entail the emergence of a Debye warp on the kinks for kl/Ql and 1/Ql . +Where will the additional peak be located for realistic values of the relaxation timescale +τ ? What values for the mean viscosity will it entail? +In the end of Section 3.4, we introduced the relative relaxation time as trel ≡ τ/τM . +Figure 2 illustrates specifically the effect of trel in the Sundberg-Cooper model on the +position of the additional Debye peak for a homogeneous lunar interior with an arbitrar- +ily chosen high mean viscosity ηMoon = 1022 Pa s. The emergence of another local max- +imum in the k2/Q2 and 1/Q2 functions may naturally explain the decrease in dissipa- +tion (or increase in the quality factor Q) with frequency, even within a homogeneous and +highly viscous model. +Figure 2. +The negative imaginary part of the Love number (left) and the inverse quality fac- +tor (right) for different ratios between the timescale τ and the Maxwell time τM (indicated by +the shades of blue). The yellow and red vertical lines show the Q2 values given by Williams and +Boggs (2015) for the annual and the monthly component, respectively. In this case, we consider +a homogeneous lunar interior model governed by the Sundberg-Cooper rheology. The mantle +viscosity was set to 1022 Pa s and the mantle rigidity to 80 GPa. +5.3 Constructing a Multi-layered Model +Section 4 introduced the complex Love number ¯kl(χ) for an arbitrary linear anelas- +tic or viscoelastic rheology assuming a homogeneous incompressible sphere. While such +a model can reasonably approximate the response of the Moon with a homogeneous man- +tle and a small core (see also Figure 4), its application to a body with a highly dissipa- +tive basal layer would not be accurate (Bolmont et al., 2020). Planetary interior with +a highly dissipative layer can still be approximated by a homogeneous model with an ad- +ditional absorption peak or band in the underlying rheological law. However, we would +need to know the mapping between the parameters of the dissipative layer and the pa- +rameters of the additional peak (Gevorgyan, 2021). +Therefore, in the following sections, we will complement the homogeneous model +with three models consisting of two or three layers and we will calculate the correspond- +ing complex Love numbers numerically, using a matrix method based on the normal mode +–17– + +-2.0 +-0.5 +-2.5 +-1.0 +-3.0 +Q +301 +3.5 +1.0 +0.0001 +2.0 +0.1 +1e-05 +-4.0 +0.01 +1e-06 +0.001 +1e-07 +1 month +yr +4.5 +2.5 +-10 +-10 +-9 +-8 +7 +-5 +-6 +-8 +-6 +-5 +4 +logx [rad/s] +logx[rad/s]manuscript submitted to JGR: Planets +theory (e.g., Takeuchi & Saito, 1972; Wu & Peltier, 1982; Sabadini & Vermeersen, 2004). +For the sake of simplicity, we consider all layers in the numerical model (linearly) vis- +coelastic and we model the response of liquid layers by the Maxwell model with Je in +equation (7) approaching 0. This method has also been tested against another imple- +mentation of the same model, in which the liquid layers were inputted through differ- +ent boundary conditions; the results obtained within the two approaches are virtually +the same. Using the outputted complex Love numbers for various rheological parame- +ters, we then proceed by fitting the empirical values. If not stated differently for illus- +trative purposes, the three alternative models will always comprise a liquid core with a +low viscosity (ηc = 1 Pa s), a constant density (ρc = 5000 kg m−3), and an outer ra- +dius identical to the mean value reported by Weber et al. (2011), Rc = 330 km. +Although the existence of an inner core is possible and even indicated by the stacked +seismograms presented by Weber et al. (2011), its response to tidal loading would be de- +coupled from the rest of the mantle, and it would contribute to the resulting tidal de- +formation only negligibly. Therefore, we do not include the inner core in our modelling. +Subsection 5.4 makes use of a two-layered model consisting of the liquid core and +a homogeneous mantle, the response of which is described by the Andrade rheology. For +the mantle density, we prescribe a constant value of ρm = 3300 kg m−3, and Andrade +parameter ζ is set to 1, implying comparable timescales for viscous and anelastic relax- +ation. Other values of ζ were also tested and their effect on the results is discussed in +the aforementioned Subsection. The viscosity ηm, rigidity µm, and Andrade parameter +α of the mantle are treated as free parameters and fitted to the data. +The second model, considered in Subsection 5.5, comprises a liquid core and a Sundberg- +Cooper homogeneous mantle. The mantle density is always set to the average value ρm = +3300 kg m−3. Rheological parameters ηm, µm, τ, and ∆ are fitted, while the Andrade em- +pirical parameters α and ζ are held constant during each run of the inversion. We have +also tested the effect of varying α in the range [0.1, 0.4] and of magnifying or reducing +ζ by one order of magnitude. +The model with a basal dissipative layer, which is discussed in Subsection 5.6, con- +tains a core and a two-layered mantle. Each layer of the mantle is assumed to be homo- +geneous. The basal layer is decribed by the Maxwell model with fitted parameters µLVZ +and ηLVZ; additionally, we fit its outer radius RLVZ. For the overlying bulk mantle, we +consider the Andrade model with free (fitted) parameters ηm, µm and with α, ζ kept con- +stant during each run of the inversion. Both mantle layers have a prescribed density of +ρLVZ = ρm = 3300 kg m−3. The reason for using the simple Maxwell model instead +of the Andrade model in the basal layer is the following: in order to fit the measured tidal +quality factor Q at the monthly and the annual frequency, the peak dissipation from the +basal layer should be located either between these frequencies, or above the monthly fre- +quency. At the same time, in the vicinity of the peak dissipation, the Andrade and Maxwell +rheologies are almost indistinguishable from each other. (Comparing the last two terms +on the final line of equation (19), we observe that the viscous term exceeds the Andrade +term when τMχ ≪ (τA/τM)α/(1−α) . In realistic situations, τMχpeak satisfies this con- +dition safely. So, near the peak the Andrade term is virtually irrelevant, and the regime +is almost Maxwell.) Hence, we chose the simpler of the two rheological models. This de- +cision will also facilitate the comparison of our results for the basal layer’s characteris- +tics with the predictions by Harada et al. (2014, 2016), and Matsumoto et al. (2015), who +likewise modeled the basal layer with the Maxwell model. In contrast to our study, they +applied the same model to the mantle as well. +In this work, we are not predicting the mineralogy of the mantle — and the com- +position of the basal layer, if present, is only briefly discussed in Subsection 6.2. Our use +of a homogeneous mantle layer (or two homogeneous mantle layers) reflects our lack of +information on the exact chemical and mineralogical composition, the grain size, the ther- +mal structure, and the presence of water. Instead, we characterise the mantle with a sin- +–18– + +manuscript submitted to JGR: Planets +gle, “effective”, rigidity and viscosity, which can be later mapped to a detailed interior +structure (see also Dumoulin et al., 2017; Bolmont et al., 2020, who discussed the effect +of approximating a radially stratified mantle with a homogeneous one for Venus and ter- +restrial exoplanets). Furthermore, we neglect any lateral heterogeneities in the lunar in- +terior. We also assume that the lunar mantle is incompressible and can be reasonably +described by a linear viscoelastic model — which is valid at low stresses. Given the mag- +nitude of tidal stresses in the Moon, this assumption might have to be lifted in future +works, though (Karato, 2013). +Since the radial structure of our models is deliberately simplified, we do not attempt +to fit either the mean density or the moment of inertia given for the Moon. (The mean +density of our lunar toy-models is less than 1% lower than the actual value.) The inver- +sions presented below are only performed for the tidal parameters, namely k2 and tidal +Q at the monthly frequency, k2/Q at the annual frequency, and k3, h2 at the monthly +frequency. A list of the model parameters in the reference cases discussed in the follow- +ing sections is presented in Table 1. The empirical values considered are then given in +Table 2. +Parameter +Type +Value +Unit +Common parameters +Core size Rc +const. +330 +km +Core viscosity ηc +const. +1 +Pa s +Core density ρc +const. +5, 000 +kg m−3 +Mantle viscosity ηm +fitted +1015 − 1030 +Pa s +Mantle rigidity µm +fitted +109 − 1012 +Pa +Mantle density ρm +const. +3, 300 +kg m−3 +Andrade parameter ζ +const. +1 +— +Two-layered model I (Andrade mantle) +Andrade parameter α +fitted +0 − 0.5 +— +Two-layered model II (Sundberg-Cooper mantle) +Andrade parameter α +const. +0.2 +— +Relaxation strength ∆ +fitted +10−5 − 100 +— +Relative relaxation time trel +fitted +10−7 − 100 +— +Three-layered model (Andrade mantle) +Andrade parameter α +const. +0.2 +— +Thickness of the basal layer DLVZ +fitted +0 − 370 +km +Viscosity of the basal layer ηLVZ +fitted +100 − 1030 +Pa s +Rigidity of the basal layer µLVZ +fitted +0 − µm +Pa +Table 1. +Parameters of the three models considered in this work. +5.4 Applicability of the Andrade Model +Before discussing the two interior models able to fit the anomalous frequency de- +pendence of lunar tidal dissipation, we first attempt to use the full set of tidal param- +eters given in Table 2 to constrain a simpler model, which only contains a liquid core and +a viscoelastic mantle governed by the Andrade rheology (equation (11)). Such a model, +accounting neither for a basal dissipative layer nor for elastically-accommodated GBS, +might still be able to fit the data. Thanks to the large uncertainty on the lunar qual- +–19– + +manuscript submitted to JGR: Planets +Parameter +Value +Reference +k2, monthly +0.02422 ± 0.00022 +Williams et al. (2014) +Q, monthlya +38 ± 4 +Williams and Boggs (2015) +k2/Q, annuala +(6.2 ± 1.4) × 10−4 +Williams and Boggs (2015) +k3, monthlyb +0.0081 ± 0.0018 +Konopliv et al. (2013); Lemoine et al. (2013) +h2, monthly +0.0387 ± 0.0025 +Thor et al. (2021) +a The standard deviations from this table are only used in Subsection 5.4. In the rest of the paper, we +arbitrarily set the uncertainties to 1% of the mean value. b Listed is the unweighted mean of the values +given in references. +Table 2. +Observational constraints used in this work. +ity factor (more than 10% at the monthly frequency and 20% at the annual frequency, +Williams & Boggs, 2015), we may not need to introduce any additional complexities to +interpret the tidal response of the Moon. The error bars of the tidal quality factors are +so wide that they allow, at least in principle, for a situation where Q2, annual is smaller +than Q2, monthly . +To find the parameters of this preliminary model, we performed a Bayesian inver- +sion using the MCMC approach and assuming Gaussian distributions of observational +uncertainties (e.g., Mosegaard & Tarantola, 1995). In particular, we employed the em- +cee library for Python (Foreman-Mackey et al., 2013), which is based on the sampling +methods proposed by Goodman and Weare (2010). The algorithm was instructed to look +for the mantle viscosity ηm, the mantle rigidity µm, and the Andrade parameter α fit- +ting the empirical values of k2,monthly, k3,monthly, h2,monthly, Q2,monthly, and (k2/Q2)annual , +while the other Andrade parameter was set to ζ = 1. We generated ∼ 30, 000 random +samples until the model converged. Specifically, the convergence was tested against the +autocorrelation time of each variable in the ensemble, the total length of all chains be- +ing required to exceed 100 times the longest autocorrelation time. Moreover, in order +to filter out the influence of initial conditions, we neglected the first ∼ 3, 000 samples +(our burn-in period was, therefore, 10 times the autocorrelation time). +The posterior probabilities of the fitted parameters are depicted in Figure 3, us- +ing the Python library corner (Foreman-Mackey, 2016). In line with a similar model by +Nimmo et al. (2012), we find a relatively high lunar mantle viscosity of log η[Pa s] = 22.99+0.89 +−1.35 +and rigidity of log µ[Pa] = 10.92±0.06, the Andrade parameter α being as low as 0.06+0.04 +−0.02. +Treating the Andrade parameter ζ as a free parameter in the Bayesian inversion +has a negligible effect on the predicted values of α and µm. However, it essentially de- +termines the fitted mantle viscosity. If the transient deformation prevails over the vis- +cous creep (ζ ≪ 1), the response of the lunar mantle to tidal loading is almost elastic +(with viscosity up to η ≈ 1027 Pa s). On the other hand, if the dissipation is preferen- +tially due to viscous creep (ζ ≫ 1), the mantle viscosity allowed by the observational +data has to be much lower, η ≈ 1021 Pa s. This latter case is equivalent to the assump- +tion that the mantle is governed by the Maxwell rheology, followed by Harada et al. (2014, +2016); Matsumoto et al. (2015); Tan and Harada (2021), and Kronrod et al. (2022). +If we compare the resulting Andrade parameter α = 0.06+0.04 +−0.02 with the typical +values reported in the literature (0.1 < α < 0.4; see, e.g., the overview by Castillo- +Rogez et al., 2011; Efroimsky, 2012a, 2012b), we may notice that it is unusually small. +This discrepancy between our prediction and the laboratory data already indicates that +although it is, in principle, possible to fit the lunar tidal response with a simple model +assuming Andrade rheology in the mantle, the required parameters of this model might +–20– + +manuscript submitted to JGR: Planets +not be realistic. A similar point has been made by Khan et al. (2014) and used as an ar- +gument in favour of their interior model containing basal partial melt. Following the same +line of argumentation, we will now focus our study on the Sundberg-Cooper model. +Figure 3. +Posterior probabilities of the effective mantle rigidity µm, the mantle viscosity +ηm, and the Andrade parameter α satisfying the full set of observational constraints (k2, k3, h2, +and Q at the monthly period; k2/Q at the annual period). A model with a liquid core and a +viscoelastic mantle governed by the Andrade rheology, assuming ζ = 1. +5.5 Lunar Mantle Governed by the Sundberg-Cooper Model +In the present Subsection, as well as in Subsection 5.6, we will specifically search +for lunar interior models that exhibit a second dissipation peak in the spectra of k2/Q2 +and Q−1 +2 . Since the current error bars of the empirical Qs allow for both a decrease and +increase of dissipation with frequency, and since our study focuses on the latter case, we +consider a hypothetical situation in which the uncertainty in Q2 is comparable with the +present-day uncertainty in k2. The standard deviations of Q2 at the monthly frequency +and k2/Q2 at the annual frequency are thus arbitrarily set to 1% of the mean value. As +in the previous inversion with Andrade mantle, we again employ the MCMC approach +and seek the parameters of the Sundberg-Cooper model (ηm, µm, ∆, and trel) fitting the +empirical tidal parameters. Values of α and ζ are kept constant. For illustration pur- +poses, we consider both 1) a two-layered interior structure consisting of a liquid core and +a viscoelastic (Sudberg-Cooper) mantle and 2) a homogeneous lunar interior. As we shall +see, the effect of the small lunar core (Rc = 330 km) on the results is negligible. +–21– + +log nm +0.24 +810 +0.12 +0.06 +98 +.92 +10.86 +10.9 +1og μm +log nm +αmanuscript submitted to JGR: Planets +In contrast with the previous inversion, and mainly due to the greater dimension +of the explored parameter space, the model only succeeded to converge after generat- +ing ∼ 700, 000 random samples. The posterior distributions of the tidal quality factors +typically presented two peaks: a higher one with Q2,monthly > Q2,annual and a lower one +with Q2,monthly < Q2,annual. Here, we only discuss the model parameters correspond- +ing to the latter case. +Figure 4 illustrates the results of the inversion with Andrade parameters specifi- +cally set to α = 0.2 and ζ = 1. Similarly as before, to filter-out the influence of ini- +tial conditions, we neglected the first 70, 000 samples. Then, 16% of the remaining, anal- +ysed samples fulfilled the condition of quality factor decreasing with frequency. The mean +value of the predicted mantle viscosity lies close to 3.5×1022 Pa s and the predicted un- +relaxed rigidity is around 60 − 120 GPa. In particular, for the nominal case with α = +0.2 and ζ = 1 and for the arbitrarily chosen small standard deviation of empirical Q +and k2/Q, the decadic logarithms of the predicted mantle viscosity and rigidity are log ηm[Pa s] = +22.55+0.15 +−0.54 and log µm[Pa] = 10.84+0.14 +−0.02. Increasing α by 0.1 or ζ by the factor of 10 re- +sults in decreasing the mantle viscosity approximately by an order of magnitude (and +the same trend pertains to the other direction, when decreasing α or ζ ). On the other +hand, the mantle rigidity, being dictated by the magnitude of k2, seems relatively robust +and its inverted value does not depend on α. +The parameters of the Debye peak are, in this story, the key to fitting the unex- +pected slope of the frequency-dependent tidal dissipation. Independently of the consid- +ered Andrade parameters, the relaxation timescale τ lies between 104 and 106 s (log τ[s] = +4.89+0.62 +−0.72), while the relaxation strength falls into the interval between 0.03 and 1 (log ∆ = +−1.17+0.84 +−0.35). The exact values depend on the predicted viscosity and rigidity, which de- +fine the position of the first peak, corresponding to the attenuation in the overlying man- +tle. Such short relaxation timescales would indicate that the elastically accommodated +GBS is much faster than diffusion creep. For comparison, Sundberg and Cooper (2010) +mention a GBS relaxation timescale of 0.1 s as a reasonable value in their experiments, +using a material with τM ∼ 10 – 100 s. Our τM in this specific case is in the order of 1010− +1013 s; hence, the ratio of the two time scales for α = 0.2 and ζ = 1 reaches trel = 10−7− +10−6. A more detailed discussion of this result will be provided in Subsection 6.1. +5.6 Comparison of a Sundberg-Cooper Moon with an Andrade Moon +Having a Weak Basal Layer +As was recently shown by Gevorgyan (2021), the tidal response of a homogeneous +Sundberg-Cooper planet mimics the response of a body consisting of two Andrade lay- +ers with different relaxation times. This kind of aliasing may, in principle, be demonstrated +by the Moon. Figure 5 depicts the imaginary part of the tidal Love number (equal to +k2/Q2) and the inverse quality factor 1/Q2 as functions of frequency, for a homogeneous +Sundberg-Cooper moon and for a differentiated lunar interior with a rheologically weak +layer at the base of the mantle. In the second case, the basal layer is described by the +Maxwell model and the overlying mantle by the Andrade model. Both cases follow the +same frequency dependence, implying that the existence of a weak basal layer cannot +be confirmed unequivocally by the tidal data. In a layered model containing a core, a +Sundberg-Cooper mantle, and a Maxwell basal semi-molten layer, the tidal response would +be characterised by three peaks (Figure 6). +For comparison with other models presented in the literature, we also sought for +the parameters of a three-layered lunar model comprising a liquid core, an Andrade man- +tle, and a Maxwell basal low-viscosity layer that would fit the empirical constraints. As +in the previous Subsection, in order to reduce the number of unknowns, the parameters +α and ζ of the Andrade model were kept constant. We also prescribed the same constant +core radius of 330 km. The remaining quantities were treated as free parameters: we thus +varied the rigidity and viscosity of the mantle and of the basal layer, and the outer ra- +–22– + +manuscript submitted to JGR: Planets +Figure 4. +Best-fitting models and the corresponding model parameters for a melt-free Moon +with a liquid core and a Sundberg-Cooper mantle. Upper row: the real (left) and negative imagi- +nary (right) parts of the complex Love number ¯k2, as functions of frequency. The red and yellow +lines indicate the values provided by Williams and Boggs (2015). Lower row: model samples plot- +ted in the parameter space, with the mantle rigidity µm depicted against viscosity ηm (left), the +relaxation strength ∆ depicted against the characteristic time τ of the elastically-accommodated +GBS (centre), and the Maxwell time τM versus the characteristic time τ (right). The Andrade +parameters are kept constant at α = 0.2 and ζ = 1. Gray dots in the lower left panel show the +results obtained with a homogeneous model consisting only of a Sundberg-Cooper mantle, while +black dots represent the default two-layered model. +Figure 5. +The negative imaginary part of the Love number (left) and inverse quality factor +(right) for three model cases: a homogeneous Andrade model (dashed red line), a homogeneous +Sundberg-Cooper model (blue line), and a three-layered model (solid red line) comprising a core, +an Andrade mantle and a Maxwell semi-molten layer at the base of the mantle. +–23– + +0.028 +2.0 +0.026 +1.5 +F 0.022 +S- +0.5 +0.020 +1 month +1 month +l yr +0.018 +0.0 +-5 +-7 +5 +-7 +4 +-8 +8 +6 +-4 +6 +logx [rad/s] +logx [rad/s] +6.0 +6.0 +23.5 +5.5 +5.5 +23.0 +T +T +5.0 +a05.0 +0 +4.5 +4.5 +22.0 +21.5 +4.0 +4.0 +10.8 +10.9 +11.0 +11.1 +-1.5 +-1.0 +-0.5 +0.0 +10 +11 +12 +13 +log △ +1og TM0 +0 +Q +3 +Q +-2 +301 +-3 +-3 +Andrade, homogeneous +Andrade, layered +4 +Sundberg-Cooper +15.0 +-7.5 +2.5 +-10.0 +-5.0 +0.0 +-12.5 +-10.0 +-2.5 +-12.5 +-5.0 +0.0 +logx [rad/s] +logx [rad/s]manuscript submitted to JGR: Planets +Figure 6. +The negative imaginary part of the Love number (left) and inverse quality fac- +tor (right) of a three-layered lunar model comprising a core, a Sundberg-Cooper mantle, and a +Maxwell semi-molten basal layer. Different shades of blue correspond to different ratios between +the timescale τ and the Maxwell time τM. For illustrative purposes, the semi-molten basal layer +is made unrealistically thick (500 km). +dius of the basal layer. Due to the higher dimensionality of the parameter space, the in- +verse problem took longer to converge; therefore, we generated 10, 000, 000 random sam- +ples satisfying all constraints from Table 2. Since the longest autocorrelation time in this +case was 500, 000 steps, we discarded the first 5, 000, 000 samples and then applied the +condition Q2,monthly < Q2,annual , being left with 11% of the generated samples. +As illustrated in Figure 7, and in line with the discussion above, the frequency de- +pendencies of ℜ[¯k2] and −ℑ[¯k2] in the model with a low-viscosity basal layer closely re- +semble those of the previous one, in which we considered the Sundberg-Cooper model. +Similarly to the earlier predictions of the basal layer’s viscosity and thickness (e.g., Harada +et al., 2014, 2016; Matsumoto et al., 2015), we find that the observed frequency depen- +dence of lunar Q−1 +2 +can be explained by the viscosity ηLVZ in the range from ∼ 1015 to +∼ 3×1016 Pa s and the thickness DLVZ in the range from 70 km to the maximum value +considered in our model (370 km). The parameter dependencies of all model samples are +plotted on Figure 8. For the nominal case with α = 0.2 and ζ = 1, and considering +the condition on Q mentioned in the above paragraph, we obtain the following rigidity +and viscosity of the overlying mantle and of the LVZ: log ηm[Pa s] = 22.79+0.19 +−0.06, µm[Pa] = +10.89±0.03, ηLVZ[Pa s] = 15.20+0.53 +−0.21, µLVZ[Pa] = 10.23+0.37 +−0.34. The corresponding outer +radius of the LVZ is RLVZ[km] = 599.39+65.83 +−84.46. +Similarly to the “melt-free” case with the Sundberg-Cooper model, increasing α +to 0.3 results in an order-of-magnitude decrease in the fitted mantle viscosity. Decreas- +ing α to 0.1 leads to a mantle viscosity two orders of magnitude greater. On the other +hand, the predicted properties of the semi-molten layer remain almost the same. +6 Discussion +In the previous section, we have compared the frequency dependence of lunar Q +within the widely accepted lunar interior model containing a highly dissipative layer at +the base of the mantle (e.g., Nakamura et al., 1973; Williams et al., 2001; Harada et al., +2014) and within an alternative model taking into account the time relaxation of the elas- +tic compliance Je. On the following lines, we discuss the implications of each of the con- +sidered models for the lunar interior properties. Keep in mind that the inversions per- +–24– + +0 +0 +Q +Q +2 +301 +301 +1.0 +0.0001 +3 +0.1 +1e-05 +0.01 +1e-06 +4 +0.001 +1e-07 +15.0 +-7.5 +2.5 +-10.0 +-5.0 +0.0 +-12.5 +-10.0 +7.5 +5.0 +-2.5 +-12.5 +0.0 +50 +logx [rad/s] +logx [rad/s]manuscript submitted to JGR: Planets +Figure 7. +Overview of best-fitting models for the case with a basal low-viscosity zone. The +red and yellow lines indicate the values provided by Williams and Boggs (2015). As in the previ- +ous inversion, the Andrade parameters are kept constant at α = 0.2 and ζ = 1, and the core size +is fixed to 330 km. +Figure 8. +Model samples corresponding to Figure 7, plotted in the parameter space. The +intensity indicates the sample count. Upper row: the rigidity vs. viscosity of the LVZ (left), the +rigidity vs. viscosity of the mantle (centre), and the outer radius vs. viscosity of the LVZ (right). +Lower row: the rigidity of the LVZ vs. rigidity of the mantle (left), the viscosity of the LVZ vs. +viscosity of the mantle (centre), and the outer radius vs. rigidity of the LVZ (right). +–25– + +0.028 +2.0 +0.026 +1.5 +H 0.022 +0.5 +0.020 +1 month +1 month +yr +0.018 +0.0 +-6 +8 +-5 +-4 +8 +-5 +4 +logx [rad/s] +logx [rad/s]24.00 +16.5. +16.5. +23.75 +S +s 23.50 +[Pa + edl +16.0 +nLVZ +nLVZ +10g 1 +15.5 +23.00 +22.75 +15.0 +15.0 +22.50 +10.0 +10.5 +11.0 +10.80 +9.5 +10.95 +500 +600 +700 +10.85 +10.90 +400 +RLvz [km] +log μLVz [Pa] +log μm [Pa] +24.00 +11.00 +10.95 +23.75 +10.75 +S23.50 +0.50 + edl +VZ +23.25 +10.25 +20 10.85 +23.00 +22.75 +9.75 +10.80 +22.50 +9.5 +10.0 +10.5 +11.0 +15.0 +600 +15.5 +16.0 +16.5 +400 +500 +700 +RLvz [km] +log μLvz [Pa] +log nLVz [Pa s]manuscript submitted to JGR: Planets +formed in our study explicitly assumed that the value of Q at the monthly frequency and +k2/Q at the annual frequency are known with a high precision. This is not the case in +reality. However, as we have seen in Subsection 5.4, a lunar mantle governed by the An- +drade model without a basal dissipative layer can fit the data with the actual uncertain- +ties only for unrealistically low values of parameter α. +6.1 Melt-free Lunar Interior +In the model cases considering a two-layered, “melt-free” lunar interior, where the +negative slope of the frequency dependence of k2/Q is explained by a secondary dissi- +pation peak induced by elastically accommodated GBS, we found that the logarithm of +the relaxation timescale, log τ, falls into the range of [4, 6], corresponding to τ between +3 and 300 hours. In the reference case depicted in Figure 4, this would imply a ratio of +the characteristic timescales for the elastic and diffusional accommodation trel = τ/τM +to be of order from 10−7 to 10−6. Are such ratios of the characteristic times observed +in any natural materials? +According to Jackson et al. (2014), grain boundary sliding comprises three processes. +The relative contribution of each of them to the energy dissipation in a sample depends +on the temperature and loading frequency. The processes are: (i) elastically accommo- +dated GBS with a characteristic time τ, at high frequencies/low temperatures; (ii) dif- +fusionally assisted GBS described by the power-law frequency-dependence of the seis- +mic quality factor, Q ∝ χp ; and (iii) diffusionally accommodated GBS at timescales +greater than the Maxwell time τM, where the seismic Q is a linear function of frequency, +Q ∝ χ. The value of trel thus determines the range of frequencies over which the dif- +fusionally assisted sliding on spacial scales smaller than grain size occurs. Experimen- +tal data for fine-grained polycrystals indicate that trel ≪ 1 (Morris & Jackson, 2009). +Jackson et al. (2014) presented results of laboratory experiments on fine-grained +olivine subjected to torsional oscillations at high pressures (P = 200 MPa) and rela- +tively low temperatures (T < 900 ◦C), i.e., around the threshold between elastic response +and elastically accommodated GBS. They found a GBS relaxation timescale of log τR = +1.15±0.07 s, where the subscript “R” now stands for “reference”. Considering the ref- +erence temperature TR = 900 ◦C, reference pressure PR = 200 MPa, reference grain +size dR = 10 µm, activation volume V ∗ = 10 cm3 mol−1, and activation energy E∗ = +259 kJ mol−1, as given by Jackson et al. (2014), we can extrapolate τ to the conditions +of the lunar mantle with the Arrhenius law (Jackson et al., 2010): +τ = τR +� d +dR +�m +exp +�E∗ +R +� 1 +T − 1 +TR +�� +exp +�V ∗ +R +�P +T − PR +TR +�� +. +(46) +In addition to the parameters introduced earlier, d is the grain size and m char- +acterises the grain-size dependence of the process in question. We adopt the value m = +1.31, found by Jackson et al. (2010) for anelastic processes. Figure 9 illustrates the ex- +trapolation of τR of Jackson et al. (2014) to lunar interior conditions, considering our +melt-free model and two depth-independent grain sizes. Over the colour-coded maps, we +also plot the steady-state heat conduction profiles of Nimmo et al. (2012). We note that +the conduction profiles were only chosen for illustration purposes: the discussion of the +thermal regime (conductive vs. convective) in the lunar mantle is beyond the scope of +this paper. +The laboratory measurements of Jackson et al. (2014) were performed on a single +sample of fine-grained polycrystalline olivine under constant pressure PR and the Arrhe- +nian extrapolation of τ was only tested for temperature dependence. Nevertheless, if we +accept the assumption that these results are applicable to the Moon, Figure 9 and the +fitted relaxation time from Figure 4 (log τ ∈ [4, 6]) can help us to identify the minimum +depth in which elastically accomodated GBS contributes to the tidal dissipation. For the +–26– + +manuscript submitted to JGR: Planets +Figure 9. +Relaxation time τ (colour-coded) of elastically accommodated GBS, as given by +Jackson et al. (2014) and extrapolated to lunar interior conditions using the Arrhenian equation +(46). White lines demarcate the relaxation times resulting from our inversion. Blue lines indicate +analytically-calculated conduction profiles proposed by Nimmo et al. (2012) for three different +mantle heat productions (8, 9.5, and 11 nW m−3), crustal heat production of 160 nW m−3 crustal +thickness of 45 km, and no heat exchange between core and mantle. Other parameters, such as +the core size, core density, and mantle density, are adjusted to our melt-free model. Grain sizes +are given in the upper right corner of each plot. +smaller grain size (d = 0.1 mm) and the reference profile of Nimmo et al. (2012) (solid +line, mantle heat production of 9.5 nW m−3), we predict the minimum depth of 400–500 km. +For the larger grain size (d = 1 cm), the minimum depth is 600–800 km. A conductive +profile corresponding to lower heat production than illustrated here would push the min- +imum depth to even greater values. The occurrence of elastically accommodated GBS +in shallower depths would give rise to a relaxation peak (or to an onset of a relaxation +band) at lower loading frequencies, which would not fit the observed annual and monthly +tidal Q. Although the MCMC inversion from the previous section was performed for a +model with a homogeneous mantle, i.e., assuming the occurrence of elastically-accommodated +GBS at all depths from the surface down to the core, we also checked that a model de- +scribed by the Andrade rheology above the derived depths and by the Sundberg-Cooper +model below the derived depths would fit the considered observables under the condi- +tion that log τ ≳. For shorter τ, the estimated minimum depth of applicability of the +Sundberg-Cooper model would not match the Love numbers at monthly frequency. +Besides the timescale τ, we have derived the relaxation strength of the hypothet- +ical secondary peak: log ∆ ∈ [−1.5, 0], or ∆ ∈ [0.03, 1]. Parameter ∆ controls the height +of the secondary dissipation peak in the Sundberg-Cooper model. Figure 10 shows the +dependence of this Q−1 on the relaxation strength for all our models from Figure 4. Are +these values consistent with theoretical prediction and laboratory data? +Sundberg and Cooper (2010) reported relaxation strengths of polycrystalline olivine +between 0.23 and 1.91, as found in different sources and under different assumptions on +the grain shapes (Kˆe, 1947; Raj & Ashby, 1971; Ghahremani, 1980). Their own mechan- +ical tests on peridotite (olivine-orthopyroxene) at temperatures between 1200 and 1300 ◦C +were best fitted with ∆ = 0.43 and the corresponding dissipation associated with elastically- +accommodated GBS in their sample was Q−1 = 0.25−0.3. On the other hand, Jackson +et al. (2014), who performed torsion oscillation experiments on olivine, found a relatively +low dissipation peak with Q−1 ≤ 0.02. Low secondary dissipation peaks with Q−1 ∼ +10−2 were also predicted theoretically by Lee and Morris (2010) for a grain boundary +slope of 30◦, while smaller slopes seem to allow Q−1 exceeding 1, especially when the in- +dividual grains are of comparable sizes and the grain boundary viscosity does not vary +–27– + +20 +d=10 +d=10 +m +m +1500 +1500 +16 +1250 +1250 +12 +K +K +T +log +T 1000 +T 1000 +8 +750 +750 +4 +500 +500 +0 +400 +700 +1000 +1300 +1600 +400 +700 +1000 +1300 +1600 +r [km] +r [km]manuscript submitted to JGR: Planets +too much. Accordingly, Lee et al. (2011) note that Q−1 in the secondary peak depends +strongly on the slope of the grain boundaries. +Following this brief discussion of dissipation arising due to elastically accommo- +dated GBS, we can conclude that the relaxation strength ∆ (or Q−1 in the secondary +dissipation peak) is not well constrained and the values found in literature permit any +of the ∆s predicted in our Subsection 5.5. +Figure 10. +Seismic Q−1 of the mantle at the frequency of the secondary peak, plotted as a +function of the relaxation strength ∆ for models from Figure 4. +6.2 Highly Dissipative Basal Layer +Figure 11. +Shear modulus prediction compared to seismic measurements. Shear modulus +µLVZ for RLVZ += +400, 500 and 700 km (gray, yellow and orange areas). Shear modulus derived +from seismic velocities and densities: green (Weber et al., 2011), red (Khan et al., 2014) and blue +(Matsumoto et al., 2015), dashed lines: errors. +A highly dissipative layer located at any depth could also produce the desired sec- +ondary peak needed to explain the anomalous Q dependence. (Note, however, that a pres- +ence of a highly dissipative layer at a shallow depth may lead to changes in the body’s +response to tides and might be incompatible with the measured values of the Love num- +bers.) Petrological considerations combined with an indication of a basal low-velocity +–28– + +peak +0.30 +in the secondary +0.25 +0.20 +0.15 +1 +0.10 +seismic ( +0.05 +0.00 +-1.5 +-1.0 +-0.5 +0.0 +log △11.00 +10.75 +10.50 +log μ [Pa] +10.25 +Weber et al. (2011) +Khan et al. (2014) +10.00 +Matsumoto et al. (2015) +this study, RLvz = 400 km +9.75 +this study, RLvz = 500 km +this study, RLvz = 700 km +9.50 +400 +600 +800 +1000 +r [km]manuscript submitted to JGR: Planets +zone point to the presence of this anomalous layer in the deep interior. Therefore, as an +alternative to the “melt-free” model, we tested the popular hypothesis of a putative highly +dissipative layer at the base of the lunar mantle. +The derived rheological properties of the mantle and of the basal layer as well as +the layer’s thickness are poorly constrained and can be strongly biased. Firstly, the outer +radius RLVZ of the basal layer is correlated with the value of the mantle rigidity µm; the +thicker the basal layer, the larger mantle rigidity can be expected to satisfy the model +constraints. The mantle viscosity ηm depends on the empirical Andrade parameters, and +an increase of α by 0.1 leads to a reduction of the fitted mantle viscosity approximately +by one order of magnitude. On the other hand, the viscosity of the basal layer remains +independent of the empirical Andrade parameters. The predicted contrast in viscosity +between the two layers thus decreases with increasing α and/or ζ. Secondly, the range +of acceptable basal rigidities µLVZ widens with the basal layer thickness (Figure 11). We +do not find an acceptable solution for RLVZ ≲ 400 km due to our a priori requirement +on the relationship between the mantle and basal layer’s rigidities (µLVZ ≤ µm). The +range of acceptable µLVZ values increases with the basal layer radius up to one and a half +order of magnitude for the maximum RLVZ = 700 km considered here. Interestingly, +the predicted rigidities of a basal layer with thickness ∼ 170 km (RLVZ ≈ 500 km) cor- +responds well with the seismic observations. Lastly, the basal viscosity is correlated with +the basal layer thickness: the viscosity ηLVZ decreases from 3·1016 Pa s for a thin weak +layer (RLVZ = 400 km) to < 1015 Pa s for the greatest considered thickness (RLVZ = +700 km). The basal layer viscosity is, therefore, always considerably lower than the man- +tle viscosity. However, this is not surprising, as the low viscosity of this layer is essen- +tial to predict the anomalous frequency dependence of the tidal quality factor, when the +rest of the high-viscosity mantle is set to obey the Andrade law. +Figure 12. +Basal viscosity prediction compared to rheological properties. Predicted ranges +of viscosities ηLVZ for RLVZ += +400, 500 and 700 km are indicated by gray, yellow, and orange +areas, respectively. Over the predicted ranges is plotted the temperature dependence of viscosity +of ilmenite (blue, Dygert et al., 2016), dry olivine (red, Hirth & Kohlstedt, 1996), and ilmenite- +olivine aggregate (2 +− +16 %), the latter corresponding either to isostress (blue area, harmonic +mean, suggested for high strain) or Tullis (red area, geometric mean, suggested for low strain) +models. Errors of experimentally determined viscosities not included; ilmenite error factor is ∼ 5. +Vertical lines delimit solidus temperatures for peridotite (Katz et al., 2003) and ilmenite-bearing +material (Wyatt, 1977) at radii 330 km and 700 km. Left panel: temperature dependence for +σD = 1 MPa, dry olivine. Right panel: temperature dependence for σD = 1 MPa, wet olivine. +Rigidity and viscosity magnitudes, and their contrast between the mantle and the +basal layer values, can be indicative of the variations in the composition, in the presence +of melt, and in temperature. A stable partially molten zone in the lunar interior would +pose strong constraints on the composition (Khan et al., 2014). Given the absence of ge- +–29– + +Wet olivine +Dry olivine +20- +20- + olivine, Hirth and Kohlstedt (1996) +1 +ilmenite, Dygert et al. (2016) +18 +18 +isostress/Reuss model +S +[Pa +Tullis model +ii + peridotite, solidus, Katz et al. (2003) +ilmenite-bearing, solidus, Wyatt (1977) +16 +this study, RLVz + 400 km +I +H +this study, RLvz = 500 km +this study, RLvz = 700 km +I +14 +14 +I +ii +ii +2000 +2000 +1200 +1400 +1600 +1800 +1200 +1400 +1600 +1800 +T [K] +T [K]manuscript submitted to JGR: Planets +ologically recent volcanic activity, any melt residing in the deep lunar interior would have +to be neutrally or negatively buoyant. Using an experimental approach on the synthetic +equivalent of Moon samples, van Kan Parker et al. (2012) concluded that the condition +on the buoyancy below 1000 km is satisfied if high content of titanium dioxide is present +in the melt. We can expect the presence of a partially molten layer at any depth below +this neutral buoyancy level. +Moreover, evolutionary models suggest that high-density ilmenite bearing cumu- +lates enriched with TiO2 and FeO are created towards the end of the shallow lunar magma +ocean crystallisation, resulting in near-surface gravitational anomalies. This instability, +combined with the low viscosity of those cumulates, might have eventually facilitated +the mantle overturn, creating an ilmenite-rich layer at the base of the mantle (e.g., Zhang +et al., 2013; Zhao et al., 2019; Li et al., 2019). Recently, Kraettli et al. (2022) suggested +an alternative compositional model: a ∼ 70 km thick layer of garnetite could have been +created at the base of the mantle if two independently evolving melt reservoirs were present. +The resulting high-density garnet, olivine, and FeTi-oxide assemblage is gravitationally +stable and can contain a neutrally or negatively buoyant Fe-rich melt. The scenario of +Kraettli et al. (2022) can also be accompanied by the mantle overturn, as suggested for +the ilmenite-rich layer created at shallow depths. +Rheologically weak ilmenite combined with appropriate lower-mantle temperature +can help to explain the low basal viscosity (Figure 12). If the lower mantle were only made +of dry olivine, the predicted viscosity would require temperature ≳ 1800 K, whereas for +wet olivine, the temperature range between ∼ 1500 and ∼ 1800 K would be sufficient. +Creep experiments (Dygert et al., 2016) conclude that the viscosity of ilmenite is more +than three orders of magnitude lower than dry olivine. Consequently, a lower-mantle tem- +perature (1400 − 1700 K) might be acceptable to explain the predicted viscosities for +pure ilmenite. The properties of ilmenite-olivine aggregates introduce yet another com- +plexity. The viscosity of aggregates is suggested to depend on the value of the strain: it +follows the Tullis model for low strain, whereas it tends to follow the lower bound on Fig- +ure 12 (isostress model) for large strain (see, e.g., Dygert et al., 2016, for a deeper dis- +cussion). The acceptable temperature range for olivine-ilmenite aggregate is close to the +values for the pure olivine in the case of the Tullis model. The prediction for the isostress +model (minimum bound, Reuss model) is consistent with temperature values between +1500−1800 K. Another obstacle in interpretation originates in the stress-sensitivity of +the relevant creep. The viscosity can decrease by ∼ 2.5 orders of magnitude while de- +creasing the differential stress by one order of magnitude. In terms of acceptable ther- +mal state, the temperature consistent with our prediction would decrease roughly by ∼ +100 K considering two-fold higher differential stress and increase by the same value for +two-fold lower stress, respectively. +Consequently, we find acceptable solutions both below and above the solidus. Our +three-layered model thus cannot exclude or confirm a possible partial melt presence. An +alternative explanation for the viscosity reduction can be the presence of water (see also +Karato, 2013, for a deeper dicussion), which would also reduce the solidus temperature +and facilitate partial melting. Both the enrichment in ilmenite and elevated water con- +tent can lead to the desired value of viscosity at lower temperatures compared to the dry +and/or ilmenite-free models (Figure 12). +Focusing now on the elastic properties, we note that the rigidities of olivine (e.g. +Mao et al., 2015), ilmenite (Jacobs et al., 2022), and garnetite (Kraettli et al., 2022) are +comparable. The temperature has only a limited impact on their value (−0.01 GPa/K +for olivine and ilmenite). Also, dependence on the water content (olivine-brucite) is only +moderate (−1.3 GPa/wt%; Jacobsen et al., 2008). The magnitude of rigidity is, there- +fore, rather insensitive to possible constituents, temperature and water content. The up- +per bound of basal layer’s rigidity predicted here (∼ 60 GPa for RLVZ = 400 km, ∼ 70 GPa +for RLVZ = 500 km and ∼ 85 GPa for RLVZ = 700 km) fits the elastic properties of all +–30– + +manuscript submitted to JGR: Planets +considered minerals—ilmenite, olivine, and garnet. However, the lower bound values (for +RLVZ > 500 km) are difficult to explain by the changes in composition, high temper- +ature, and/or water content. +Figure 13. +Impact of melt on the viscosity and rigidity contrast. The viscosity and rigidity +contrast expressed as a function of the φ/φc (φ denotes the porosity and φc the critical porosity +and parameterised using Kervazo et al. (2021); ηsolid and µsolid represents values with no melt +present at the solidus temperature; no change in composition is considered. The shaded areas +depict the predicted contrasts. +The magnitude of rigidity (Figure 13) is, nevertheless, sensitive to the presence of +melt around or above the disintegration point (characterised by the critical porosity φc), +which describes the transition from the solid to liquid behaviour and its typical values +lie between 25−40%. Similarly, the viscosity value is very sensitive to the presence of +melt for porosity higher than φc. For low porosities, it follows an exponential (Arrhe- +nian) dependence. Figure 13 suggest that the predicted rheological contrasts in the nom- +inal case are consistent with φ ≲ 1.1φc for shear modulus contrast and with φ > 1.1φc +for the viscosity contrast. This apparent inconsistency may be accounted for by the pres- +ence of melt accompanied by the changes in composition of the basal layer and by the +susceptibility of viscosity to these changes. Consequently, the knowledge of the contrasts +in both rheological parameters (rigidity and viscosity) could help tackle the trade-offs +between porosity content and composition/temperature. Nevertheless, we must empha- +sise that the viscosity contrast predicted by our models is sensitive to the Andrade pa- +rameters of the mantle, leading to another uncertainty. +The presence of a partially molten material would pose a strong constraint on the +temperature and possible mode of the heat transfer in the lower mantle of the Moon, al- +lowing only models that reach the temperature between the solidus and liquidus (Fig- +ure 14). The traditional advective models predict stagnant-lid convection with a rela- +tively thick lid at present (e.g. Zhang et al., 2013). Below the stagnant lid, the temper- +ature follows the adiabatic or, for large internal heating, sub-adiabatic gradient. We es- +timate the temperature increase across the entire mantle due to the adiabatic gradient +to be bounded by 100 K. Within those traditional models, it is plausible to reach solidus +only in the lowermost thermal-compositional boundary layer. In the case of conductive +models (e.g. Nimmo et al., 2012), the temperature gradient is steeper than the solidus +gradient and the solidus temperature can be reached in the entire basal layer, given ap- +propriate internal heating (as demonstrated in Figure 14). Interestingly, the lunar se- +lenotherm determined by the inversions of lunar geophysical data combined with phase- +equilibrium computations (Khan et al., 2014) lies between the conductive and adiabatic +gradients. +–31– + +0.0 +0 +2.5 +1 +log n(d) / N solid +-5.0 +2 +7.5 +3 +Kervazo et al. (2021) +-10.0 +this study, Rvz =400 km +4 +this study, RLvz = 500 km +-12.5 +this study, RLvz = 700 km +-5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +relative porosity / Φ +relative porosity Φ / Φcmanuscript submitted to JGR: Planets +Figure 14. +Comparison of temperature profiles. Colour scale: conductive profile, calculated +with the matrix propagator method; parameters as in Figure 9. Individual branches correspond +to average heating 8, 9.5 and 11 nW/m2 in the mantle. The coefficient f denotes the enrich- +ment in the radiogenic elements of the basal layer (RLVZ += +500 km) compared to the rest of the +mantle. Gray area is the temperature profile adapted from Khan et al. (2014); darker blue lines: +peridotite solidus (solid), water-saturated solidus (dotted), and liquidus (dashed) according to +Katz et al. (2003); light blue lines: clinopyroxene+ilmenite solidus (solid) and liquidus (dashed) +according to Wyatt (1977). +–32– + +5.0 +2500 +4.5 +4.0 +2000 +3.5 +basal enrichment +1500 +3.0 +T +2.5 +solidus, dry peridotite, Katz et al. (2003) +1000 +solidus, water saturated peridotite, Katz et al. (2003) +2.0 +liquidus, peridotite, Katz et al. (2003) +solidus, ilmenite bearing, Wyatt (1977) +500 +1.5 +liquidus, ilmenite bearing, Wyatt (1977) +Khan et al. (2014) +1.0 +400 +600 +800 +1000 +1200 +1400 +1600 +r [km]manuscript submitted to JGR: Planets +In the future, distinct sensitivity of rigidity, viscosity, and other transport prop- +erties to temperature, melt fraction, and composition may provide a way to separate the +interior thermal and composition structure. At present, inversion errors and the uncer- +tainties on material properties cannot confirm or rule out the existence of a partially molten +basal layer. It therefore remains a valid hypothesis. +6.3 Other Sources of Information +The two models discussed here — one with a highly dissipative basal layer and the +other with elastically-accommodated GBS in the mantle — cannot be distinguished from +each other by the available selenodetic measurements. To answer the question stated in +the title of our paper, one would need to resort to other types of empirical data. Among +all geophysical methods devised for the exploration of planetary interiors, seismology is +of foremost importance. Therefore, a question that cannot be solved by the interpreta- +tion of lunar tidal response might be answered by comparing the arrival times and the +phases detected at individual seismic stations. +As we mentioned in Introduction, the Moon demonstrates a nearside-farside seis- +mic asymmetry. Judging by the currently available seismic data collected on the near +side, the deep interior of the far side is virtually aseismic or, alternatively, the seismic +waves emanating from it are strongly attenuated or deflected. The existence of an aseis- +mic area on the farside might not be entirely inconceivable. First, as pointed out by Nakamura +(2005), there are large zones with no located nests of deep moonquakes even on the near- +side; and, in fact, most of the known deep seismic nests are part of an extended belt reach- +ing from the south-west to the north-east of the lunar face. Second, there exists a pro- +nounced dichotomy between the near side and far side of the Moon in terms of the crustal +thickness, gravity field, and surface composition, which might point to a deeper, inter- +nal dichotomy as predicted by some evolutionary models (e.g., Laneuville et al., 2013; +Zhu et al., 2019; Jones et al., 2022). +An obvious way to illuminate the lack of deep farside moonquakes detected by the +Apollo seismic stations would be to place seismometers on the far side of the Moon. They +would observe the far side activity, and record the known repeating nearside moonquakes +or events determined from impact flash observations. The Farside Seismic Suite (FSS) +mission, recently selected for flight as part of the NASA PRISM program and planned +for launch in 2024 or 2025, might provide such a measurement by delivering two seismome- +ters to Schr¨odinger Crater (Panning et al., 2021). While this crater is far from the an- +tipode (in fact, close to the South pole), a seismometer residing in it should still be able +to detect events from the far side, thereby addressing the hemispheric asymmetry in the +Apollo observations. However, resolving polarisation of arrivals may be challenging for +many moonquakes, meaning that many events will only have distance estimated, but not +azimuth. (We are grateful to Mark P. Panning for an enlightening consultation on this +topic.) +A better site for this science objective would be the far side Korolev crater resid- +ing by the equator, about 23 degrees from the antipode (by which we understand the +centre of the farside). It is now considered as one of the possible landing sites for the Lu- +nar Geophysical Network (LGN) mission proposed to arrive on the Moon in 2030 and +to deploy packages at four locations to enable geophysical measurements for 6 - 10 years +(Fuqua Haviland et al., 2022). +Still, having a station or even an array of seismic stations at or near the antipode +would be ideal. Observed by such a station or stations, all events at distances less than +90 degrees from the antipode could be confidently assigned to the far side. So we would +recommend the near-antipode zone (that close to the centre of the farside) as a high-priority +landing site for some future mission, a perfect area to monitor the seismic activity on +–33– + +manuscript submitted to JGR: Planets +the far side and, especially, to observe if and how seismic waves proliferate through the +base of the mantle. +In addition to seismic measurements, and similarly to what is predicted for Jupiter’s +volcanic moon Io or for icy moons with subsurface oceans, the presence of a highly dis- +sipative or a partially molten layer might be reflected in the tidal heating pattern on lu- +nar surface (e.g., Segatz et al., 1988; Tobie et al., 2005). However, as illustrated in the +upper row of Figure 15, the positioning of the layer at the base of the mantle results in +a very small difference between the surface heating patterns corresponding to the two +alternative models. Both models show maxima of the average surface tidal heat flux Φtide +on the lunar poles and minima on the “subterranean” point (ϕ = 0) and its antipode +(ϕ = π). Moreover, the magnitude of Φtide is generally very small, about three orders +of magnitude lower than the flux produced by radiogenic heating of lunar interior (e.g., +Siegler & Smrekar, 2014). The detection of any differences between the surface heat flux +of the two models would be extremely challenging, if not impossible. +Figure 15. +Average surface tidal heat flux (top) and volumetric tidal heating (bottom) for +a specific realisation of each of the two models discussed in this work: the model considering +elastically-accommodated GBS through the Sundberg-Cooper rheological model (left) and the +model with a basal low-viscosity zone (right). In particular, the volumetric tidal heating is plot- +ted as a function of relative radius r/R and colatitude ϑ with longitude ϕ equal to 0. +The lower row of Figure 15 illustrates volumetric heat production due to tidal dis- +sipation. As pointed out by Harada et al. (2014), the presence of a low-viscosity zone +at the base of the mantle results in considerable local increase of tidal heating with re- +spect to the rest of the mantle or to the model without the basal layer. While the tidal +contribution to heat production in the high-viscosity parts of the mantle is around 10−11 W m−3, +the tidal heat production in the basal layer reaches ∼ 10−8 W m−3. For comparison, the +global average of mantle heat production by all sources (radiogenic and tidal) is estimated +to be 6.3×10−9 W m−3 (Siegler & Smrekar, 2014). The predicted tidal dissipation in +the basal layer can help to locally increase the temperature and exceed the solidus, es- +pecially if conductive heat transfer prevails in the lunar mantle. Combined with a high +enrichment of the basal layer in heat producing elements, it may then contribute to main- +taining the presence of melt. +–34– + +Sundberg-Cooper mantle +Mantle with a low-viscosity layer +儿 +元 +0.021 +0.019 2 +[mW /m +0.017 +0.015 +0.013 +0.011 +0 +0 +元-2 +-2 +2元 +元-2 +3-2 +0 +0 +2元 +元 +1.0 +1.0 +-8 +0.8 +0.8 +9 +0.6 +10 +-11 +0.4 +0.4 +-12 +0.2 +0.2 +元-4 +π14 +元-2 +π-2 +0 +元 +0 +元 +9 +6manuscript submitted to JGR: Planets +Although virtually discarded in the beginning of this Subsection, let us neverthe- +less discuss possible insights provided by future high-precision tidal measurements. At +present, the quality factor Q at tidal frequencies is obtained exclusively from fitting the +lunar physical libration, empirically determined by LLR. However, increased precision +of satellite tracking (Dirkx et al., 2019; Hu et al., 2022; Stark et al., 2022) might even- +tually enable the determination of lunar tidal phase lag from the gravity field. Having +an independent determination of tidal Q, which is related to the phase lag, would serve +as a verification of the method used for fitting the LLR time series. +Among the quantities that we used in the inversion was degree-3 potential Love +number k3. This parameter is currently only known with a large error bar but its refine- +ment would only help to discern between the two alternative models considered here if +the elastically-accommodated GBS was contributing to the dissipation throughout the +entire mantle (and not only in greater depths, as tentatively derived in Subsection 6.1). +This is a consequence of a degree-dependent sensitivity of Love numbers to the interior +structure. While degree-2 Love numbers and quality factors probe the lunar interior down +to the core, higher-order quantities are only sensitive to shallower depths. The Love num- +ber k3—or the quality factor Q3—would thus not “see” the basal low-viscosity layer, but +it might sense complex tidal response in the upper mantle. As a result, the detection of +the unexpected frequency dependence of tidal dissipation even in Q3 (accompanied by +a relatively high k3 ∼ 0.01) would clearly point at a mechanism acting in shallow depths. +Interestingly, the two alternative models can be better distinguished from each other +in case the secondary peak of tidal dissipation, resulting either from the existence of a +weak basal layer or from the Sundberg-Cooper model, lies at frequencies close to 10−4 rad s−1. +Then, provided that the elastically-accommodated GBS is only active below distinct depths +(400−600 km), one could see a difference in predicted h2 of the two models. Indepen- +dently on that depth, the models with secondary dissipation peak close to 10−4 rad s−1 +also differ in elastic Love number k2,e, which can be calculated for interior structures ob- +tained from the inversion of seismic waves (as was done by Weber et al., 2011). Specif- +ically, k2,e in the melt-free model is then much lower that that of the model with a weak +basal layer. The value reported by Weber et al. (2011), which is k2,e = 0.0232, is at- +tained by both the alternative models for a secondary tidal dissipation peak lying at ∼ +10−5.5 rad s−1. In that case, the models are already indistinguishable. Seismic Q in the +melt-free part of the mantle (at 1 Hz) for the models mentioned in the previous sentence +is around 800 − 1000. +Finally, we would like to note that any increase in the precision of Q determina- +tion will greatly help in answering the question whether any specific source of additional +dissipation, be it a weak basal layer or elastic accommodation of strain at grain bound- +aries, is necessary in the first place. Recall that in order to fit the two alternative mod- +els to the tidal data, we assumed that the uncertainty on Q is of the order of 1% the mean +value. In reality, the empirical Q at the monthly and the annual frequencies present an +uncertainty between 10 and 20%. Keeping the original uncertainties, we were still able +to fit the tidal data with the standard Andrade model, although with an unrealistically +small exponential factor. +7 Conclusions +Tidal effects strongly depend not only on the interior density, viscosity, and rigid- +ity profiles of celestial bodies, but also on the implied deformation mechanisms, which +are reflected in the rheological models adopted. In this work, we attempted to illustrate +that the unexpected frequency dependence of the tidal Q measured by LLR (Williams +& Boggs, 2015) can be explained by lunar interior models both with and without a par- +tially molten basal layer, and that each of the considered models leads to a different set +of constraints on the interior properties. +–35– + +manuscript submitted to JGR: Planets +As a first guess, we fitted the lunar tidal parameters (k2, k3, h2, Q at the monthly +frequency and k2/Q at the annual frequency) with a model consisting of a fluid core and +a viscoelastic mantle governed by the Andrade rheology. Within that model, and set- +ting ζ = 1 (i.e., the time scales of viscoelastic and anelastic deformation were consid- +ered comparable) we found a mantle viscosity of log ηm[Pa s] = 22.99+0.89 +−1.35, mantle rigid- +ity of log µm[Pa] = 10.92±0.06, and the Andrade parameter α as low as 0.06+0.04 +−0.02. The +predicted value of α is generally lower than reported in the literature (0.1-0.4; e.g., Jack- +son et al., 2010; Castillo-Rogez et al., 2011; Efroimsky, 2012a, 2012b). This observation +leads us to the conclusion that the tidal response of the Moon probably cannot be ex- +plained by the Andrade model alone and requires either a basal low-viscosity zone (in +line with the conclusion of Khan et al., 2014) or an additional dissipation mechanism in +the mantle (similar to Nimmo et al., 2012). +Throughout Section 5, we have seen that the two alternative models expected to +explain the anomalous frequency dependence of lunar Q (assumed to be known with an +arbitrarily chosen high precision) cannot be distinguished from each other by the exist- +ing measurements of tidal deformation and dissipation alone. In the two-layered model +consisting of a liquid core and a Sundberg-Cooper mantle, the fitting of tidal parame- +ters requires the relaxation time τ associated with elastically-accommodated GBS to be +in the range from 3 to 300 hours. The corresponding relaxation strength ∆ is predicted +to lie in the interval [0.03, 1]. For a nominal case with Andrade parameters α = 0.2 and +ζ = 1, we further obtain a mantle viscosity of log ηm[Pa s] = 22.55+0.15 +−0.54 and a mantle +rigidity log µm[Pa] = 10.840.14 +−0.02. +In the three-layered model containing a liquid core, a low-rigidity basal layer, and +an Andrade mantle, the tidal parameters are consistent with a wide range of basal layer +thicknesses DLVZ and rigidities µLVZ. As a general rule, a thicker layer implies weaker +constraints on its rigidity, allowing both melt-like and solid-like behaviour. The predicted +values of µLVZ are consistent with elastic properties of all considered minerals (olivine, +ilmenite, granite) and with a wide range of lower-mantle temperatures. In contrast to +the rigidity, the viscosity ηLVZ of the basal layer is constrained relatively well and falls +into the range from about 1015 to 3×1016 Pa s, with a preference for the lower values +(log ηLVZ[Pa s] = 15.20+0.53 +−0.21). This is also in accordance with the results of Efroimsky +(2012a, 2012b); Harada et al. (2014, 2016); Matsumoto et al. (2015); Tan and Harada +(2021), and Kronrod et al. (2022). Nevertheless, even the viscosity is not able to pose +strong constraints on the lower-mantle temperature, owing to the large uncertainties both +on tidal Q and on the rheological properties of lunar minerals. For the viscosity and rigid- +ity of the overlying mantle in the nominal case, we get log ηm[Pa s] = 22.79+0.19 +−0.06 and +log µm[Pa] = 10.88 ± 0.03. +The existence of a basal weak or possibly semi-molten layer in the mantles of ter- +restrial bodies has been recently also suggested for Mercury (Steinbr¨ugge et al., 2021) +and for Mars (Samuel et al., 2021). In the case of Mercury, a lower mantle viscosity as +low as 1013 Pa s was proposed to match the latest measurements of the moment of in- +ertia and of k2; although this result was later critically reassessed by Goossens et al. (2022), +who showed that more realistic values around 1018 Pa s might still explain the observa- +tions. In the case of Mars, the putative basal semi-molten layer was introduced by Samuel +et al. (2021) to provide an alternative fit to seismic data which would not require the ex- +istence of a large core with unexpectedly high concentration of light elements (reported +in St¨ahler et al., 2021). Lastly, large provinces of decreased shear seismic velocities also +exist at the base of the Earth’s mantle. These zones form a heterogeneous pattern in the +deep terrestrial interior; however, according to numerical models, the formation of a con- +tinuous layer right above the core-mantle boundary is also possible for some values of +model parameters (e.g., Dannberg et al., 2021). A new question thus arises: is a weak +basal layer something common among terrestrial planet’s mantles? Is it a natural and +widely present outcome of magma ocean solidification and subsequent dynamical pro- +cesses? Or is it merely a popular explanation of the data available? +–36– + +manuscript submitted to JGR: Planets +Since the available tidal parameters were deemed insufficient to distinguish a weak +basal layer above the lunar core from the manifestation of elastically accommodated GBS +in the mantle, we conclude that an answer to the question stated in the title of our pa- +per awaits future lunar seismic experiments (ideally with a uniform distribution of seis- +mometers across the lunar surface) as well as a better understanding of elastic param- +eters of olivine-ilmenite assemblages near their melting point. Additionally, a tighter bound +on the hypothetical basal layer parameters or on the strength and position of the sec- +ondary Debye peak in the alternative, Sundberg-Cooper model might be given by up- +dated values of tidal Q at multiple frequencies or by an independent inference of inte- +rior dissipation from the tidal phase lag and frequency-dependent k2, theoretically mea- +surable by laser altimetry or orbital tracking data (Dirkx et al., 2019; Hu et al., 2022; +Stark et al., 2022). A combination of all those sources of information will probably still +not provide a bright picture of deep lunar interior; however, it will help us to refute at +least some of the many possible interior models. +Open Research +The software developed for the calculation of tidal Love numbers and quality fac- +tors of multi-layered bodies, the Python interface for running the MCMC inversion, and +the plotting tools used for the figures presented in this study will be made available at +the GitHub repository of the corresponding author (https://github.com/kanovami/ +Lunar Q) and preserved at [DOI to be added later during the peer review process] un- +der the licence [to be added later during the peer review process]. +Acknowledgments +The authors would like to thank James G. Williams, Mark P. Panning, and Alexander +S. Konopliv for extremely helpful conversations on various aspects of the lunar science. +M.W. is grateful to Ana-Catalina Plesa and Martin Knapmeyer for discussions about +the lunar interior and to Philipp A. Baumeister for introducing her to the Python libraries +used in this work. She also gratefully acknowledges the financial support and endorse- +ment from the DLR Management Board Young Research Group Leader Program and +the Executive Board Member for Space Research and Technology. M.B. received fund- +ing from Czech Science Foundation grant no. GA22-20388S. +References +Bagheri, A., Efroimsky, M., Castillo-Rogez, J., Goossens, S., Plesa, A.-C., Rambaux, +N., . . . Giardini, D. +(2022, August). +Tidal insights into rocky and icy bodies: +An introduction and overview. 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Kawano, +N. +(2012, March). +CEGM02: An improved lunar gravity model using +Chang’E-1 orbital tracking data. +Planetary and Space Science, 62(1), 1-9. +doi: 10.1016/j.pss.2011.11.010 +Yan, J., Liu, S., Xiao, C., Ye, M., Cao, J., Harada, Y., . . . Barriot, J.-P. +(2020, +April). +A degree-100 lunar gravity model from the Chang’e 5T1 mission. +Astronomy & Astrophysics, 636, A45. doi: 10.1051/0004-6361/201936802 +Zhang, N., Parmentier, E. M., & Liang, Y. +(2013, September). +A 3-D numerical +study of the thermal evolution of the Moon after cumulate mantle overturn: +The importance of rheology and core solidification. +Journal of Geophysical +Research. Planets, 118(9), 1789-1804. doi: 10.1002/jgre.20121 +Zhao, Y., de Vries, J., van den Berg, A. P., Jacobs, M. H. G., & van Westrenen, +W. +(2019, April). +The participation of ilmenite-bearing cumulates in lu- +nar mantle overturn. +Earth and Planetary Science Letters, 511, 1-11. +doi: +10.1016/j.epsl.2019.01.022 +–43– + +manuscript submitted to JGR: Planets +Zharkov, V. N., & Gudkova, T. V. +(2005, September). +Construction of Martian In- +terior Model. Solar System Research, 39(5), 343-373. doi: 10.1007/s11208-005 +-0049-7 +Zhu, M.-H., W¨unnemann, K., Potter, R. W. K., Kleine, T., & Morbidelli, A. +(2019, +August). +Are the Moon’s Nearside-Farside Asymmetries the Result of a Giant +Impact? Journal of Geophysical Research (Planets), 124(8), 2117-2140. doi: 10 +.1029/2018JE005826 +–44– + diff --git a/R9E0T4oBgHgl3EQfkwEq/content/tmp_files/load_file.txt b/R9E0T4oBgHgl3EQfkwEq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d3295c09a9d9a8b2b41cabff4c53dcbf0526bc83 --- /dev/null +++ b/R9E0T4oBgHgl3EQfkwEq/content/tmp_files/load_file.txt @@ -0,0 +1,2582 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf,len=2581 +page_content='manuscript submitted to JGR: Planets Is there a semi-molten layer at the base of the lunar mantle?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Michaela Walterov´a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Marie Bˇehounkov´a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Michael Efroimsky3 1Institute of Planetary Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' German Aerospace Center (DLR),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Berlin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Germany 2Department of Geophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Faculty of Mathematics and Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Charles University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Prague,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Czech Republic 3US Naval Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Washington DC 20392 USA Key Points: A lunar mantle governed by the Andrade model fits selenodetic constraints only with a very weak frequency dependence of tidal dissipation We seek the parameters of the Sundberg-Cooper model that would explain the anoma- lous frequency dependence of tidal Q measured by LLR Both a dissipative basal layer and elastically-accommodated grain-boundary slid- ing in the deep mantle result in the same tidal response Corresponding author: Michaela Walterov´a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' kanovami@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='com –1– arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='02476v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='EP] 6 Jan 2023 manuscript submitted to JGR: Planets Abstract Parameterised by the Love number k2 and the tidal quality factor Q, and inferred from lunar laser ranging (LLR), tidal dissipation in the Moon follows an unexpected frequency dependence often interpreted as evidence for a highly dissipative, melt-bearing layer en- compassing the core-mantle boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Within this, more or less standard interpretation, the basal layer’s viscosity is required to be of order 1015 to 1016 Pa s and its outer ra- dius is predicted to extend to the zone of deep moonquakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' While the reconciliation of those predictions with the mechanical properties of rocks might be challenging, alterna- tive lunar interior models without the basal layer are said to be unable to fit the frequency dependence of tidal Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The purpose of our paper is to illustrate under what conditions the frequency-dependence of lunar tidal Q can be interpreted without the need for deep-seated partial melt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' De- vising a simplified lunar model, in which the mantle is described by the Sundberg-Cooper rheology, we predict the relaxation strength and characteristic timescale of elastically- accommodated grain boundary sliding in the mantle that would give rise to the desired frequency dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Along with developing this alternative model, we test the tra- ditional model with a basal partial melt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' and we show that the two models cannot be distinguished from each other by the available selenodetic measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Additional in- sight into the nature of lunar tidal dissipation can be gained either by measurements of higher-degree Love numbers and quality factors or by farside lunar seismology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Plain Language Summary As the Moon raises ocean tides on the Earth, the Earth itself gives rise to periodic deformation of the Moon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Precise measurements of lunar shape and motion can reveal those deformations and even relate them to our natural satellite’s interior structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' In this work, we discuss two interpretations of those measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' According to the first one, the lunar interior is hot and there is a thick layer of partial melt or other weak ma- terial buried more than 1000 km deep under the lunar surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' According to the second one, there is no such layer, and the measured deformation can be explained by the be- haviour of solid rocks at relatively low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' We show that the two possibili- ties cannot be distinguished from each other by the existing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' 1 Motivation Fitting of the lunar laser ranging (LLR) data to the quality-factor power scaling law Q ∼ χp rendered a small negative value of the exponential: p = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='19 (Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Further attempts by the JPL team to reprocess the data led to p = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='07 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' According to Williams and Boggs (2009), “ Q for rock is expected to have a weak dependence on tidal period, but it is ex- pected to decrease with period rather than increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' ” The most recent estimates of the tidal contribution to the lunar physical librations (Williams & Boggs, 2015) still predict a mild increase of Q with period: from Q = 38± 4 at one month to Q = 41 ± 9 at one year, yielding p = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Efroimsky (2012a, 2012b) suggested that since the frequency-dependence of k2/Q always has a kink shape, like in Figure 1, the negative slope found by the LLR measure- ments could be consistent with the peak of the kink residing between the monthly and annual frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' This interpretation entails, for a Maxwell or Andrade moon, very low values of the mean viscosity, indicating the presence of partial melt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' –2– manuscript submitted to JGR: Planets Our goal now is to devise an interpretation based on the Sundberg-Cooper model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Within that model, the kink contains not one but two peaks, and we are considering the possibility that the negative slope of our interest is due to the monthly and annual fre- quencies bracketing either this peak or the local inter-peak minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (It is unlikely that both of these frequencies are located on the negative-slope side of the peak, because the slope of that peak is too steep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=') 2 Introduction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='1 Overview of Previous Works The knowledge of the interior structure of the Moon is essential for understand- ing its thermal, geochemical, and orbital evolution as well as the coupled evolution of the Earth-Moon system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The proximity of our natural satellite to the Earth has also made it a frequent target of geophysical exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' A large amount of data was collected by lunar seismic stations, deployed by the Apollo missions, that were functional for several years between 1972 and 1977 (for a review, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', Garcia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Nunn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Other constraints are being placed by selenodetic measurements or by geochemical and petrological considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' However, the deepest interior of the Moon still remains some- what mysterious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Although different models based on the inversion of seismic travel times generally agree on the lunar mantle structure down to ∼ 1200 km, below these depths they start to diverge greatly (Garcia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' After the acquisition of the first data by the lunar seismic network, it was pointed out by Nakamura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (1973, 1974) that direct shear-waves from the farside of the Moon are not being detected by some of the near-side seismometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Moreover, deep moonquakes, a class of tidally-triggered seismic events originating at around 1000 km depth, were al- most absent on the farside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' This puzzling phenomenon was interpreted by Nakamura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (1973) as an indication for a shear-wave shadow zone caused by a highly attenuat- ing region around the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Later, Nakamura (2005) reported his further efforts to find farside moonquakes among the discovered nests of deep moonquakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Having had iden- tified about 30 nests likely to be on the farside, his updated analysis still demonstrated that either the region of the Moon’s deep interior within about 40 degrees from the an- tipode (the centre of the farside) is nearly aseismic or a portion of lunar lower mantle severely attenuates or deflects seismic waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Lunar seismic data were also reprocessed by Weber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2011) and Garcia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' However, while Weber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2011) also found evidence for deep mantle layering and a strongly attenuating zone at the mantle base, Garcia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2011) did not include such a feature in their lunar interior model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The discussion about the seismic evidence for a strongly attenuating zone is thus still ongoing (Garcia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Several authors argued for the existence of a low-velocity zone (LVZ) at the base of the mantle also on other than seismological grounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' They linked it to partial melt- ing in deep lunar interior, which might be triggered either by tidal dissipation (Harada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2014), or by the presence of incompatible, radiogenic elements buried after an an- cient mantle overturn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The idea of an overturn has been suggested by numerical mod- elling of magma ocean solidification with the emplacement of ilmenite-bearing cumulates above core-mantle boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Moreover, it is potentially supported by observations of near-surface gravity anomalies (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Evidence for a low-rigidity/low-viscosity zone has also been sought in the lunar li- bration signal obtained by LLR (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Williams & Boggs, 2015), and in selenodetic measurements (including orbiter tracking) that are sensitive to the lunar gravity field and tidal deformation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', Konopliv et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Lemoine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' One of the most surprising findings resulting from fitting the LLR data was the low value and unexpected frequency dependence of the tidal quality factor Q, as mentioned in Sec- tion 1 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The inferred frequency dependence can be explained by a low effective vis- –3– manuscript submitted to JGR: Planets cosity of the Moon (Efroimsky, 2012a, 2012b), or by the presence of a secondary peak in the dissipation spectrum (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', Williams & Boggs, 2015), possibly caused by the pu- tative basal layer (Harada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Matsumoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Earlier results from LLR indicated that the lunar core-mantle boundary (CMB) might still be out of equilibrium, which would imply long relaxation times and high lower-mantle viscosities, in contra- diction to the presence of a partial melt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' However, this hypothesis is not supported by more recent evaluations of LLR data (Viswanathan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2019), showing a CMB at hy- drostatic equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Despite relative consistency of the evidence for and the theoretical expectation of a highly dissipative basal layer, alternative models of a “melt-free” Moon have been pro- posed (Nimmo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Karato, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Matsuyama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' One argument for high values of lower-mantle viscosities comes from the observations of deep moonquakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Kawamura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2017) reevaluated an ensemble of moonquakes occurring at depths be- tween 750 and 1200 km and found a brittle-ductile transition temperature of approxi- mately 1240 – 1275 K, implying a cold lunar interior with temperatures below solidus of dry peridotite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Moreover, the employment of a realistic, microphysically substantiated models of the tidal response (Nimmo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2012) can explain the low tidal Q and the observed k2 of the Moon without requiring the existence of a weak basal layer, which is necessitated in some of the other studies by the model settings and the simplified rhe- ological assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' A feature of the selenodetic measurements that is difficult to explain without the existence of a highly dissipative basal layer is the aforementioned frequency dependence of the lunar Q, repeatedly derived from LLR measurements in the series of works by Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2001);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Williams and Boggs (2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2014), and Williams and Boggs (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Even an independent implementation of the LLR software by Pavlov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2016) predicts the same value of Q for the monthly period as for the annual period, which is still not consistent with the expected frequency dependence of tidal dissipation in melt- free silicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' In the absence of other than LLR-based data on the lunar Q, the most plausible explanation for the unexpected frequency dependence might still be an observational un- certainty, rather than an effect contained in a tidal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Nevertheless, in this work, we shall explore two possible implications of the frequency dependence under the explicit assumption that the fitted values are a result of a natural phenomenon and not of a model’s limitations or an observation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='2 A Putative Weak Basal Layer: Pros and Contras The following paragraphs review the last ten years of discussion about the pres- ence or absence of a low-viscosity basal layer, with the argumentation derived mainly from the lunar tidal response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' We begin by noting that a negative value of the exponent in Q ∼ χp is impossi- ble for the seismic quality factor of rocks obeying simple rheologies like the Maxwell or Andrade models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' This can be easily understood if we express the seismic Q via the real and imaginary parts of the complex compliance (Efroimsky, 2015, eqn 46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' By insert- ing into this expression either the Maxwell model or any other simple model lacking peaks, we obtain a monotonic function Q(χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' On the other hand, even for simple rheologies the exponential p can assume negative values if we are fitting to the Q ∼ χp law not a seis- mic but a tidal quality factor (Efroimsky, 2015, eqn 45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The tidal Q tends to zero at both very low and very high loading frequencies χ, and has a maximum in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The maximum is called into being by interplay of rheology and self-gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' This theoretical frequency dependence of the tidal quality factor motivated Efroimsky (2012a, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='2) to hypothesise that the small negative exponent p reported by Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2001) and Williams and Boggs (2009) may result from a proximity of the major –4– manuscript submitted to JGR: Planets tidal frequencies in the Moon to the frequency delimiting the peak dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Efroimsky (2012a, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='7) also noted that this interpretation would imply a low effective vis- cosity of the Moon (modeled with a homogeneous body governed by the Maxwell or the combined Maxwell-Andrade rheology), with an estimated value of η = 3 × 1015 Pa s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Such a low viscosity would support seismic models containing a layer of partial melt (Nakamura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 1974;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Weber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Nimmo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2012) aimed at answering the question whether basal partial melt is indeed required for reproducing the tidal data, and studied the effect of lunar ther- mal structure on the seismic and tidal Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' They described the rheology of the lunar in- terior with the extended Burgers model of Jackson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2010), which contains an ab- sorption band corresponding to high-temperature background, as well as an additional low-temperature peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The peak represents the elastically-accommodated grain bound- ary sliding, a phenomenon that will be considered also in our work, although within an- other rheology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Nimmo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2012) further considered a radially heterogeneous elastic structure of the mantle and accounted for the temperature-, pressure-, and grain-size- dependence of the characteristic relaxation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Using this model, they were able to match the tidal Love numbers k2 and h2 and the monthly quality factor, and they also deduced that the lower-mantle viscosity should be as high as 1023 Pa s and must be in- creasing towards the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' However, the model used did not succeed in fitting the un- expected slope of Q as a function of frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Although the authors showed that a model case with grain size of 1 mm (instead of their baseline value of 1 cm) would imply a neg- ative value of the exponential, p = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='02, they dismissed this model as a poor fit to both k2 and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Moreover, they argued that the smaller grain size would not match the tentative observation of unrelaxed CMB (Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' An original explanation of the high tidal dissipation in the Moon was provided by Karato (2013), who linked the measurements of electrical conductivity and Q to the wa- ter content in the lunar mantle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' That the water content might not be as low as had been presumed in earlier models was illustrated by geochemical studies of lunar samples, and Karato (2013) combined this observation with his own results to propose a new theory of lunar formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Using the observational constraints on Q and electrical conductiv- ity, he further concluded that the temperature at an 800 km depth of the lunar mantle is ∼ 1200–1500 K for a water content between 10−3 and 10−2 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Karato (2013) was sceptical to the idea of partial melting at the base of the lunar mantle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' He argued that the melt-bearing seismic model of Weber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2011) would require more than ∼ 1% of melt and that retaining such an amount of melt would be difficult due to efficient com- paction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Regarding the frequency-dependence of Q, Karato (2013) rejected the models of Efroimsky (2012a) and Nimmo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2012) and suggested that the negative exponent p might be caused by non-linear anelasticity of the monthly tide and linear anelasticity of the annual tide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' However, this idea was partly based on the incorrect assumption that the tide at the annual frequency is due to Sun-raised tidal deformation of the Moon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' As explained by Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2001), the annual modulation is produced by solar pertur- bations to lunar orbit only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The annual tide is thus raised by the Earth, just as the monthly tide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Still, the remark on a possible non-linearity of the lunar tide remains valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Adopting the density and rigidity profiles from a 10-layer structural model by Weber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2011), Harada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2014) explored the possible effects of a low-viscosity layer at the base of the mantle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' To keep the number of unknowns reasonable, the authors set constant viscosity values for the lithosphere, mantle, low-viscosity layer, outer core, and inner core, and applied the Maxwell rheological model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' They then calculated the tidal parameters for various thicknesses (outer radii 450–500 km) and viscosities (109–1021 Pa s) of the basal layer, at both the monthly and annual tidal frequencies, assuming that the rest of the mantle has a constant viscosity of η = 1021 Pa s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' With the highest consid- ered basal layer thickness (DLVZ = 170 km) and a viscosity of about 2×1016 Pa s, Harada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2014) were able to reproduce the quality factors given by Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2001) as well as their frequency dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Their value for the Love number at the monthly period falls into the interval k2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0242±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0004 suggested by Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2012), while –5– manuscript submitted to JGR: Planets their value of the Love number at the annual period fits into the interval k2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0255± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0016 observed by Goossens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Viscoelastic, the model of Harada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2014) rendered different values of k2 at the monthly and annual frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' This said, neither Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2012) nor Goossens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2011) considered frequency-dependence of their empirical values of k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' An updated version of the forward-modelling approach by Harada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2014) was presented in Harada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Using the improved set of tidal parameters (limits on Q at four tidal frequencies and the values of k2, k3, and h2 at the monthly frequency), the estimate of the basal layer’s outer radius was expanded from 500 km to 540−560 km (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', layer thickness DLVZ = 210 − 230 km for a core radius of 330 km) and the corre- sponding basal viscosity slightly changed to 3×1016 Pa s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' In a recent follow-up study, Tan and Harada (2021) considered full radial profile of the lunar interior (Weber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Garcia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2011) and assumed a temperature-dependent viscosity structure of the basal layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The viscosity structure either followed a convective temperature profile (viscosity almost constant with depth) or a conductive profile (linear decrease of viscos- ity with depth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Since the former model was shown to match the selenodetic data bet- ter, the authors argued that the low-viscosity layer should be locally convecting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' More- over, they concluded that the layer’s outer radius reaches 560 or 580 km (that is, to the depths of ∼ 1160 km) and that the viscosity is the same as found by Harada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The question whether a basal partial melt is required by the selenodetic data was also raised by Khan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2014), though with an answer different from Nimmo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Khan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2014) concentrated on detailed modelling of the lunar mantle petrol- ogy, and performed a Bayesian inversion of the mean density, the moment of inertia, the apparent resistivity, and the tidal data (k2 and Q) at the monthly period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' To model the tidal response of the lunar mantle within a purely elastic model, they calculated an anelas- tic correction to k2 based on a homogeneous spherical model and the power-law depen- dence of tidal dissipation, which is valid for large seismic quality factors (or weak seis- mic wave attenuation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Zharkov & Gudkova, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' For cases with the Andrade param- eter α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='1, the resulting elastic k2 clearly implied the existence of a partial melt in a basal layer with the thickness of 150−200 km (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', a depth range ∼ 1250−1400 km or the outer radii between ∼340-490 km).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Khan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2014) also found that, in order to be neutrally buoyant, the partially molten material should be enriched in FeO and TiO2 with respect to the bulk mantle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' In addition to the models with a partially molten layer, the authors tested a model with a fully solid mantle: this model still fitted all ob- servations, except for the anelastically-corrected k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Similarly, Matsumoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2015) performed a Bayesian inversion of seismic travel times and a set of available selenodetic data (mean density, moment of inertia, k2, and Q at the monthly and annual frequencies), to infer the interior structure of an eight-layered lunar model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' As in Harada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2014), the authors considered the Maxwell rheolog- ical model, in which the existence of a low-viscosity layer is required not only by the slope of Q’s frequency dependence but also by the magnitude of k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The viscosity of the solid mantle was always set to 1021 Pa s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' otherwise, Matsumoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2015) varied a wide range of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' While their inverted structure of the shallow mantle agrees with the re- sults of Weber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2011) and Garcia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2011), the lower mantle, mainly constrained by selenodetic data, slightly differs from the melt-containing model of Weber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The outer radius of the highly dissipative layer is around 570 km and the predicted vis- cosity in that region reaches 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='9×1016 Pa s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The authors noted that with the model used, k2 and the annual Q are slightly biased from the observed values, although not be- yond 1σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Matsumoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2015) also reported a trade-off between the outer core ra- dius and the LVZ thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The thickness of the LVZ corresponding to the calculated outer radius is at least 170 km and, for the core size estimate of Weber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2011), it may reach 240 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' In a paper presenting their interpretation of LLR data, Williams and Boggs (2015) compared several rheological models and endeavoured to fit the lunar k2/Q at the monthly –6– manuscript submitted to JGR: Planets and annual tidal periods, considering physical libration at five periods (1 month, 206 days, 1 year, 3 years, and 6 years).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Aware of the complex properties of the lunar interior and the possible unmodelled effects of its lateral heterogeneity, the authors proposed a model consisting of an absorption band and a narrow Debye peak: the former characterising the dissipation in the solid mantle, the latter describing the contribution of the partially molten layer suggested by Harada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' For the thickness of the partially molten layer, Williams and Boggs (2015) obtained DLVZ ≥ 205 km, placing its outer radius at ≥ 535 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The results of Williams and Boggs (2015) are relatively consistent with the pre- dictions by Harada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Matsumoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2015), and Harada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' As in the other studies containing a LVZ, they indicate that if partial melt is present, it might extend to the zone of deep moonquakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' On the one hand, the coexistence of partially molten material with seismic sources is hard to imagine: while the former requires that the lower-mantle temperatures exceed solidus, the latter should be concentrated in re- gions where the mantle rocks undergo brittle deformation, limited to lower temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' On the other hand, the movement of small amounts of melt to the zone of moonquake nests might be considered one of the mechanisms triggering seismic events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Frohlich and Nakamura (2009) proposed an explanation for the periodic occurrence of deep moonquakes, which combines dehydration embrittlement due to partial melting and crack opening by moving fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The authors pointed out the correlation between tidal loading and seis- mic events associated with magma movements in terrestrial volcanoes and remarked that a similar process may be active in the lunar interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Tentative evidence for a link be- tween deep moonquakes and magma movements might also be seen in the correlation between the locations of deep moonquake nests and lunar maria (Qin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' How- ever, a definitive answer to the question of whether a rheologically weak layer and seis- mic sources can exist at comparable depths awaits further modelling efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The specific effect of a partially-molten basal layer on the elastic Love number k2,e was discussed in the study of Raevskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2015), which combined seismic and geode- tic data with models of lunar mantle composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Depending on the model used, the rigidity of the basal layer was required to be 20–50% lower than the rigidity of the over- lying solid mantle and the outer radius of that zone was determined to reach 530–550 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' From the petrological perspective, the authors argued that partial melting of a peridotite/harzburgite mantle above the core-mantle boundary (CMB) would require temperatures in the depth of 1000 km to be in the range of 1350–1400 ◦C, unless the temperature gradients in the lower mantle become steeper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Furthermore, they concluded that the seismic velocities of Weber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2011) are inconsistent with temperature profiles approaching solidus at the CMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Although the models of Raevskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2015) assume elastic response of the Moon, the authors also mentioned that anelasticity might explain the observed Love num- ber without the need for a basal semi-molten layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Matsuyama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2016) constrained their lunar interior models by the elastic Love numbers k2 and h2 (calculated using the same anelastic correction for Q at the monthly period as in Khan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2014), the mean density of the Moon, and the moment of in- ertia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' After carrying out MCMC-type inversion, the authors concluded that although the chosen observables do not rule out the existence of a semi-molten layer, there is a strong preference for higher, solid-mantle-like values of the lower-mantle rigidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' If the semi-molten layer exists, its thickness calculated by Matsuyama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2016) is DLVZ = 194+66 −186 km, its rigidity is µLVZ = 43+26 −9 GPa, and its density may reach exceptionally high values, ρLVZ = 4676+410 −1179 kg m−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Recently, the combined geochemical, seismic, and selenogetic ensemble of Raevskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2015) was further studied by Kronrod et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2022), who extended the former work by considering explicitly a viscoelastic lunar interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Regarding the division into inte- rior layers and the adopted rheological model, the authors followed Matsumoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', they assumed the Maxwell model for the mantle and included a semi-molten basal –7– manuscript submitted to JGR: Planets layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Besides the main results of their Bayesian analysis, indicating a major difference in the chemical composition of the bulk silicate Earth and the Moon, Kronrod et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2022) presented probability distributions for the seismic wave velocities, mean density, and the thickness of the basal layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The resulting distributions are wide, constraining the basal layer’s density to 3400–3800 kg m−3 and the thickness to 100–350 km, depending on the mantle composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' As in Khan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2014), the authors conclude that the layer should be enriched in TiO2 and FeO, if it is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' In summary, the literature discussing the unexpected frequency dependence of lu- nar tidal Q as well as the properties of a hypothetical semi-molten layer atop the lunar core is rich, and the proposed values of the layer’s thickness range from 0 to 350 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Mod- els considering linear viscoelastic Maxwell rheology (both for the basal layer and for the bulk mantle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Harada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2014, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Matsumoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Tan & Harada, 2021) typically arrive at viscosities of order 1016 Pa s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' If the semi-molten layer exists, its up- per radius extends to the depths of ∼ 1150 km, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', just below the regions that are rel- atively well mapped by seismological studies and contain the nests of tidally-triggered deep moonquakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Nevertheless, the existence of a low-viscosity layer is not necessarily required by selenodetic measurements at the best accessible, monthly period (Nimmo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Matsuyama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The main advantage of melt-bearing models lies in their ability to explain the possible increase in tidal Q from the monthly to the an- nual period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='3 Lunar k2 and Q Here, we shall use the potential tidal Love number derived from the GRAIL mis- sion tracking data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Two independent analyses performed by the JPL group (Konopliv et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2013, the GL0660B solution) and the GSFC group (Lemoine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2013, the GRGM660PRIM solution) yielded two possible values of the parameter: k2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='02405 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='000176 and k2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='02427±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='00026, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The unweighted mean of the two alternative val- ues is k2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='02416 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='000222 for a reference radius of 1738 km, and k2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='02422 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='000222 for the actual mean radius of 1737.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='151 km (Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' For compar- ison, the recent analysis of the data from the Chang’e 5T1 mission gives k2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='02430± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0001 (Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' We note that the value obtained from satellite tracking data corresponds, in particular, to the real part of the complex Love number introduced later in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The GRAIL data are dominated by data arcs collected throughout a one-month time interval, and the resulting k2 is thus interpreted as indicative of the deformation at monthly frequency (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Konopliv, private communication).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The tidal quality factor Q was obtained by fitting tidal contribution to lunar phys- ical libration measured by LLR (Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2001, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Williams & Boggs, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Interpreting the measurements of physical libration presents a highly complex problem, depending on cross interactions of tides raised by the Earth and the Sun, precise mod- eling of the lunar orbit and of the instantaneous positions of the Earth-based stations and the Moon-based retroreflectors, and on an adequate incorporation of the lunar core- mantle friction (Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' In practice, the tidal time delay at a monthly pe- riod and the dissipation-related corrections to the periodic latitudinal and longitudinal variations in the Moon’s orientation are outputted and related analytically to linear com- binations of k2/Q at a number of loading frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Since many of the loading frequen- cies are close to each other, the periodic corrections enable approximate estimation of the leading dissipation terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Specifically, the strongest correction (compared to its un- certainty) is related to the annual longitudinal libration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Assuming a fixed k2 at the monthly frequency, equal to the above-mentioned unweighted average, and using a complex rhe- ological model best fitting the dissipation-related corrections to libration angles, Williams and Boggs (2015) derived the following frequency-dependent values of tidal quality fac- tor: Q = 38 ± 4 at the period of 1 month, Q = 41 ± 9 at 1 year, and lower bounds of Q ≥ 74 at 3 years and Q ≥ 58 at 6 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The tidal quality factors at other than the monthly frequency are model-dependent because the actual quantities extracted from –8– manuscript submitted to JGR: Planets the dissipation-related corrections to libration angles are the ratios (k2/Q)χ/(k2/Q)monthly, where χ denotes frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Williams and Boggs (2015) also attempted to find the frequency-dependence of k2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' however, the effect could not be detected by existing measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' We note that in con- trast to the unexpected frequency dependence of Q found with the JPL-based software (Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2001, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Williams & Boggs, 2015), an independent implementation of the fitting tool with different preset solutions for part of the geophysical phenomena (Pavlov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2016) predicted Q = 45 at both the monthly and the annual frequen- cies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' As an additional, though a relatively weak constraint on the lunar interior struc- ture, we consider the degree-3 potential tidal Love number k3 and the degree-2 defor- mational Love number h2 corresponding to radial deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The k3 number has been derived from GRAIL mission tracking data and, as with k2 above, we adopt the unweighted average of the two existing independent solutions (Lemoine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Konopliv et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2013): k3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0081±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The h2 number has been measured by LLR and by laser altimetry (Mazarico et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Pavlov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Viswanathan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Thor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2021), the most recent value, presented by Thor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2021), being h2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0387± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' We would finally mention the reason why the constraints on the lunar interior from the measurements of k3 are weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' A degree-l component of the internal tidal potential is proportional to rl, where r is the distance between the centres of mass of the tidally perturbed body and the perturber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' For this reason, with increasing degree l, the shal- lower depths contribute more and more to the Love numbers kl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The sensitivity of the higher-degree Love numbers to the deep interior is, therefore, limited as compared to de- gree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='4 Outline of This Work After an overview of the models and interpretations proposed in recent literature (with the focus on the last ten years of the discussion), we are ready to continue with the central part of this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Our plan is to provide an interpretation of the unexpected frequency dependence of tidal Q which does not require partial melting (in a way sim- ilar to Nimmo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2012) and compare it with a model containing a highly dissipative basal layer (Harada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Matsumoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Section 3 introduces and gives a justification for the rheological model employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Namely, it discusses the Sundberg- Cooper extension of the Andrade model and the dissipation related to elastically accom- modated grain-boundary sliding (GBS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The following Section 4 links the non-elastic rhe- ology to Love numbers and tidal quality factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' In Section 5, we first illustrate the ex- pected position of a secondary peak in the dissipation spectrum of a homogeneous Moon, and then attempt to find the parameters of two- or three-layered lunar models that would produce the values of the monthly tidal Q and annual k2/Q reported by Williams and Boggs (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' At the same time, we fit the empirical values of lunar k2, k3, and h2 given in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Section 6 discusses implication of both our models, and the results are briefly summarised in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' 3 General Facts on Rheologies 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='1 Constitutive Equation Rheological properties of a material are encoded in a constitutive equation inter- connecting the present-time deviatoric strain tensor uγν(t) with the values that have been assumed by the deviatoric stress σγν(t ′) over the time period t ′ ≤ t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Under lin- –9– manuscript submitted to JGR: Planets ear deformation, the equation has the form of convolution, in the time domain: 2 uγν(t) = ˆJ(t) σγν = � t −∞ � J (t − t ′) σγν(t ′) dt ′ , (1) and the form of product, in the frequency domain: 2 ¯uγν(χ) = ¯J(χ) ¯σγν(χ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2) Here ¯uγν(χ) and ¯σγν(χ) are the Fourier images of strain and stress, while the complex compliance ¯J(χ) is a Fourier image of the kernel ˙J(t−t ′) of the integral operator (1), see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', Efroimsky (2012a, 2012b) for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='2 The Maxwell and Andrade Models At low frequencies, deformation of most minerals is viscoelastic and obeys the Maxwell model: � U = 1 2 µ � S + 1 2 η S (3a) or, equivalently: � S + 1 τM S = 2 µ � U , (3b) U and S being the deviatoric strain and stress;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' η and µ denoting the viscosity and rigidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (Below, we shall address the question as to whether µ is the unrelaxed or re- laxed rigidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=') The Maxwell time is introduced as τM ≡ η µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (4) For this rheological model, the kernel of the convolution operator (1) is a time deriva- tive of the compliance function (M)J(t − t ′) = � Je + (t − t ′) 1 η � Θ(t − t ′) , (5) where Θ(t − t ′) is the Heaviside step function, while the elastic compliance Je is the inverse of the shear rigidity µ : Je ≡ 1 µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (6) In the frequency domain, equation (3) can be cast into form (2), with the complex com- pliance given by (M) ¯J (χ) = Je − i ηχ = Je � 1 − i χ τM � , (7) and the terms Je and − i/(ηχ) being the elastic and viscous parts of deformation, cor- respondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' So a Maxwell material is elastic at high frequencies, viscous at low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' More general is the combined Maxwell-Andrade rheology, often referred to simply as the Andrade rheology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' It comprises inputs from elasticity, viscosity, and anelastic pro- cesses: (A)J(t − t ′) = � Je + β (t − t ′)α + t − t ′ η � Θ(t − t ′) , (8) the corresponding complex compliance being (A) ¯J (χ) = Je + β (iχ)−α Γ (1 + α) − i ηχ (9a) = Je + β (iχ)−α Γ (1 + α) − i J (χ τM)−1 , (9b) –10– manuscript submitted to JGR: Planets where Γ is the Gamma function, while α and β denote the dimensionless and dimen- sional Andrade parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Expressions (9a - 9b) suffer an inconvenient feature, the fractional dimensions of the parameter β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' It was therefore suggested in Efroimsky (2012a, 2012b) to shape the compliance into a more suitable form (A)J(t − t ′) = � Je + Je �t − t ′ τA �α + Je t − t ′ τM � Θ(t − t ′) , (10) (A) ¯J (χ) = Je � 1 + (i χ τA)−α Γ (1 + α) − i (χ τM)−1� , (11) with the parameter τA christened as the Andrade time and linked to β through β = Je τ −α A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (12) Compliance (11) is identical to (9a) and (9b), but is spared of the parameter β of frac- tional dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='3 Why the Maxwell and Andrade Models Require Refinement In the literature, it is common to postulate that both the rigidity and compliance assume their unrelaxed values denoted with µU and JU .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' This convention is reasonable for sufficiently high frequencies: χ is high =⇒ µ = µU and Je = JU .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (13) The convention, however, becomes unjustified for low frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' In that situation, the material has, at each loading cycle, enough time to relax, wherefore both the rigidity mod- ulus and its inverse assume values different from the unrelaxed ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' In the zero-frequency limit, they must acquire the relaxed values: χ → 0 =⇒ µ → µR and Je → JR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (14) This fact must be taken care of, both within the Maxwell and Andrade models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='4 Generalisation of the Maxwell and Andrade Models, according to Sundberg and Cooper (2010) The simplest expression for the time relaxation of the elastic part of the compli- ance is Je(t) = JU + (JR − JU) � 1 − e−t/τ � (15a) = JU � 1 + ∆ � 1 − e− t/τ�� , (15b) where the so-called relaxation strength is introduced as ∆ ≡ JR JU − 1 , (16) while τ is the characteristic relaxation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' When relaxation of Je is due to elastically accommodated grain-boundary sliding, this time can be calculated as τ = τgbs = ηgb d µU δ , (17) where ηgb is the grain-boundary viscosity, d is the grain size, while δ is the structural width of the grain boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' –11– manuscript submitted to JGR: Planets In the frequency domain, this compliance is written as ¯Je(χ) = JU � 1 + ∆ 1 + χ2 τ 2 + i χ τ ∆ 1 + χ2 τ 2 � , (18) its imaginary part demonstrating a Debye peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Our goal is to trace how this Debye peak translates into the frequency-dependence of the inverse tidal quality factor 1/Q and of k2/Q of a near-spherical celestial body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Substitution of formula (18) into the overall expression (11) for the Andrade com- plex compliance will produce the Sundberg and Cooper (2010) rheology: ¯J (χ) = JU � 1 + ∆ 1 + χ2τ 2 − i χ τ ∆ 1 + χ2τ 2 + (iχτA)−α Γ(1 + α) − i(χτM)−1 � (19a) = JU � 1 + ∆ 1 + χ2 τ 2 + Γ(1 + α) ζ−α (χτM)−α cos �απ 2 � � (19b) − i JU � χ τ ∆ 1 + χ2 τ 2 + Γ(1 + α) ζ−α (χτM)−α sin �απ 2 � + (χτM)−1 � , where we introduced the dimensionless Andrade time ζ = τA τM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (20) Be mindful that in expression (10) it is only the first term, Je, that is changed to func- tion (15b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Accordingly, in equation (11), it is only the first term, Je, that is substituted with function (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' In the other terms, both the Maxwell and Andrade times are still introduced through the unrelaxed value Je = JU : τM ≡ η JU , τA ≡ �JU β �1/α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (21) Had we combined the elastic relaxation rule (18) with the Maxwell model (7) in- stead of Andrade, we would have arrived at the Burgers model — which would be equa- tion (19) with the Andrade terms omitted, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' with τA −→ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Simply speaking, in the absence of transient processes, Andrade becomes Maxwell, while Sundberg-Cooper be- comes Burgers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The presently standard term “Sundberg-Cooper rheology” was coined by Renaud and Henning (2018) who studied tidal heating in mantles obeying this rheological law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Soon thereafter, this law was later employed for Mars (Bagheri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2019) and for Pluto and Charon (Bagheri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Along with the dimensionless Andrade time ζ, below we shall employ the relative relaxation time trel = τ τM (22) relating the relaxation timescale for the compliance Je to the Maxwell time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 Further Options The characteristic relaxation time τ can be replaced with a distribution D(τ) of times spanning an interval from a lower bound τL to an upper bound τH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' So the relax- ation of the elastic part of the compliance will be not Je(t) = JU � 1 + ∆ � 1 − e− t/τ�� (23) –12– manuscript submitted to JGR: Planets but Je(t) = JU � 1 + ∆ � τH τL D(τ) � 1 − exp � − t τ �� dτ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (24) If the relaxation is due to elastically-accommodated GBS, this distribution would be a consequence of variable grain-boundary viscosity, grain sizes and shapes, and non-uniform orientation of grain boundaries with respect to the applied stress (see also Lee & Mor- ris, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Insertion of expression (24) in the Maxwell model (5) or in the Andrade model (10) produces the extended Burgers model or the extended Sundberg-Cooper model, correspond- ingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' For details, see Bagheri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2022) and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' 4 Complex Love Numbers and Quality Functions The perturbing potential wherewith the Earth is acting on the Moon can be de- composed in series over Fourier modes ωlmpq parameterised with four integers lmpq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' If the tidal response of the Moon is linear, both the produced deformation and the result- ing additional tidal potential of the Moon are expandable over the same Fourier modes, as proved in Efroimsky and Makarov (2014, Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The proof is based on the fact that a linear integral operator (convolution) in the time domain corresponds to a prod- uct of Fourier images in the frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' While the Fourier modes can be of either sign, the physical forcing frequencies in the body are χlmpq = |ωlmpq | .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (25) An extended discussion of this fact can be found in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='3 of Efroimsky and Makarov (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Wherever this causes no confusion, we omit the subscript to simplify the notation: ω ≡ ωlmpq , χ ≡ χlmpq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (26) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='1 The Complex Love Number Writing the degree-l complex Love number as ¯kl(ω) = ℜ �¯kl(ω) � + i ℑ �¯kl(ω) � = |¯kl(ω)| e −iϵl(ω) , (27) we conventionally denote the phase as − ϵl , with a “minus” sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' This convention im- parts ϵl with the meaning of phase lag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' We also introduce the so-called dynamical Love number kl(ω) = |¯kl(ω)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (28) A key role in the tidal theory is played by the quality functions Kl(ω) ≡ − ℑ � ¯kl(ω) � = ¯kl(ω) sin ϵl(ω) (29a) entering the series expansions for tidal forces, torques, dissipation rate (Efroimsky & Makarov, 2014), and orbital evolution (Bou´e & Efroimsky, 2019) Since Sign ϵl(ω) = Sign ω (Efroimsky & Makarov, 2013), they can be written as Kl(ω) ≡ − ℑ � ¯kl(ω) � = kl(ω) Ql(ω) Sign ω , (29b) where the tidal quality factor is introduced via Q−1 l (ω) = | sin ϵl(ω)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (30) –13– manuscript submitted to JGR: Planets The dependency sin ϵl(ω) being odd, the function Ql(ω) is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Also, even is the function kl(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Therefore, for any sign of ω and ϵl, it is always possible to treat both Ql(ω) and kl(ω) as functions of the forcing frequency χ ≡ |ω| : Ql(ω) = Ql(χ) , kl(ω) = kl(χ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (31) Often attributed to Biot (1954), though known yet to Sir George Darwin (1879), the so-called correspondence principle, or the elastic-viscoelastic analogy, is a valuable key to numerous problems of viscoelasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' It enables one to derive solutions to these problems from the known solutions to analogous static problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' In application to bod- ily tides,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' this principle says that the complex Love number of a uniform spherical vis- coelastic body,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' ¯kl(χ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' is linked to the complex compliance ¯J(χ) by the same algebraic expression through which the static Love number kl of that body is linked to the relaxed compliance JR : ¯kl(χ) = 3 2 (l − 1) 1 1 + Bl/ ¯J(χ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (32) where Bl ≡ (2 l 2 + 4 l + 3) l g ρ R = 3 (2 l 2 + 4 l + 3) 4 l π G ρ2 R2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (33) ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' and g being the density,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' radius,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' and surface gravity of the body,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' and G being New- ton’s gravitational constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' As an aside, we would mention that while −ℑ [kl(ω)] emerges in the tidal torque, the real part of the complex Love number, ℜ [kl(ω)] = kl(ω) cos ϵl(ω), shows up in the expansion for the tidal potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Not considered further in the present study, the gen- eral expression for this product and its version for the Maxwell and other rheologies can be found in Efroimsky (2015, Appendix A6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='2 kl(χ)/Ql(χ) and 1/Ql(χ) for an Arbitrary Rheology Expression (32) entails: Kl(χ) = kl(χ) sin ϵl(χ) = − 3 2(l − 1) Bl ℑ � ¯J(χ) � � ℜ � ¯J(χ) � + Bl �2 + � ℑ � ¯J(χ) ��2 , (34) the coefficients Bl rendered by equation (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' We see that for a homogeneous incom- pressible sphere, the information needed to calculate the quality function comprises the radius, the density, and the rheological law ¯J(χ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The inverse tidal quality factor of degree l is given by (Efroimsky, 2015) Ql(χ)−1 ≡ | sin ϵl(χ)| , (35) sin ϵl(χ) = − Bl ℑ � ¯J(χ) � �� ℜ � ¯J(χ) � �2 + � ℑ � ¯J(χ) � �2 �� ℜ � ¯J(χ) � + Bl �2 + � ℑ � ¯J(χ) ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (36) All new is well-forgotten old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' As we were writing this paper, it became known to us that for the Maxwell rheology the frequency-dependence of sin ϵ2 was studied yet by Gerstenkorn (1967, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' 2) in a work that went virtually unnoticed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Because of different notation and Gerstenkorn’s terse style, it is not apparent if his values for the peak’s magnitude and location are the same as ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' However, the overall shape of the frequency-dependence of sin ϵ2 obtained by Gerstenkorn (1967) seems right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' –14– manuscript submitted to JGR: Planets 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='3 Notational Point: Q and Q2 In publications where both seismic and tidal dissipation are considered, it is nec- essary to distinguish between the seismic and tidal quality factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' In that situation, the letter Q without a subscript is preserved for the seismic factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' In the literature on tides, it is common to employ Q as a shorter notation for the quadrupole tidal factor Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' We shall follow the latter convention: Q ≡ Q2 , (37) and shall use the two notations intermittently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='4 The frequency-dependencies of kl/Ql and 1/Ql for the Maxwell and Andrade models For a homogeneous sphere composed of a Maxwell or Andrade material, the qual- ity function Kl(ω) has a kink form, as in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The function sin ϵl(ω) is shaped sim- ilarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='10 tidal mode ω kl (ω) sin εl (ω) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' A typical shape of the quality function Kl(ω) = kl(ω) sin ϵl(ω) , where ω is a shortened notation for the tidal Fourier mode ωlmpq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (From Noyelles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Insertion of expression (7) into equation (34) shows that for a spherical Maxwell body the extrema of the kink Kl(ω) are located at ωpeakl = ± τ −1 M 1 + Bl µ (38) the corresponding extrema assuming the values K(peak) l = ± 3 4(l − 1) Bl µ 1 + Bl µ , (39) wherefrom |Kl| < 3 4(l − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Inside the interval between peaks, the quality functions are near-linear in ω : |ω| < |ωpeakl | =⇒ Kl(ω) ≃ 3 2(l − 1) Bl µ 1 + Bl µ ω |ωpeakl | .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (40) –15– manuscript submitted to JGR: Planets Outside the inter-peak interval, they fall off as about ω−1 : |ω| > |ωpeakl | =⇒ Kl(ω) ≃ 3 2(l − 1) Bl µ 1 + Bl µ |ωpeakl | ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (41) While the peak magnitudes (39) are ignorant of the viscosity η, the spread between the peaks scales as the inverse η, as evident from expression (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The lower the mean viscosity, the higher the peak frequency |ωpeakl|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' It can be demonstrated using equation (36) that for a homogeneous Maxwell body the extrema of sin ϵl(ω) are located at ωpeak of sin ϵl = ± τ −1 M √1 + Blµ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (42) For the Moon, this peak is located within a decade from its counterpart for Kl given by formula (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' In many practical situations, only the quadrupole (l = 2) terms matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The cor- responding peaks are located at ωpeak2 = ± τ −1 M 1 + B2 µ ≈ ± 1 B2 η = ± 8 π G ρ2 R2 57 η .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (43) The approximation in this expression relies on the inequality Bl µ ≫ 1, fulfilment whereof depends on the size of the body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' For a Maxwell Moon with µ = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='4×1010 Pa and G(ρR)2 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='24 × 109 Pa, we have B2 µ ≈ 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5, so the approximation works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' While for the Maxwell and Andrade models each of the functions Kl(ω) and sin ϵl(ω) possesses only one peak for a positive argument, the situation changes for bodies of a more complex rheology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' For example, the existence of an additional peak is ensured by the insertion of the Sundberg-Cooper compliance (19) into expressions (34) or (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' 5 Application to the Moon 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='1 The “Wrong” Slope Interpreted with the Maxwell Model As we explained in Section 1, fitting of the LLR-obtained quadrupole tidal qual- ity factor Q = Q2 to the power law Q ∼ χp resulted in small negative value of the exponential p (Williams & Boggs, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' An earlier attempt to explain this phenomenon implied an identification of this slightly negative slope with the incline located to the left of the maximum of the quality function (k2/Q2)(χ), see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Within this interpre- tation, χpeak ≡ |ωpeak| should be residing somewhere between the monthly and annual frequencies explored in Williams and Boggs (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' As was explained in Efroimsky (2012a) , this sets the mean viscosity of the Moon as low as η ≈ 3 × 1015 Pa s , (44) The extrema of (1/Q2)(χ) are close to those of (k2/Q2)(χ), as can be observed from equations (19) and (45) Efroimsky (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Therefore, had we used instead of the max- imum of k2/Q2 given by (43) the maximum of 1/Q2 given by (42), the ensuing value would have been only an order higher: η ≈ 4 × 1016 Pa s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (45) Such values imply a high concentration of the partial melt in the mantle – quite in ac- cordance with the seismological models by Nakamura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (1974) and Weber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' However, employment of a rheology more realistic than Maxwell may entail not so low a viscosity — in which case the existence of a semi-molten layer may be questioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' –16– manuscript submitted to JGR: Planets 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='2 Frequency Dependence of Tidal Dissipation in the Sundberg-Cooper Model The Debye peak emerging in the imaginary part of ¯Je (equation (18)) will, obvi- ously, show itself also in the shape of the imaginary part of the overall ¯J , the bottom line of equation (19b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Consequently, substitution of expression (19) in equations (34) and (36) will entail the emergence of a Debye warp on the kinks for kl/Ql and 1/Ql .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Where will the additional peak be located for realistic values of the relaxation timescale τ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' What values for the mean viscosity will it entail?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' In the end of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='4, we introduced the relative relaxation time as trel ≡ τ/τM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Figure 2 illustrates specifically the effect of trel in the Sundberg-Cooper model on the position of the additional Debye peak for a homogeneous lunar interior with an arbitrar- ily chosen high mean viscosity ηMoon = 1022 Pa s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The emergence of another local max- imum in the k2/Q2 and 1/Q2 functions may naturally explain the decrease in dissipa- tion (or increase in the quality factor Q) with frequency, even within a homogeneous and highly viscous model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The negative imaginary part of the Love number (left) and the inverse quality fac- tor (right) for different ratios between the timescale τ and the Maxwell time τM (indicated by the shades of blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The yellow and red vertical lines show the Q2 values given by Williams and Boggs (2015) for the annual and the monthly component, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' In this case, we consider a homogeneous lunar interior model governed by the Sundberg-Cooper rheology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The mantle viscosity was set to 1022 Pa s and the mantle rigidity to 80 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='3 Constructing a Multi-layered Model Section 4 introduced the complex Love number ¯kl(χ) for an arbitrary linear anelas- tic or viscoelastic rheology assuming a homogeneous incompressible sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' While such a model can reasonably approximate the response of the Moon with a homogeneous man- tle and a small core (see also Figure 4), its application to a body with a highly dissipa- tive basal layer would not be accurate (Bolmont et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Planetary interior with a highly dissipative layer can still be approximated by a homogeneous model with an ad- ditional absorption peak or band in the underlying rheological law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' However, we would need to know the mapping between the parameters of the dissipative layer and the pa- rameters of the additional peak (Gevorgyan, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Therefore, in the following sections, we will complement the homogeneous model with three models consisting of two or three layers and we will calculate the correspond- ing complex Love numbers numerically, using a matrix method based on the normal mode –17– 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 Q 301 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0001 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='1 1e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='01 1e-06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='001 1e-07 1 month yr 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 10 10 9 8 7 5 6 8 6 5 4 logx [rad/s] logx[rad/s]manuscript submitted to JGR: Planets theory (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', Takeuchi & Saito, 1972;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Wu & Peltier, 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Sabadini & Vermeersen, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' For the sake of simplicity, we consider all layers in the numerical model (linearly) vis- coelastic and we model the response of liquid layers by the Maxwell model with Je in equation (7) approaching 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' This method has also been tested against another imple- mentation of the same model, in which the liquid layers were inputted through differ- ent boundary conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' the results obtained within the two approaches are virtually the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Using the outputted complex Love numbers for various rheological parame- ters, we then proceed by fitting the empirical values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' If not stated differently for illus- trative purposes, the three alternative models will always comprise a liquid core with a low viscosity (ηc = 1 Pa s), a constant density (ρc = 5000 kg m−3), and an outer ra- dius identical to the mean value reported by Weber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2011), Rc = 330 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Although the existence of an inner core is possible and even indicated by the stacked seismograms presented by Weber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2011), its response to tidal loading would be de- coupled from the rest of the mantle, and it would contribute to the resulting tidal de- formation only negligibly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Therefore, we do not include the inner core in our modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='4 makes use of a two-layered model consisting of the liquid core and a homogeneous mantle, the response of which is described by the Andrade rheology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' For the mantle density, we prescribe a constant value of ρm = 3300 kg m−3, and Andrade parameter ζ is set to 1, implying comparable timescales for viscous and anelastic relax- ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Other values of ζ were also tested and their effect on the results is discussed in the aforementioned Subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The viscosity ηm, rigidity µm, and Andrade parameter α of the mantle are treated as free parameters and fitted to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The second model, considered in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5, comprises a liquid core and a Sundberg- Cooper homogeneous mantle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The mantle density is always set to the average value ρm = 3300 kg m−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Rheological parameters ηm, µm, τ, and ∆ are fitted, while the Andrade em- pirical parameters α and ζ are held constant during each run of the inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' We have also tested the effect of varying α in the range [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='4] and of magnifying or reducing ζ by one order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The model with a basal dissipative layer, which is discussed in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='6, con- tains a core and a two-layered mantle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Each layer of the mantle is assumed to be homo- geneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The basal layer is decribed by the Maxwell model with fitted parameters µLVZ and ηLVZ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' additionally, we fit its outer radius RLVZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' For the overlying bulk mantle, we consider the Andrade model with free (fitted) parameters ηm, µm and with α, ζ kept con- stant during each run of the inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Both mantle layers have a prescribed density of ρLVZ = ρm = 3300 kg m−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The reason for using the simple Maxwell model instead of the Andrade model in the basal layer is the following: in order to fit the measured tidal quality factor Q at the monthly and the annual frequency, the peak dissipation from the basal layer should be located either between these frequencies, or above the monthly fre- quency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' At the same time, in the vicinity of the peak dissipation, the Andrade and Maxwell rheologies are almost indistinguishable from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (Comparing the last two terms on the final line of equation (19), we observe that the viscous term exceeds the Andrade term when τMχ ≪ (τA/τM)α/(1−α) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' In realistic situations, τMχpeak satisfies this con- dition safely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' So, near the peak the Andrade term is virtually irrelevant, and the regime is almost Maxwell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=') Hence, we chose the simpler of the two rheological models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' This de- cision will also facilitate the comparison of our results for the basal layer’s characteris- tics with the predictions by Harada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2014, 2016), and Matsumoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2015), who likewise modeled the basal layer with the Maxwell model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' In contrast to our study, they applied the same model to the mantle as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' In this work, we are not predicting the mineralogy of the mantle — and the com- position of the basal layer, if present, is only briefly discussed in Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Our use of a homogeneous mantle layer (or two homogeneous mantle layers) reflects our lack of information on the exact chemical and mineralogical composition, the grain size, the ther- mal structure, and the presence of water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Instead, we characterise the mantle with a sin- –18– manuscript submitted to JGR: Planets gle, “effective”, rigidity and viscosity, which can be later mapped to a detailed interior structure (see also Dumoulin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Bolmont et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2020, who discussed the effect of approximating a radially stratified mantle with a homogeneous one for Venus and ter- restrial exoplanets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Furthermore, we neglect any lateral heterogeneities in the lunar in- terior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' We also assume that the lunar mantle is incompressible and can be reasonably described by a linear viscoelastic model — which is valid at low stresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Given the mag- nitude of tidal stresses in the Moon, this assumption might have to be lifted in future works, though (Karato, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Since the radial structure of our models is deliberately simplified, we do not attempt to fit either the mean density or the moment of inertia given for the Moon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (The mean density of our lunar toy-models is less than 1% lower than the actual value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=') The inver- sions presented below are only performed for the tidal parameters, namely k2 and tidal Q at the monthly frequency, k2/Q at the annual frequency, and k3, h2 at the monthly frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' A list of the model parameters in the reference cases discussed in the follow- ing sections is presented in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The empirical values considered are then given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Parameter Type Value Unit Common parameters Core size Rc const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' 330 km Core viscosity ηc const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' 1 Pa s Core density ρc const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' 5, 000 kg m−3 Mantle viscosity ηm fitted 1015 − 1030 Pa s Mantle rigidity µm fitted 109 − 1012 Pa Mantle density ρm const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' 3, 300 kg m−3 Andrade parameter ζ const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' 1 — Two-layered model I (Andrade mantle) Andrade parameter α fitted 0 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 — Two-layered model II (Sundberg-Cooper mantle) Andrade parameter α const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='2 — Relaxation strength ∆ fitted 10−5 − 100 — Relative relaxation time trel fitted 10−7 − 100 — Three-layered model (Andrade mantle) Andrade parameter α const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='2 — Thickness of the basal layer DLVZ fitted 0 − 370 km Viscosity of the basal layer ηLVZ fitted 100 − 1030 Pa s Rigidity of the basal layer µLVZ fitted 0 − µm Pa Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Parameters of the three models considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='4 Applicability of the Andrade Model Before discussing the two interior models able to fit the anomalous frequency de- pendence of lunar tidal dissipation, we first attempt to use the full set of tidal param- eters given in Table 2 to constrain a simpler model, which only contains a liquid core and a viscoelastic mantle governed by the Andrade rheology (equation (11)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Such a model, accounting neither for a basal dissipative layer nor for elastically-accommodated GBS, might still be able to fit the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Thanks to the large uncertainty on the lunar qual- –19– manuscript submitted to JGR: Planets Parameter Value Reference k2, monthly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='02422 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='00022 Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2014) Q, monthlya 38 ± 4 Williams and Boggs (2015) k2/Q, annuala (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='4) × 10−4 Williams and Boggs (2015) k3, monthlyb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0081 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0018 Konopliv et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Lemoine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2013) h2, monthly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0387 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0025 Thor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2021) a The standard deviations from this table are only used in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' In the rest of the paper, we arbitrarily set the uncertainties to 1% of the mean value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' b Listed is the unweighted mean of the values given in references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Observational constraints used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' ity factor (more than 10% at the monthly frequency and 20% at the annual frequency, Williams & Boggs, 2015), we may not need to introduce any additional complexities to interpret the tidal response of the Moon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The error bars of the tidal quality factors are so wide that they allow, at least in principle, for a situation where Q2, annual is smaller than Q2, monthly .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' To find the parameters of this preliminary model, we performed a Bayesian inver- sion using the MCMC approach and assuming Gaussian distributions of observational uncertainties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', Mosegaard & Tarantola, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' In particular, we employed the em- cee library for Python (Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2013), which is based on the sampling methods proposed by Goodman and Weare (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The algorithm was instructed to look for the mantle viscosity ηm, the mantle rigidity µm, and the Andrade parameter α fit- ting the empirical values of k2,monthly, k3,monthly, h2,monthly, Q2,monthly, and (k2/Q2)annual , while the other Andrade parameter was set to ζ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' We generated ∼ 30, 000 random samples until the model converged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Specifically, the convergence was tested against the autocorrelation time of each variable in the ensemble, the total length of all chains be- ing required to exceed 100 times the longest autocorrelation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Moreover, in order to filter out the influence of initial conditions, we neglected the first ∼ 3, 000 samples (our burn-in period was, therefore, 10 times the autocorrelation time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The posterior probabilities of the fitted parameters are depicted in Figure 3, us- ing the Python library corner (Foreman-Mackey, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' In line with a similar model by Nimmo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2012), we find a relatively high lunar mantle viscosity of log η[Pa s] = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='99+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='89 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='35 and rigidity of log µ[Pa] = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='92±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='06, the Andrade parameter α being as low as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='06+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Treating the Andrade parameter ζ as a free parameter in the Bayesian inversion has a negligible effect on the predicted values of α and µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' However, it essentially de- termines the fitted mantle viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' If the transient deformation prevails over the vis- cous creep (ζ ≪ 1), the response of the lunar mantle to tidal loading is almost elastic (with viscosity up to η ≈ 1027 Pa s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' On the other hand, if the dissipation is preferen- tially due to viscous creep (ζ ≫ 1), the mantle viscosity allowed by the observational data has to be much lower, η ≈ 1021 Pa s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' This latter case is equivalent to the assump- tion that the mantle is governed by the Maxwell rheology, followed by Harada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2014, 2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Matsumoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Tan and Harada (2021), and Kronrod et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' If we compare the resulting Andrade parameter α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='06+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='02 with the typical values reported in the literature (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='1 < α < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', the overview by Castillo- Rogez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Efroimsky, 2012a, 2012b), we may notice that it is unusually small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' This discrepancy between our prediction and the laboratory data already indicates that although it is, in principle, possible to fit the lunar tidal response with a simple model assuming Andrade rheology in the mantle, the required parameters of this model might –20– manuscript submitted to JGR: Planets not be realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' A similar point has been made by Khan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2014) and used as an ar- gument in favour of their interior model containing basal partial melt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Following the same line of argumentation, we will now focus our study on the Sundberg-Cooper model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Posterior probabilities of the effective mantle rigidity µm, the mantle viscosity ηm, and the Andrade parameter α satisfying the full set of observational constraints (k2, k3, h2, and Q at the monthly period;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' k2/Q at the annual period).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' A model with a liquid core and a viscoelastic mantle governed by the Andrade rheology, assuming ζ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 Lunar Mantle Governed by the Sundberg-Cooper Model In the present Subsection, as well as in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='6, we will specifically search for lunar interior models that exhibit a second dissipation peak in the spectra of k2/Q2 and Q−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Since the current error bars of the empirical Qs allow for both a decrease and increase of dissipation with frequency, and since our study focuses on the latter case, we consider a hypothetical situation in which the uncertainty in Q2 is comparable with the present-day uncertainty in k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The standard deviations of Q2 at the monthly frequency and k2/Q2 at the annual frequency are thus arbitrarily set to 1% of the mean value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' As in the previous inversion with Andrade mantle, we again employ the MCMC approach and seek the parameters of the Sundberg-Cooper model (ηm, µm, ∆, and trel) fitting the empirical tidal parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Values of α and ζ are kept constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' For illustration pur- poses, we consider both 1) a two-layered interior structure consisting of a liquid core and a viscoelastic (Sudberg-Cooper) mantle and 2) a homogeneous lunar interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' As we shall see, the effect of the small lunar core (Rc = 330 km) on the results is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' –21– log nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='24 810 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='06 98 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='92 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='86 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='9 1og μm log nm αmanuscript submitted to JGR: Planets In contrast with the previous inversion, and mainly due to the greater dimension of the explored parameter space, the model only succeeded to converge after generat- ing ∼ 700, 000 random samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The posterior distributions of the tidal quality factors typically presented two peaks: a higher one with Q2,monthly > Q2,annual and a lower one with Q2,monthly < Q2,annual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Here, we only discuss the model parameters correspond- ing to the latter case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Figure 4 illustrates the results of the inversion with Andrade parameters specifi- cally set to α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='2 and ζ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Similarly as before, to filter-out the influence of ini- tial conditions, we neglected the first 70, 000 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Then, 16% of the remaining, anal- ysed samples fulfilled the condition of quality factor decreasing with frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The mean value of the predicted mantle viscosity lies close to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5×1022 Pa s and the predicted un- relaxed rigidity is around 60 − 120 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' In particular, for the nominal case with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='2 and ζ = 1 and for the arbitrarily chosen small standard deviation of empirical Q and k2/Q, the decadic logarithms of the predicted mantle viscosity and rigidity are log ηm[Pa s] = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='55+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='54 and log µm[Pa] = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='84+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Increasing α by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='1 or ζ by the factor of 10 re- sults in decreasing the mantle viscosity approximately by an order of magnitude (and the same trend pertains to the other direction, when decreasing α or ζ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' On the other hand, the mantle rigidity, being dictated by the magnitude of k2, seems relatively robust and its inverted value does not depend on α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The parameters of the Debye peak are, in this story, the key to fitting the unex- pected slope of the frequency-dependent tidal dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Independently of the consid- ered Andrade parameters, the relaxation timescale τ lies between 104 and 106 s (log τ[s] = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='89+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='62 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='72), while the relaxation strength falls into the interval between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='03 and 1 (log ∆ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='17+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='84 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The exact values depend on the predicted viscosity and rigidity, which de- fine the position of the first peak, corresponding to the attenuation in the overlying man- tle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Such short relaxation timescales would indicate that the elastically accommodated GBS is much faster than diffusion creep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' For comparison, Sundberg and Cooper (2010) mention a GBS relaxation timescale of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='1 s as a reasonable value in their experiments, using a material with τM ∼ 10 – 100 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Our τM in this specific case is in the order of 1010− 1013 s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' hence, the ratio of the two time scales for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='2 and ζ = 1 reaches trel = 10−7− 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' A more detailed discussion of this result will be provided in Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='6 Comparison of a Sundberg-Cooper Moon with an Andrade Moon Having a Weak Basal Layer As was recently shown by Gevorgyan (2021), the tidal response of a homogeneous Sundberg-Cooper planet mimics the response of a body consisting of two Andrade lay- ers with different relaxation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' This kind of aliasing may, in principle, be demonstrated by the Moon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Figure 5 depicts the imaginary part of the tidal Love number (equal to k2/Q2) and the inverse quality factor 1/Q2 as functions of frequency, for a homogeneous Sundberg-Cooper moon and for a differentiated lunar interior with a rheologically weak layer at the base of the mantle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' In the second case, the basal layer is described by the Maxwell model and the overlying mantle by the Andrade model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Both cases follow the same frequency dependence, implying that the existence of a weak basal layer cannot be confirmed unequivocally by the tidal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' In a layered model containing a core, a Sundberg-Cooper mantle, and a Maxwell basal semi-molten layer, the tidal response would be characterised by three peaks (Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' For comparison with other models presented in the literature, we also sought for the parameters of a three-layered lunar model comprising a liquid core, an Andrade man- tle, and a Maxwell basal low-viscosity layer that would fit the empirical constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' As in the previous Subsection, in order to reduce the number of unknowns, the parameters α and ζ of the Andrade model were kept constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' We also prescribed the same constant core radius of 330 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The remaining quantities were treated as free parameters: we thus varied the rigidity and viscosity of the mantle and of the basal layer, and the outer ra- –22– manuscript submitted to JGR: Planets Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Best-fitting models and the corresponding model parameters for a melt-free Moon with a liquid core and a Sundberg-Cooper mantle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Upper row: the real (left) and negative imagi- nary (right) parts of the complex Love number ¯k2, as functions of frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The red and yellow lines indicate the values provided by Williams and Boggs (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Lower row: model samples plot- ted in the parameter space, with the mantle rigidity µm depicted against viscosity ηm (left), the relaxation strength ∆ depicted against the characteristic time τ of the elastically-accommodated GBS (centre), and the Maxwell time τM versus the characteristic time τ (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The Andrade parameters are kept constant at α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='2 and ζ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Gray dots in the lower left panel show the results obtained with a homogeneous model consisting only of a Sundberg-Cooper mantle, while black dots represent the default two-layered model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The negative imaginary part of the Love number (left) and inverse quality factor (right) for three model cases: a homogeneous Andrade model (dashed red line), a homogeneous Sundberg-Cooper model (blue line), and a three-layered model (solid red line) comprising a core, an Andrade mantle and a Maxwell semi-molten layer at the base of the mantle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' –23– 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='028 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='026 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='022 S- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='020 1 month 1 month l yr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 5 7 5 7 4 8 8 6 4 6 logx [rad/s] logx [rad/s] 6.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='8 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='9 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 10 11 12 13 log △ 1og TM0 0 Q 3 Q 2 301 3 3 Andrade, homogeneous Andrade, layered 4 Sundberg-Cooper 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 logx [rad/s] logx [rad/s]manuscript submitted to JGR: Planets Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The negative imaginary part of the Love number (left) and inverse quality fac- tor (right) of a three-layered lunar model comprising a core, a Sundberg-Cooper mantle, and a Maxwell semi-molten basal layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Different shades of blue correspond to different ratios between the timescale τ and the Maxwell time τM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' For illustrative purposes, the semi-molten basal layer is made unrealistically thick (500 km).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' dius of the basal layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Due to the higher dimensionality of the parameter space, the in- verse problem took longer to converge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' therefore, we generated 10, 000, 000 random sam- ples satisfying all constraints from Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Since the longest autocorrelation time in this case was 500, 000 steps, we discarded the first 5, 000, 000 samples and then applied the condition Q2,monthly < Q2,annual , being left with 11% of the generated samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' As illustrated in Figure 7, and in line with the discussion above, the frequency de- pendencies of ℜ[¯k2] and −ℑ[¯k2] in the model with a low-viscosity basal layer closely re- semble those of the previous one, in which we considered the Sundberg-Cooper model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Similarly to the earlier predictions of the basal layer’s viscosity and thickness (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', Harada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2014, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Matsumoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2015), we find that the observed frequency depen- dence of lunar Q−1 2 can be explained by the viscosity ηLVZ in the range from ∼ 1015 to ∼ 3×1016 Pa s and the thickness DLVZ in the range from 70 km to the maximum value considered in our model (370 km).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The parameter dependencies of all model samples are plotted on Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' For the nominal case with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='2 and ζ = 1, and considering the condition on Q mentioned in the above paragraph, we obtain the following rigidity and viscosity of the overlying mantle and of the LVZ: log ηm[Pa s] = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='79+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='19 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='06, µm[Pa] = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='89±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='03, ηLVZ[Pa s] = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='20+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='53 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='21, µLVZ[Pa] = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='23+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='37 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The corresponding outer radius of the LVZ is RLVZ[km] = 599.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='39+65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='83 −84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Similarly to the “melt-free” case with the Sundberg-Cooper model, increasing α to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='3 results in an order-of-magnitude decrease in the fitted mantle viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Decreas- ing α to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='1 leads to a mantle viscosity two orders of magnitude greater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' On the other hand, the predicted properties of the semi-molten layer remain almost the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' 6 Discussion In the previous section, we have compared the frequency dependence of lunar Q within the widely accepted lunar interior model containing a highly dissipative layer at the base of the mantle (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', Nakamura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 1973;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Harada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2014) and within an alternative model taking into account the time relaxation of the elas- tic compliance Je.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' On the following lines, we discuss the implications of each of the con- sidered models for the lunar interior properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Keep in mind that the inversions per- –24– 0 0 Q Q 2 301 301 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0001 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='1 1e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='01 1e-06 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='001 1e-07 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 50 logx [rad/s] logx [rad/s]manuscript submitted to JGR: Planets Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Overview of best-fitting models for the case with a basal low-viscosity zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The red and yellow lines indicate the values provided by Williams and Boggs (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' As in the previ- ous inversion, the Andrade parameters are kept constant at α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='2 and ζ = 1, and the core size is fixed to 330 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Model samples corresponding to Figure 7, plotted in the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The intensity indicates the sample count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Upper row: the rigidity vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' viscosity of the LVZ (left), the rigidity vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' viscosity of the mantle (centre), and the outer radius vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' viscosity of the LVZ (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Lower row: the rigidity of the LVZ vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' rigidity of the mantle (left), the viscosity of the LVZ vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' viscosity of the mantle (centre), and the outer radius vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' rigidity of the LVZ (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' –25– 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='028 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='026 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 H 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='020 1 month 1 month yr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 6 8 5 4 8 5 4 logx [rad/s] logx [rad/s]24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='00 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='75 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='75 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='75 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='80 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='50 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 600 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 400 500 700 RLvz [km] log μLvz [Pa] log nLVz [Pa s]manuscript submitted to JGR: Planets formed in our study explicitly assumed that the value of Q at the monthly frequency and k2/Q at the annual frequency are known with a high precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' This is not the case in reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' However, as we have seen in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='4, a lunar mantle governed by the An- drade model without a basal dissipative layer can fit the data with the actual uncertain- ties only for unrealistically low values of parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='1 Melt-free Lunar Interior In the model cases considering a two-layered, “melt-free” lunar interior, where the negative slope of the frequency dependence of k2/Q is explained by a secondary dissi- pation peak induced by elastically accommodated GBS, we found that the logarithm of the relaxation timescale, log τ, falls into the range of [4, 6], corresponding to τ between 3 and 300 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' In the reference case depicted in Figure 4, this would imply a ratio of the characteristic timescales for the elastic and diffusional accommodation trel = τ/τM to be of order from 10−7 to 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Are such ratios of the characteristic times observed in any natural materials?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' According to Jackson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2014), grain boundary sliding comprises three processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The relative contribution of each of them to the energy dissipation in a sample depends on the temperature and loading frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The processes are: (i) elastically accommo- dated GBS with a characteristic time τ, at high frequencies/low temperatures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (ii) dif- fusionally assisted GBS described by the power-law frequency-dependence of the seis- mic quality factor, Q ∝ χp ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' and (iii) diffusionally accommodated GBS at timescales greater than the Maxwell time τM, where the seismic Q is a linear function of frequency, Q ∝ χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The value of trel thus determines the range of frequencies over which the dif- fusionally assisted sliding on spacial scales smaller than grain size occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Experimen- tal data for fine-grained polycrystals indicate that trel ≪ 1 (Morris & Jackson, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Jackson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2014) presented results of laboratory experiments on fine-grained olivine subjected to torsional oscillations at high pressures (P = 200 MPa) and rela- tively low temperatures (T < 900 ◦C), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', around the threshold between elastic response and elastically accommodated GBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' They found a GBS relaxation timescale of log τR = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='15±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='07 s, where the subscript “R” now stands for “reference”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Considering the ref- erence temperature TR = 900 ◦C, reference pressure PR = 200 MPa, reference grain size dR = 10 µm, activation volume V ∗ = 10 cm3 mol−1, and activation energy E∗ = 259 kJ mol−1, as given by Jackson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2014), we can extrapolate τ to the conditions of the lunar mantle with the Arrhenius law (Jackson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2010): τ = τR � d dR �m exp �E∗ R � 1 T − 1 TR �� exp �V ∗ R �P T − PR TR �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (46) In addition to the parameters introduced earlier, d is the grain size and m char- acterises the grain-size dependence of the process in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' We adopt the value m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='31, found by Jackson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2010) for anelastic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Figure 9 illustrates the ex- trapolation of τR of Jackson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2014) to lunar interior conditions, considering our melt-free model and two depth-independent grain sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Over the colour-coded maps, we also plot the steady-state heat conduction profiles of Nimmo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' We note that the conduction profiles were only chosen for illustration purposes: the discussion of the thermal regime (conductive vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' convective) in the lunar mantle is beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The laboratory measurements of Jackson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2014) were performed on a single sample of fine-grained polycrystalline olivine under constant pressure PR and the Arrhe- nian extrapolation of τ was only tested for temperature dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Nevertheless, if we accept the assumption that these results are applicable to the Moon, Figure 9 and the fitted relaxation time from Figure 4 (log τ ∈ [4, 6]) can help us to identify the minimum depth in which elastically accomodated GBS contributes to the tidal dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' For the –26– manuscript submitted to JGR: Planets Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Relaxation time τ (colour-coded) of elastically accommodated GBS, as given by Jackson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2014) and extrapolated to lunar interior conditions using the Arrhenian equation (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' White lines demarcate the relaxation times resulting from our inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Blue lines indicate analytically-calculated conduction profiles proposed by Nimmo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2012) for three different mantle heat productions (8, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5, and 11 nW m−3), crustal heat production of 160 nW m−3 crustal thickness of 45 km, and no heat exchange between core and mantle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Other parameters, such as the core size, core density, and mantle density, are adjusted to our melt-free model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Grain sizes are given in the upper right corner of each plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' smaller grain size (d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='1 mm) and the reference profile of Nimmo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2012) (solid line, mantle heat production of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 nW m−3), we predict the minimum depth of 400–500 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' For the larger grain size (d = 1 cm), the minimum depth is 600–800 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' A conductive profile corresponding to lower heat production than illustrated here would push the min- imum depth to even greater values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The occurrence of elastically accommodated GBS in shallower depths would give rise to a relaxation peak (or to an onset of a relaxation band) at lower loading frequencies, which would not fit the observed annual and monthly tidal Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Although the MCMC inversion from the previous section was performed for a model with a homogeneous mantle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', assuming the occurrence of elastically-accommodated GBS at all depths from the surface down to the core, we also checked that a model de- scribed by the Andrade rheology above the derived depths and by the Sundberg-Cooper model below the derived depths would fit the considered observables under the condi- tion that log τ ≳.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' For shorter τ, the estimated minimum depth of applicability of the Sundberg-Cooper model would not match the Love numbers at monthly frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Besides the timescale τ, we have derived the relaxation strength of the hypothet- ical secondary peak: log ∆ ∈ [−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5, 0], or ∆ ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='03, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Parameter ∆ controls the height of the secondary dissipation peak in the Sundberg-Cooper model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Figure 10 shows the dependence of this Q−1 on the relaxation strength for all our models from Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Are these values consistent with theoretical prediction and laboratory data?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Sundberg and Cooper (2010) reported relaxation strengths of polycrystalline olivine between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='23 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='91, as found in different sources and under different assumptions on the grain shapes (Kˆe, 1947;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Raj & Ashby, 1971;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Ghahremani, 1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Their own mechan- ical tests on peridotite (olivine-orthopyroxene) at temperatures between 1200 and 1300 ◦C were best fitted with ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='43 and the corresponding dissipation associated with elastically- accommodated GBS in their sample was Q−1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='25−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' On the other hand, Jackson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2014), who performed torsion oscillation experiments on olivine, found a relatively low dissipation peak with Q−1 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Low secondary dissipation peaks with Q−1 ∼ 10−2 were also predicted theoretically by Lee and Morris (2010) for a grain boundary slope of 30◦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' while smaller slopes seem to allow Q−1 exceeding 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' especially when the in- dividual grains are of comparable sizes and the grain boundary viscosity does not vary –27– 20 d=10 d=10 m m 1500 1500 16 1250 1250 12 K K T log T 1000 T 1000 8 750 750 4 500 500 0 400 700 1000 1300 1600 400 700 1000 1300 1600 r [km] r [km]manuscript submitted to JGR: Planets too much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Accordingly, Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2011) note that Q−1 in the secondary peak depends strongly on the slope of the grain boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Following this brief discussion of dissipation arising due to elastically accommo- dated GBS, we can conclude that the relaxation strength ∆ (or Q−1 in the secondary dissipation peak) is not well constrained and the values found in literature permit any of the ∆s predicted in our Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Seismic Q−1 of the mantle at the frequency of the secondary peak, plotted as a function of the relaxation strength ∆ for models from Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='2 Highly Dissipative Basal Layer Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Shear modulus prediction compared to seismic measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Shear modulus µLVZ for RLVZ = 400, 500 and 700 km (gray, yellow and orange areas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Shear modulus derived from seismic velocities and densities: green (Weber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2011), red (Khan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2014) and blue (Matsumoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2015), dashed lines: errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' A highly dissipative layer located at any depth could also produce the desired sec- ondary peak needed to explain the anomalous Q dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (Note, however, that a pres- ence of a highly dissipative layer at a shallow depth may lead to changes in the body’s response to tides and might be incompatible with the measured values of the Love num- bers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=') Petrological considerations combined with an indication of a basal low-velocity –28– peak 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='30 in the secondary 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='15 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='10 seismic ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 log △11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='00 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='75 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='50 log μ [Pa] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='25 Weber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2011) Khan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2014) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='00 Matsumoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2015) this study, RLvz = 400 km 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='75 this study, RLvz = 500 km this study, RLvz = 700 km 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='50 400 600 800 1000 r [km]manuscript submitted to JGR: Planets zone point to the presence of this anomalous layer in the deep interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Therefore, as an alternative to the “melt-free” model, we tested the popular hypothesis of a putative highly dissipative layer at the base of the lunar mantle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The derived rheological properties of the mantle and of the basal layer as well as the layer’s thickness are poorly constrained and can be strongly biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Firstly, the outer radius RLVZ of the basal layer is correlated with the value of the mantle rigidity µm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' the thicker the basal layer, the larger mantle rigidity can be expected to satisfy the model constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The mantle viscosity ηm depends on the empirical Andrade parameters, and an increase of α by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='1 leads to a reduction of the fitted mantle viscosity approximately by one order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' On the other hand, the viscosity of the basal layer remains independent of the empirical Andrade parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The predicted contrast in viscosity between the two layers thus decreases with increasing α and/or ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Secondly, the range of acceptable basal rigidities µLVZ widens with the basal layer thickness (Figure 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' We do not find an acceptable solution for RLVZ ≲ 400 km due to our a priori requirement on the relationship between the mantle and basal layer’s rigidities (µLVZ ≤ µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The range of acceptable µLVZ values increases with the basal layer radius up to one and a half order of magnitude for the maximum RLVZ = 700 km considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Interestingly, the predicted rigidities of a basal layer with thickness ∼ 170 km (RLVZ ≈ 500 km) cor- responds well with the seismic observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Lastly, the basal viscosity is correlated with the basal layer thickness: the viscosity ηLVZ decreases from 3·1016 Pa s for a thin weak layer (RLVZ = 400 km) to < 1015 Pa s for the greatest considered thickness (RLVZ = 700 km).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The basal layer viscosity is, therefore, always considerably lower than the man- tle viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' However, this is not surprising, as the low viscosity of this layer is essen- tial to predict the anomalous frequency dependence of the tidal quality factor, when the rest of the high-viscosity mantle is set to obey the Andrade law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Basal viscosity prediction compared to rheological properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Predicted ranges of viscosities ηLVZ for RLVZ = 400, 500 and 700 km are indicated by gray, yellow, and orange areas, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Over the predicted ranges is plotted the temperature dependence of viscosity of ilmenite (blue, Dygert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2016), dry olivine (red, Hirth & Kohlstedt, 1996), and ilmenite- olivine aggregate (2 − 16 %), the latter corresponding either to isostress (blue area, harmonic mean, suggested for high strain) or Tullis (red area, geometric mean, suggested for low strain) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Errors of experimentally determined viscosities not included;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' ilmenite error factor is ∼ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Vertical lines delimit solidus temperatures for peridotite (Katz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2003) and ilmenite-bearing material (Wyatt, 1977) at radii 330 km and 700 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Left panel: temperature dependence for σD = 1 MPa, dry olivine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Right panel: temperature dependence for σD = 1 MPa, wet olivine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Rigidity and viscosity magnitudes, and their contrast between the mantle and the basal layer values, can be indicative of the variations in the composition, in the presence of melt, and in temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' A stable partially molten zone in the lunar interior would pose strong constraints on the composition (Khan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Given the absence of ge- –29– Wet olivine Dry olivine 20- 20- olivine, Hirth and Kohlstedt (1996) 1 ilmenite, Dygert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2016) 18 18 isostress/Reuss model S [Pa Tullis model ii peridotite, solidus, Katz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2003) ilmenite-bearing, solidus, Wyatt (1977) 16 this study, RLVz 400 km I H this study, RLvz = 500 km this study, RLvz = 700 km I 14 14 I ii ii 2000 2000 1200 1400 1600 1800 1200 1400 1600 1800 T [K] T [K]manuscript submitted to JGR: Planets ologically recent volcanic activity, any melt residing in the deep lunar interior would have to be neutrally or negatively buoyant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Using an experimental approach on the synthetic equivalent of Moon samples, van Kan Parker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2012) concluded that the condition on the buoyancy below 1000 km is satisfied if high content of titanium dioxide is present in the melt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' We can expect the presence of a partially molten layer at any depth below this neutral buoyancy level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Moreover, evolutionary models suggest that high-density ilmenite bearing cumu- lates enriched with TiO2 and FeO are created towards the end of the shallow lunar magma ocean crystallisation, resulting in near-surface gravitational anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' This instability, combined with the low viscosity of those cumulates, might have eventually facilitated the mantle overturn, creating an ilmenite-rich layer at the base of the mantle (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Recently, Kraettli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2022) suggested an alternative compositional model: a ∼ 70 km thick layer of garnetite could have been created at the base of the mantle if two independently evolving melt reservoirs were present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The resulting high-density garnet, olivine, and FeTi-oxide assemblage is gravitationally stable and can contain a neutrally or negatively buoyant Fe-rich melt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The scenario of Kraettli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2022) can also be accompanied by the mantle overturn, as suggested for the ilmenite-rich layer created at shallow depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Rheologically weak ilmenite combined with appropriate lower-mantle temperature can help to explain the low basal viscosity (Figure 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' If the lower mantle were only made of dry olivine, the predicted viscosity would require temperature ≳ 1800 K, whereas for wet olivine, the temperature range between ∼ 1500 and ∼ 1800 K would be sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Creep experiments (Dygert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2016) conclude that the viscosity of ilmenite is more than three orders of magnitude lower than dry olivine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Consequently, a lower-mantle tem- perature (1400 − 1700 K) might be acceptable to explain the predicted viscosities for pure ilmenite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The properties of ilmenite-olivine aggregates introduce yet another com- plexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The viscosity of aggregates is suggested to depend on the value of the strain: it follows the Tullis model for low strain, whereas it tends to follow the lower bound on Fig- ure 12 (isostress model) for large strain (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', Dygert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2016, for a deeper dis- cussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The acceptable temperature range for olivine-ilmenite aggregate is close to the values for the pure olivine in the case of the Tullis model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The prediction for the isostress model (minimum bound, Reuss model) is consistent with temperature values between 1500−1800 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Another obstacle in interpretation originates in the stress-sensitivity of the relevant creep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The viscosity can decrease by ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 orders of magnitude while de- creasing the differential stress by one order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' In terms of acceptable ther- mal state, the temperature consistent with our prediction would decrease roughly by ∼ 100 K considering two-fold higher differential stress and increase by the same value for two-fold lower stress, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Consequently, we find acceptable solutions both below and above the solidus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Our three-layered model thus cannot exclude or confirm a possible partial melt presence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' An alternative explanation for the viscosity reduction can be the presence of water (see also Karato, 2013, for a deeper dicussion), which would also reduce the solidus temperature and facilitate partial melting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Both the enrichment in ilmenite and elevated water con- tent can lead to the desired value of viscosity at lower temperatures compared to the dry and/or ilmenite-free models (Figure 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Focusing now on the elastic properties, we note that the rigidities of olivine (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2015), ilmenite (Jacobs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2022), and garnetite (Kraettli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2022) are comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The temperature has only a limited impact on their value (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='01 GPa/K for olivine and ilmenite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Also, dependence on the water content (olivine-brucite) is only moderate (−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='3 GPa/wt%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Jacobsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The magnitude of rigidity is, there- fore, rather insensitive to possible constituents, temperature and water content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The up- per bound of basal layer’s rigidity predicted here (∼ 60 GPa for RLVZ = 400 km, ∼ 70 GPa for RLVZ = 500 km and ∼ 85 GPa for RLVZ = 700 km) fits the elastic properties of all –30– manuscript submitted to JGR: Planets considered minerals—ilmenite, olivine, and garnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' However, the lower bound values (for RLVZ > 500 km) are difficult to explain by the changes in composition, high temper- ature, and/or water content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Impact of melt on the viscosity and rigidity contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The viscosity and rigidity contrast expressed as a function of the φ/φc (φ denotes the porosity and φc the critical porosity and parameterised using Kervazo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' ηsolid and µsolid represents values with no melt present at the solidus temperature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' no change in composition is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The shaded areas depict the predicted contrasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The magnitude of rigidity (Figure 13) is, nevertheless, sensitive to the presence of melt around or above the disintegration point (characterised by the critical porosity φc), which describes the transition from the solid to liquid behaviour and its typical values lie between 25−40%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Similarly, the viscosity value is very sensitive to the presence of melt for porosity higher than φc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' For low porosities, it follows an exponential (Arrhe- nian) dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Figure 13 suggest that the predicted rheological contrasts in the nom- inal case are consistent with φ ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='1φc for shear modulus contrast and with φ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='1φc for the viscosity contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' This apparent inconsistency may be accounted for by the pres- ence of melt accompanied by the changes in composition of the basal layer and by the susceptibility of viscosity to these changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Consequently, the knowledge of the contrasts in both rheological parameters (rigidity and viscosity) could help tackle the trade-offs between porosity content and composition/temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Nevertheless, we must empha- sise that the viscosity contrast predicted by our models is sensitive to the Andrade pa- rameters of the mantle, leading to another uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The presence of a partially molten material would pose a strong constraint on the temperature and possible mode of the heat transfer in the lower mantle of the Moon, al- lowing only models that reach the temperature between the solidus and liquidus (Fig- ure 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The traditional advective models predict stagnant-lid convection with a rela- tively thick lid at present (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Below the stagnant lid, the temper- ature follows the adiabatic or, for large internal heating, sub-adiabatic gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' We es- timate the temperature increase across the entire mantle due to the adiabatic gradient to be bounded by 100 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Within those traditional models, it is plausible to reach solidus only in the lowermost thermal-compositional boundary layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' In the case of conductive models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Nimmo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2012), the temperature gradient is steeper than the solidus gradient and the solidus temperature can be reached in the entire basal layer, given ap- propriate internal heating (as demonstrated in Figure 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Interestingly, the lunar se- lenotherm determined by the inversions of lunar geophysical data combined with phase- equilibrium computations (Khan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2014) lies between the conductive and adiabatic gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' –31– 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 1 log n(d) / N solid 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 3 Kervazo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2021) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 this study, Rvz =400 km 4 this study, RLvz = 500 km 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 this study, RLvz = 700 km 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='2 relative porosity / Φ relative porosity Φ / Φcmanuscript submitted to JGR: Planets Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Comparison of temperature profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Colour scale: conductive profile, calculated with the matrix propagator method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' parameters as in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Individual branches correspond to average heating 8, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 and 11 nW/m2 in the mantle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The coefficient f denotes the enrich- ment in the radiogenic elements of the basal layer (RLVZ = 500 km) compared to the rest of the mantle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Gray area is the temperature profile adapted from Khan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' darker blue lines: peridotite solidus (solid), water-saturated solidus (dotted), and liquidus (dashed) according to Katz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' light blue lines: clinopyroxene+ilmenite solidus (solid) and liquidus (dashed) according to Wyatt (1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' –32– 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 2500 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 2000 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 basal enrichment 1500 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 solidus, dry peridotite, Katz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2003) 1000 solidus, water saturated peridotite, Katz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2003) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 liquidus, peridotite, Katz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2003) solidus, ilmenite bearing, Wyatt (1977) 500 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 liquidus, ilmenite bearing, Wyatt (1977) Khan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2014) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 400 600 800 1000 1200 1400 1600 r [km]manuscript submitted to JGR: Planets In the future, distinct sensitivity of rigidity, viscosity, and other transport prop- erties to temperature, melt fraction, and composition may provide a way to separate the interior thermal and composition structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' At present, inversion errors and the uncer- tainties on material properties cannot confirm or rule out the existence of a partially molten basal layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' It therefore remains a valid hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='3 Other Sources of Information The two models discussed here — one with a highly dissipative basal layer and the other with elastically-accommodated GBS in the mantle — cannot be distinguished from each other by the available selenodetic measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' To answer the question stated in the title of our paper, one would need to resort to other types of empirical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Among all geophysical methods devised for the exploration of planetary interiors, seismology is of foremost importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Therefore, a question that cannot be solved by the interpreta- tion of lunar tidal response might be answered by comparing the arrival times and the phases detected at individual seismic stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' As we mentioned in Introduction, the Moon demonstrates a nearside-farside seis- mic asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Judging by the currently available seismic data collected on the near side, the deep interior of the far side is virtually aseismic or, alternatively, the seismic waves emanating from it are strongly attenuated or deflected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The existence of an aseis- mic area on the farside might not be entirely inconceivable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' First, as pointed out by Nakamura (2005), there are large zones with no located nests of deep moonquakes even on the near- side;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' and, in fact, most of the known deep seismic nests are part of an extended belt reach- ing from the south-west to the north-east of the lunar face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Second, there exists a pro- nounced dichotomy between the near side and far side of the Moon in terms of the crustal thickness, gravity field, and surface composition, which might point to a deeper, inter- nal dichotomy as predicted by some evolutionary models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', Laneuville et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' An obvious way to illuminate the lack of deep farside moonquakes detected by the Apollo seismic stations would be to place seismometers on the far side of the Moon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' They would observe the far side activity, and record the known repeating nearside moonquakes or events determined from impact flash observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The Farside Seismic Suite (FSS) mission, recently selected for flight as part of the NASA PRISM program and planned for launch in 2024 or 2025, might provide such a measurement by delivering two seismome- ters to Schr¨odinger Crater (Panning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' While this crater is far from the an- tipode (in fact, close to the South pole), a seismometer residing in it should still be able to detect events from the far side, thereby addressing the hemispheric asymmetry in the Apollo observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' However, resolving polarisation of arrivals may be challenging for many moonquakes, meaning that many events will only have distance estimated, but not azimuth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (We are grateful to Mark P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Panning for an enlightening consultation on this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=') A better site for this science objective would be the far side Korolev crater resid- ing by the equator, about 23 degrees from the antipode (by which we understand the centre of the farside).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' It is now considered as one of the possible landing sites for the Lu- nar Geophysical Network (LGN) mission proposed to arrive on the Moon in 2030 and to deploy packages at four locations to enable geophysical measurements for 6 - 10 years (Fuqua Haviland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Still, having a station or even an array of seismic stations at or near the antipode would be ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Observed by such a station or stations, all events at distances less than 90 degrees from the antipode could be confidently assigned to the far side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' So we would recommend the near-antipode zone (that close to the centre of the farside) as a high-priority landing site for some future mission, a perfect area to monitor the seismic activity on –33– manuscript submitted to JGR: Planets the far side and, especially, to observe if and how seismic waves proliferate through the base of the mantle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' In addition to seismic measurements, and similarly to what is predicted for Jupiter’s volcanic moon Io or for icy moons with subsurface oceans, the presence of a highly dis- sipative or a partially molten layer might be reflected in the tidal heating pattern on lu- nar surface (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', Segatz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Tobie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' However, as illustrated in the upper row of Figure 15, the positioning of the layer at the base of the mantle results in a very small difference between the surface heating patterns corresponding to the two alternative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Both models show maxima of the average surface tidal heat flux Φtide on the lunar poles and minima on the “subterranean” point (ϕ = 0) and its antipode (ϕ = π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Moreover, the magnitude of Φtide is generally very small, about three orders of magnitude lower than the flux produced by radiogenic heating of lunar interior (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', Siegler & Smrekar, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The detection of any differences between the surface heat flux of the two models would be extremely challenging, if not impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Average surface tidal heat flux (top) and volumetric tidal heating (bottom) for a specific realisation of each of the two models discussed in this work: the model considering elastically-accommodated GBS through the Sundberg-Cooper rheological model (left) and the model with a basal low-viscosity zone (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' In particular, the volumetric tidal heating is plot- ted as a function of relative radius r/R and colatitude ϑ with longitude ϕ equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The lower row of Figure 15 illustrates volumetric heat production due to tidal dis- sipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' As pointed out by Harada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2014), the presence of a low-viscosity zone at the base of the mantle results in considerable local increase of tidal heating with re- spect to the rest of the mantle or to the model without the basal layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' While the tidal contribution to heat production in the high-viscosity parts of the mantle is around 10−11 W m−3, the tidal heat production in the basal layer reaches ∼ 10−8 W m−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' For comparison, the global average of mantle heat production by all sources (radiogenic and tidal) is estimated to be 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='3×10−9 W m−3 (Siegler & Smrekar, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The predicted tidal dissipation in the basal layer can help to locally increase the temperature and exceed the solidus, es- pecially if conductive heat transfer prevails in the lunar mantle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Combined with a high enrichment of the basal layer in heat producing elements, it may then contribute to main- taining the presence of melt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' –34– Sundberg-Cooper mantle Mantle with a low-viscosity layer 儿 元 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='019 2 [mW /m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='011 0 0 元-2 2 2元 元-2 3-2 0 0 2元 元 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='8 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='6 10 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='4 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='2 元-4 π14 元-2 π-2 0 元 0 元 9 6manuscript submitted to JGR: Planets Although virtually discarded in the beginning of this Subsection, let us neverthe- less discuss possible insights provided by future high-precision tidal measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' At present, the quality factor Q at tidal frequencies is obtained exclusively from fitting the lunar physical libration, empirically determined by LLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' However, increased precision of satellite tracking (Dirkx et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2022) might even- tually enable the determination of lunar tidal phase lag from the gravity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Having an independent determination of tidal Q, which is related to the phase lag, would serve as a verification of the method used for fitting the LLR time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Among the quantities that we used in the inversion was degree-3 potential Love number k3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' This parameter is currently only known with a large error bar but its refine- ment would only help to discern between the two alternative models considered here if the elastically-accommodated GBS was contributing to the dissipation throughout the entire mantle (and not only in greater depths, as tentatively derived in Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' This is a consequence of a degree-dependent sensitivity of Love numbers to the interior structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' While degree-2 Love numbers and quality factors probe the lunar interior down to the core, higher-order quantities are only sensitive to shallower depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The Love num- ber k3—or the quality factor Q3—would thus not “see” the basal low-viscosity layer, but it might sense complex tidal response in the upper mantle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' As a result, the detection of the unexpected frequency dependence of tidal dissipation even in Q3 (accompanied by a relatively high k3 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='01) would clearly point at a mechanism acting in shallow depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Interestingly, the two alternative models can be better distinguished from each other in case the secondary peak of tidal dissipation, resulting either from the existence of a weak basal layer or from the Sundberg-Cooper model, lies at frequencies close to 10−4 rad s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Then, provided that the elastically-accommodated GBS is only active below distinct depths (400−600 km), one could see a difference in predicted h2 of the two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Indepen- dently on that depth, the models with secondary dissipation peak close to 10−4 rad s−1 also differ in elastic Love number k2,e, which can be calculated for interior structures ob- tained from the inversion of seismic waves (as was done by Weber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Specif- ically, k2,e in the melt-free model is then much lower that that of the model with a weak basal layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The value reported by Weber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2011), which is k2,e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='0232, is at- tained by both the alternative models for a secondary tidal dissipation peak lying at ∼ 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='5 rad s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' In that case, the models are already indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Seismic Q in the melt-free part of the mantle (at 1 Hz) for the models mentioned in the previous sentence is around 800 − 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Finally, we would like to note that any increase in the precision of Q determina- tion will greatly help in answering the question whether any specific source of additional dissipation, be it a weak basal layer or elastic accommodation of strain at grain bound- aries, is necessary in the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Recall that in order to fit the two alternative mod- els to the tidal data, we assumed that the uncertainty on Q is of the order of 1% the mean value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' In reality, the empirical Q at the monthly and the annual frequencies present an uncertainty between 10 and 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Keeping the original uncertainties, we were still able to fit the tidal data with the standard Andrade model, although with an unrealistically small exponential factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' 7 Conclusions Tidal effects strongly depend not only on the interior density, viscosity, and rigid- ity profiles of celestial bodies, but also on the implied deformation mechanisms, which are reflected in the rheological models adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' In this work, we attempted to illustrate that the unexpected frequency dependence of the tidal Q measured by LLR (Williams & Boggs, 2015) can be explained by lunar interior models both with and without a par- tially molten basal layer, and that each of the considered models leads to a different set of constraints on the interior properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' –35– manuscript submitted to JGR: Planets As a first guess, we fitted the lunar tidal parameters (k2, k3, h2, Q at the monthly frequency and k2/Q at the annual frequency) with a model consisting of a fluid core and a viscoelastic mantle governed by the Andrade rheology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Within that model, and set- ting ζ = 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', the time scales of viscoelastic and anelastic deformation were consid- ered comparable) we found a mantle viscosity of log ηm[Pa s] = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='99+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='89 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='35, mantle rigid- ity of log µm[Pa] = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='92±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='06, and the Andrade parameter α as low as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='06+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The predicted value of α is generally lower than reported in the literature (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', Jack- son et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Castillo-Rogez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Efroimsky, 2012a, 2012b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' This observation leads us to the conclusion that the tidal response of the Moon probably cannot be ex- plained by the Andrade model alone and requires either a basal low-viscosity zone (in line with the conclusion of Khan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2014) or an additional dissipation mechanism in the mantle (similar to Nimmo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Throughout Section 5, we have seen that the two alternative models expected to explain the anomalous frequency dependence of lunar Q (assumed to be known with an arbitrarily chosen high precision) cannot be distinguished from each other by the exist- ing measurements of tidal deformation and dissipation alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' In the two-layered model consisting of a liquid core and a Sundberg-Cooper mantle, the fitting of tidal parame- ters requires the relaxation time τ associated with elastically-accommodated GBS to be in the range from 3 to 300 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The corresponding relaxation strength ∆ is predicted to lie in the interval [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='03, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' For a nominal case with Andrade parameters α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='2 and ζ = 1, we further obtain a mantle viscosity of log ηm[Pa s] = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='55+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='54 and a mantle rigidity log µm[Pa] = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='840.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' In the three-layered model containing a liquid core, a low-rigidity basal layer, and an Andrade mantle, the tidal parameters are consistent with a wide range of basal layer thicknesses DLVZ and rigidities µLVZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' As a general rule, a thicker layer implies weaker constraints on its rigidity, allowing both melt-like and solid-like behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The predicted values of µLVZ are consistent with elastic properties of all considered minerals (olivine, ilmenite, granite) and with a wide range of lower-mantle temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' In contrast to the rigidity, the viscosity ηLVZ of the basal layer is constrained relatively well and falls into the range from about 1015 to 3×1016 Pa s, with a preference for the lower values (log ηLVZ[Pa s] = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='20+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='53 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' This is also in accordance with the results of Efroimsky (2012a, 2012b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Harada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2014, 2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Matsumoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Tan and Harada (2021), and Kronrod et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Nevertheless, even the viscosity is not able to pose strong constraints on the lower-mantle temperature, owing to the large uncertainties both on tidal Q and on the rheological properties of lunar minerals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' For the viscosity and rigid- ity of the overlying mantle in the nominal case, we get log ηm[Pa s] = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='79+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='19 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='06 and log µm[Pa] = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The existence of a basal weak or possibly semi-molten layer in the mantles of ter- restrial bodies has been recently also suggested for Mercury (Steinbr¨ugge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2021) and for Mars (Samuel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' In the case of Mercury, a lower mantle viscosity as low as 1013 Pa s was proposed to match the latest measurements of the moment of in- ertia and of k2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' although this result was later critically reassessed by Goossens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2022), who showed that more realistic values around 1018 Pa s might still explain the observa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' In the case of Mars, the putative basal semi-molten layer was introduced by Samuel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2021) to provide an alternative fit to seismic data which would not require the ex- istence of a large core with unexpectedly high concentration of light elements (reported in St¨ahler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Lastly, large provinces of decreased shear seismic velocities also exist at the base of the Earth’s mantle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' These zones form a heterogeneous pattern in the deep terrestrial interior;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' however, according to numerical models, the formation of a con- tinuous layer right above the core-mantle boundary is also possible for some values of model parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', Dannberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' A new question thus arises: is a weak basal layer something common among terrestrial planet’s mantles?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Is it a natural and widely present outcome of magma ocean solidification and subsequent dynamical pro- cesses?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Or is it merely a popular explanation of the data available?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' –36– manuscript submitted to JGR: Planets Since the available tidal parameters were deemed insufficient to distinguish a weak basal layer above the lunar core from the manifestation of elastically accommodated GBS in the mantle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' we conclude that an answer to the question stated in the title of our pa- per awaits future lunar seismic experiments (ideally with a uniform distribution of seis- mometers across the lunar surface) as well as a better understanding of elastic param- eters of olivine-ilmenite assemblages near their melting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Additionally, a tighter bound on the hypothetical basal layer parameters or on the strength and position of the sec- ondary Debye peak in the alternative, Sundberg-Cooper model might be given by up- dated values of tidal Q at multiple frequencies or by an independent inference of inte- rior dissipation from the tidal phase lag and frequency-dependent k2, theoretically mea- surable by laser altimetry or orbital tracking data (Dirkx et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' A combination of all those sources of information will probably still not provide a bright picture of deep lunar interior;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' however, it will help us to refute at least some of the many possible interior models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Open Research The software developed for the calculation of tidal Love numbers and quality fac- tors of multi-layered bodies, the Python interface for running the MCMC inversion, and the plotting tools used for the figures presented in this study will be made available at the GitHub repository of the corresponding author (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='com/kanovami/ Lunar Q) and preserved at [DOI to be added later during the peer review process] un- der the licence [to be added later during the peer review process].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Acknowledgments The authors would like to thank James G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Williams, Mark P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Panning, and Alexander S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Konopliv for extremely helpful conversations on various aspects of the lunar science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' is grateful to Ana-Catalina Plesa and Martin Knapmeyer for discussions about the lunar interior and to Philipp A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Baumeister for introducing her to the Python libraries used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' She also gratefully acknowledges the financial support and endorse- ment from the DLR Management Board Young Research Group Leader Program and the Executive Board Member for Space Research and Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' received fund- ing from Czech Science Foundation grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' GA22-20388S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' References Bagheri, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', Efroimsky, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', Castillo-Rogez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', Goossens, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', Plesa, A.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Journal of Applied Physics, 25(11), 1385–1391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Bolmont, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', Breton, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', Tobie, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', Dumoulin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', Mathis, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', & Grasset, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2020, December).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Solid tidal friction in multi-layer planets: Application to Earth, Venus, a Super Earth and the TRAPPIST-1 planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Potential ap- –37– manuscript submitted to JGR: Planets proximation of a multi-layer planet as a homogeneous body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Astronomy & Astrophysics, 644, A165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='1051/0004-6361/202038204 Bou´e, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', & Efroimsky, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Tidal evolution of the keplerian elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Ce- lestial Mechanics and Dynamical Astronomy, 131, 30.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The tidal history of Iapetus: Spin dynamics in the light of a refined dissipation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Journal of Geophysical Research (Planets), 116(E9), E09008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='1029/2010JE003664 Dannberg, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', Myhill, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', Gassm¨oller, R.' 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+page_content='1088/0004-637X/746/2/150 Efroimsky, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2015, October).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Tidal Evolution of Asteroidal Binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Ruled by Viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Ignorant of Rigidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The Astronomical Journal, 150(4), 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' doi: 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The Astrophysical Journal, 764, 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='1088/0004-637X/764/1/26 Efroimsky, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', & Makarov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2014, November).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Tidal Dissipation in a Homoge- neous Spherical Body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' The Astrophysical Journal, 795(1), 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='1088/0004-637X/795/1/6 Foreman-Mackey, D.' metadata={'source': 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+page_content='gov/lw16/docs/papers/ sci\\ 1\\ Williams\\ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='pdf Williams, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=', & Boggs, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' (2015, April).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Tides on the Moon: Theory and de- termination of dissipation.' metadata={'source': 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Nearside-Farside Asymmetries the Result of a Giant Impact?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' Journal of Geophysical Research (Planets), 124(8), 2117-2140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content=' doi: 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} +page_content='1029/2018JE005826 –44–' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E0T4oBgHgl3EQfkwEq/content/2301.02476v1.pdf'} diff --git a/RtE3T4oBgHgl3EQfDQmV/content/tmp_files/2301.04285v1.pdf.txt b/RtE3T4oBgHgl3EQfDQmV/content/tmp_files/2301.04285v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..132ad61e64c331c42f3aa423e6cf7b8a1e2d6106 --- /dev/null +++ b/RtE3T4oBgHgl3EQfDQmV/content/tmp_files/2301.04285v1.pdf.txt @@ -0,0 +1,1363 @@ +TAPS: Topology-Aware Intra-Operator Parallelism Strategy Searching Algorithm +for Deep Neural Networks +Peng Liang, Hao Zheng, *Teng Su, Linbo Qiao, Dongsheng Li +National University of Defense Technology, *Huawei Technologies Co. Ltd +{peng_leung, zhengh, qiao.linbo, dsli}@nudt.edu.cn +*suteng@huawei.com * +Abstract +TAPS is a Topology-Aware intra-operator Parallelism strat- +egy Searching algorithm that generates intra-operator paral- +lelism strategies by considering both intra-node and inter-node +bandwidth. Most of the existing auto-parallelism works use +the communication volume as the communication cost directly +when generating strategies, which we prove to be sub-optimal +in multi-nodes cases. We design a topology-aware cost model +for multi-node intra-operator parallelism strategy searching. +Numerical experiments demonstrate that TAPS can generate +strategies with up to 85% fewer communication costs, which +outperform the latest baselines. +1. Introduction +Large-scale Deep Learning (DL) models have been a huge hot +topic in recent years for their great performance improvements +in fields like [3, 9, 16], which is a result of scaling up model +sizes and dataset sizes. For example, PaLM with 540 billion +parameters is trained with a corpus of 780 billion tokens that +represent a wide range of natural language use cases [5]. +As the model size significantly increases, training mod- +els with a single device or even within a node is no longer +practical. Thus, researchers use distributed deep learning to +train these models [19]. Manual strategies like [17] have been +widely used in training transformer-based models for their +good performance. However, it is often not optimal because +optimal parallelism strategies vary when the model or training +environment changes, in which case researchers and engineers +may need to redesign strategies. +To relieve us from the parallelism design procedure, re- +searchers propose auto-parallelism algorithms [4, 7, 23] that +can find decent strategies given a specific model and envi- +ronment. These algorithms first model parallelism strategies’ +communication costs and then use a dynamic programming +or an integer linear programming (ILP) method to find the +optimal strategy. +As model size grows larger, a single node can no longer +hold an entire large-scale model. Thus, using multi-nodes to +train a model becomes necessary. Our key observation is that +in a multi-node environment, the bandwidth within a node +(intra-node bandwidth) and across nodes (inter-node band- +width) are different, and the intra-node bandwidth is much +*A preprint version, change at any time. +higher than inter-node bandwidth in most cases. However, +existing searching algorithms model the communication cost +using the communication volume directly, ignoring the dif- +ference between the bandwidths and resulting in sub-optimal +strategies. Based on this observation, we propose a topology- +aware parallelism strategy searching algorithm called TAPS, +which can capture the difference between intra-node and inter- +node communication and thus generates better parallelism +strategies. +We first construct a topology-aware cost model, which can +determine the inter-node communication times as well as the +topology-aware communication cost given a communication +axis of a tensor. Then we formalize the strategy searching prob- +lem as an integer linear programming problem, after which +we use a third-party solver to solve the final strategy decision. +In summary, we make the following contributions: +• We prove that the volume-based communication cost model +is insufficient to generate optimal intra-operator parallelism +strategy in multi-nodes cases. +• We provide a heuristic solution in optimizing tensor redistri- +bution sequences. +• We analyze the communication in multi-node environments +and propose a topology-aware communication cost model, +which can calculate more accurate communication costs of +a parallelism strategy of an operator. +• We design and implement TAPS, a strategy-searching algo- +rithm that works for distributed DL. +• We numerically evaluate TAPS on several models of differ- +ent configurations. We compare TAPS with volume-based +searching. Our experiments show that TAPS can find strate- +gies with up to 85% fewer communication costs. +2. Background +2.1. Existing Parallelism Methods +Since Hinton [8] trained AlexNet using two GPUs in 2012, re- +searchers have proposed many parallelism methods, including +data parallelism (DP), model parallelism (MP), and pipeline +parallelism(PP). +2.1.1. Data Parallelism Data parallelism partition and dis- +tribute the data across devices that has a replicated model. +Each device computes the gradients using the split data and +uses communication like AllReduce or Broadcast to synchro- +nize the gradients or model parameters with other devices. So +arXiv:2301.04285v1 [cs.DC] 11 Jan 2023 + +that after every iteration, the models on all workers are the +same. +2.1.2. Model Parallelism Model parallelism partition the +model parameters across devices and make devices process +the same data. Model parallelism produces partial-sum or +sliced results when the parameter matrix is partitioned row- +wisely and column-wisely, respectively. Row-wise MP (Row- +MP) requires synchronization to unify the operator’s results +on different devices. Column-wise MP (Column-MP) does +synchronization only in backward propagation. +2.1.3. Pipeline Parallelism Pipeline parallelism partition op- +erators in a model into several stages and let devices hold only +one or a few of them. Meanwhile, PP splits a mini-batch of +data into several micro-batches and feeds them one by one into +the first stage. When a stage finishes its computation, it sends +the result to its next stage. Different stages can be handled +simultaneously; thus, PP forms a pipeline that can improve +performance. +2.2. Intra- and Inter-Operator Paralleism +Alpa [23] catalog existing parallelism methods into two orthog- +onal categories: intra-operator and inter-operator parallelism. +Intra-operator parallelisms are parallelism schemes that par- +tition an operator’s involved tensors along some dimensions, +assign the resulting partitioned computation to multiple de- +vices, and let them execute different parts of the computation +simultaneously. From this view, we can treat data parallelism +as a scheme that partitions an operator’s input and output +tensor along the batch-size axis; we can treat Row-MP as +a scheme that partitions an operator’s input and weight ten- +sor along the channel-in axis; we can treat Column-MP as a +scheme that partitions weight tensor and output tensor along +the channel-out axis. Inter-operator parallelism, including +pipeline parallelism, partitions models into several stages with +multiple operators. +This paper focuses on generating multi-dimensional intra- +operator parallelism strategies in multi-node environments. +2.3. Strategy Searching Algorithm +Researchers have proposed methods to search parallelism +strategies automatically. +ToFu[21], TensorOpt[4], and +Alpa[23] generate intra-operator parallelism strategies by mini- +mizing the overall communication cost of a computation graph +under the observation that all different strategies of an oper- +ator have the same computation cost. ToFu and TensorOpt +adapt the dynamic programming algorithm that OptCNN[7] +propose to produce better results. Alpa formalizes the search- +ing problem as an integer programming problem and uses a +solver to handle the solution progress. However, they assume +the bandwidths of clusters are equal everywhere, ignoring the +difference between the intra-node bandwidth and inter-node +bandwidth. This assumption may limit the searching algorithm +to find the optimal strategies, as, in large-scale clusters, intra- +node bandwidth is much higher than inter-node bandwidth. +In this paper, we propose a topology-aware communication +cost model aware of the intra-node and inter-node bandwidth, +which helps generate more fine-grained strategies. +3. Overview +TAPS is an algorithm that generates intra-operator parallelism +strategies by minimizing the communication cost of the com- +putation graph. TAPS takes a computation graph G = (V,E) +and device graph D = (VD,ED) as inputs, and output a partition +set P, which consists of strategy decisions of every operator +vi ∈ V in G. The computation graph contains operator infor- +mation, like shapes and operator types. The device graph +indicates the device types and the bandwidth between devices. +TAPS gives a solution in two steps: First, TAPS creates an +auxiliary graph where each node indicates an operator with a +specific strategy and computes the weights for each edge (u,v) +in the auxiliary graph, which equals the intra-operator com- +munication cost of v plus tensor redistribution communication +cost between u and v. Then, TAPS formalizes the searching +problem as an integer linear programming problem using the +information in the auxiliary graph and uses a third-party solver +to solve the optimal strategy. +4. Communication Cost Model +In this section, we give the details of our topology-aware com- +munication cost model. We first illustrate the details of the +volume-based cost model. Based on the volume-based cost +model, we calculate the corresponding topology-aware com- +munication cost using the volumes and effective bandwidth. +4.1. Volume-based cost model +Previous works [4, 18, 20] model the communication cost of +each strategy by symbolically computing the their communi- +cation volume. The communication volume of an operator +consists of intra-operator communication and inter-operator +communication. Intra-operator communication reduces the +partial sums generated in computing. Inter-operator commu- +nication transforms tensor to fit the succeeding operator’s +strategy. +4.1.1. Intra-operator communication Taking MatMul as an +example, its forward computation is shown as Eq.1, and its +backward computation is shown as Eq.2 and Eq.3. +Y = XW +(1) +δW = XTEy +(2) +Ex = EyW T +(3) +Let d, r, c denote the data parallelism (DP)[10], Row-MP, and +Column-MP [17] degrees of a MatMul operator, respectively; +p = drc denotes the total device number and is the power of 2. +2 + +Then we split the X and W matrices like: +X = +� +����� +X11 +X12 +... +X1r +X21 +X22 +... +... +... +... +... +... +Xd1 +Xd2 +... +Xdr +� +����� +, +W = +� +����� +W11 +W12 +... +W1c +W21 +W22 +... +... +... +... +... +... +Wr1 +Wr2 +... +Wrc +� +����� +. +After splitting the matrices X and W, we distribute their +sub-blocks to corresponding devices. As Figure 1.(a) shows, +where each cube represents a device, each sub-block of X is +replicated along axis c, and W is replicated along axis d. As +Figure 1.(b)(c) shows, we then compute the local results of Y +on each device and communicate them to form the final matrix +Y. The communication is a reduction operation of local results +and is mathematically equivalent to Eq. 4. +Yi j = +r +∑ +k=1 +XikWk j +(4) +Final matrix Y is split like: +Y = +� +����� +Y11 +Y12 +... +Y1c +Y21 +Y22 +... +... +... +... +... +... +Yd1 +Yd2 +... +Ydc +� +����� +, +where each sub-block Yi j is replicated along d axis. +Suppose we are using a bandwidth optimal Ring-AllReduce +algorithm [15], the communication volume of a MatMul oper- +ator accumulating results of Y on each device (i.e., the volume +of Row-MP) is: +VY +AR = 2(device_num−1) +device_num +data_size = 2(r −1)b out +drc +. +(5) +Similarly, we give the communication volume of DP and +Column-MP in a Matmul operator by computing the com- +munication volume of acuumulating results of δW and EX, +respectively, which are: +V δW +AR = 2(d −1)in out +drc +, +(6) +V EX +AR = 2(c−1)b in +drc +. +(7) +Finally, the overall communication volume of a MatMul oper- +ator is: +Volume = 2((d −1)in out +(r −1)b out +(c−1)b in) +drc +(8) +4.1.2. Inter-operator communication Inter-operator com- +munication happens when there are tensor redistributions be- +tween two operators. Tensor redistributions are sequences +that consist of several redistribution operators like All-Gather, +Slice, and All-To-All. In this subsection, we give our solution +for generating proper redistribution operator sequences. +Let Oout,Oin denote two operators and T denote the +output n-dimensional tensor of Oout and the input tensor +of Oin. +ST = [s0,s1,...,sn−1] is the shape of T before +partition. +Suppose the depths of device matrix of Oout +and Oin is hout and hin. +The device matrix in Oout and +Oin are Dout = [dout,hout−1,dout,hout−2,...,dout,0] and Din = +[din,hin−1,din,hin−2,...,din,0], respectively. +The tensor maps +of T in Oout and Oin are Min = [mout,0,mout,1,mout,n−1] and +Mout = [min,0,min,1,...,min,n−1], respectively. To do the tensor +redistribution, the device matrices and tensor shapes of Oin +and Oout must be the same. We unify them by two steps. In +step 1, we unify device matrices by factorizing some dimen- +sions in two device matrices, which may result in a shape +inconsistency of T in two operators. Thus in step 2, we need +to unify the tensor shape under the unified device matrix addi- +tionally. Note that the two-step unification does not change the +physical distribution of a Tensor. Table 1 shows an example +of unifying a 2-dimensional tensor between Oout and Oin. In +step 1, we factorize "8" in two device matrices and replace +them by the factorizing results [4,2] and [2,4] for Oout and +Oin, respectively. Meanwhile, we must change the tensor maps +and shapes as we modify device matrices. Since the tensor +shapes change in step 1, we need to unify it again before we +infer tensor redistribution operators. In step 2, we reshape the +tensor in Oin and Oout to make them have the same shape and +modify tensor maps simultaneously. +After unifying the device matrix and tensor shape, we can +infer the redistribution operators. A naive way to do the re- +distribution is to AllGather along all the workers and then +partition along axes that are not repetitive. To reduce the +communication cost, we use a heuristical algorithm 1 to gen- +erate tensor redistribution operators. Our algorithm contains +three optimizations. First, we only AllGather along the nec- +essary axes of the tensor, which are partitioned in Oout and +replicated in Oin. Second, we rearrange the redistribution se- +quence, putting dependent Slice before AllGather to reduce the +communication volume that AllGather produces. Third, we re- +place the implicit permutations (i.e., AllGather and Slice along +the same axis in the device matrix) with AllToAll operators, +thus further reducing the communication volume. In Algo- +rithm 1, InferSlice finds all necessary Slice-Op and appends +them to the operator sequence S. If there is no more SliceOp, +InferSlice sets S_ flag to False. Similarly, InferAll2All and +InferAllGather do the same things for AllToAllOp and All- +GatherOp. Table 3 shows an example of using above men- +tioned three optimizations to fine-tune the redistribution se- +quence. +Finally, we obtain the inter-operator communication volume +of such tensor redistribution by accumulating the communi- +cation volumes of redistribution operators within sequence +S. Suppose we are using bandwidth optimal Ring-AllGather +3 + +X11 +X12 +X21 +X22 +X21 +X22 +X11 +X12 +Y11 +Y21 +Y11 +Y21 +X21W11 X22W21 +X11W11 X12W21 +Y11 +Y21 +Y11 +Y21 +(a) Distribution of Tensor X and W +(b) Local computation  +(c) Communicate along the r axis  +W12 +W11 +W22 +W12 +W21 +d +c +r +X21W11 X22W21 +Y21 +Y21 +Y11 +Y11 +on each device +to form Tensor Y +Figure 1: Multi dimensional Intra-P of a MatMul operator on 8 devices where d = r = c = 2 +Table 1: Unifying Device Matrix and Tensor Shape of Tensor T +Step +Operator +Device Matrix +Tensor Map +Tensor Shape +0: Initial +Oout +[2,8] +[1,0] +[s0,s1] +Oin +[8,2] +[1,0] +[s0,s1] +1: Unifying device matrix +Oout +[2,4,2] +[2,1,0] +[s0,4,s1/4] +Oin +[2,1,0] +[2,s0/2,s1] +2: Unifying tensor shape +Oout +[2,4,2] +[2,−1,1,0] +[2,s0/2,4,s1/4] +Oin +[2,1,0,−1] +Table 2: Tensor Redistribution Between Mfrom = [−1,−1,2,−1,3] and Mto = [1,−1,−1,0,3] +Step +Operation +Tensor Map +Communication Volume +0: Initial +AllGather(d1,1) +[−1,−1,2,−1,3] +d1d2d3−1 +d1d2d3 Size(T) +AllGather(d2,2) +[−1,−1,−1,−1,3] +AllGather(d3,4) +[−1,−1,−1,−1,−1] +Slice(d1,0) +[1,−1,−1,−1,−1] +Slice(d0,3) +[1,−1,−1,0,−1] +Slice(d3,4) +[1,−1,−1,0,3] +1: Remove +AllGather(d1,1) +[−1,−1,2,−1,3] +d1d2−1 +d1d2d3 Size(T) +AllGather(d2,2) +[−1,−1,−1,−1,3] +Slice(d1,0) +[1,−1,−1,−1,3] +Slice(d0,3) +[1,−1,−1,0,3] +2: Rearrange +Slice(d0,3) +[−1,1,2,0,3] +d1+d2−2 +d0d1d2d3 Size(T) +AllGather(d1,1) +[−1,−1,2,0,3] +Slice(d1,0) +[1,−1,2,0,3] +AllGather(d2,2) +[1,−1,−1,0,3] +3: Replace +Slice(d0,3) +[−1,1,2,0,3] +d1d2−1 +d0d2 +1d2d3 Size(T) +AllToAll(d1,1 → 0) +[1,−1,2,0,3] +AllGather(d2,2) +[1,−1,−1,0,3] +4 + +algorithm; the communication volume of AllGather is: +VAG = (device_num−1)data_size; +(9) +For AllToAll operators, each device only needs to send +1 +device_num different data to each other in the communication +group. Thus, the communication volume of AllToAll is: +VA2A = (device_num−1)data_size +device_num +. +(10) +As we can see in Table 2, the communication volume of the +operator sequence that Algorithm 1 generates is much smaller. +We then blend the bandwidth difference into the volume- +based cost model to form our topology-aware cost model. +Algorithm 1: Optimized Tensor Redistribution +Data: D = [dh−1,dh−2,...,d0]; +Mfrom = [mfrom,0,mfrom,1,...,m from,n−1]; +Mto = [mto,0,mto,1,mto,n−1] +Result: Redistribution operator sequence S. +1 while Mfrom ̸= Mto do +2 +S_ flag ← True; +3 +while S_flag do +4 +S_ flag ← InferSlice(Mfrom,Mto,S); +5 +A2A_ flag ← True; +6 +while A2A_flag do +7 +A2A_ flag ← InferAll2All(Mfrom,Mto,S); +8 +end +9 +end +10 +AG_flag ← InferAllGather(Mfrom,Mto,S); +11 +if AG_ flag == False then +12 +AllGatherFirstUndoneDim(Mfrom,Mto,S); +13 end +4.2. Topology-aware cost model +Based on volume-based cost model, we develop a topology- +aware cost model that can additionally consider the bandwidth +difference when calculating communicaiton costs. TAPS uses +this topology-aware cost model to generate more fine-grained +strategies. +Our observation is that in multi-node environment, we can +do multiple intra-node communications of different commu- +nication groups simultaneously, and they can all fully utilize +the bandwidth; But for inter-node communications, they need +to share the links between nodes, thus lowering the effec- +tive bandwidth of each communication group. Figure 2.(a) +shows a 2 DGX-V100 nodes environment, where intra-node +communication uses high-bandwidth NVLink and inter-node +communication uses 100GBps InfiniBand. In Figure 2.(b), we +do communication along the axis 0. There are 8 communica- +tion groups, which are (GPU0, GPU1), (GPU2, GPU3) and so +on. Each of them has an individual NVLink to use and thus +the effective bandwidth equals the bandwidth of NVLink. The +case in Figure 2.(c) also has 8 communication groups, which +are (GPU0, GPU8), (GPU1, GPU9) and so on. However, all +of them need to transport data via the only inter-node link +(i.e., red line in the figure). Since they are communicating +simultaneously, we need to divide the bandwidth by 8. Thus, +the effective bandwidth is 12.5/8 = 1.5625GB/s in this case. +Based on this observation, TAPS computes the number the +inter-node communication groups within a node for AllRe- +duce, AllGather, and AllToAll operators to obtain the effective +bandwidth for every communication group. TAPS computes +the communication costs by dividing communication volumes +by effective bandwidths. +Algorithm 2: Infer number of inter-node communica- +tion groups within a node for AllReduce +Data: Device Matrix D = (dh−1,dh−2,...,d0), Tensor +Map M = (m0,m1,...,ms−1), device number in a +node local_device_num +Result: Number of inter-node communication group ct +1 remain_devices ← local_device_num; +2 Total_devices ← product(D); +3 parallel_degree ← Total_devices; +4 device_in ← 1; +5 for k ← 0 to s−1 do +parallel_degree ← parallel_degree÷dmk; +6 for k ← 0 to h−1 do +7 +if not M.find(k) and remain_devices > 1 then +8 +if remain_devices > dk then +9 +device_in ← device_in×dk; +10 +else +11 +device_in ← remain_devices; +12 +remain_devices ← remain_devices÷dk; +13 end +14 if device_in ≥ parallel_degree then +15 +ct ← 0; +16 else if device_in > 1 then +17 +ct ← local_device_num÷device_in; +18 else +19 +ct ← local_device_num; +4.2.1. AllReduce For an arbitrary AllReduce operator, we +first compute the inter-communication times of its input tensor +using Algorithm 2. Algorithm 2 takes the device matrix, tensor +map of the communicated tensor, and the number of devices +in a node as inputs, then infer the number of communication +groups that need to do inter-node communication. +The inter-communication times indicate how many com- +munication groups do inter-node communications for a tensor +simultaneously. For example, suppose we are executing a Mat- +Mul operator with strategy (d,r,c) = (8,2,2) with different +device maps as shown in Figure 3. The same-color cubes are +devices within a node. In Figure 3(a), there are 4 different δW +5 + +GPU3 +GPU2 +GPU0 +GPU1 +GPU4 +GPU5 +GPU7 +GPU6 +GPU11 +GPU10 +GPU8 +GPU9 +GPU12 +GPU13 +GPU15 +GPU14 +50GB/s +12.5GB/s +GPU3 +GPU2 +GPU0 +GPU1 +GPU4 +GPU5 +GPU7 +GPU6 +GPU11 +GPU10 +GPU8 +GPU9 +GPU12 +GPU13 +GPU15 +GPU14 +GPU3 +GPU2 +GPU0 +GPU1 +GPU4 +GPU5 +GPU7 +GPU6 +GPU11 +GPU10 +GPU8 +GPU9 +GPU12 +GPU13 +GPU15 +GPU14 +(a) Topology of 2 DGX-V100 nodes +(b) Device Matrix(8, 2), Tensor Map(0, 1), +Communication axis: 0 +(c) Device Matrix(2, 8), Tensor Map(1, 0), +Communication axis: 1 +25GB/s +Figure 2: Intra-communication and Inter-communication on 2 DGX-V100 nodes +d +c +r +Strategy: (d, r, c) = (8, 2, 2) +d +c +r +Strategy: (d, r, c) = (8, 2, 2) +d +c +r +Strategy: (d, r, c) = (8, 2, 2) +Device Map: (2, 1, 0), Device Matrix: (8, 2, 2) +Device Map: (0, 1, 2), Device Matrix: (2, 2, 8) +Device Map: (1, 2, 0), Device Matrix: (2, 8, 2) +(a) +(b) +(c) +Devices in +Node 3 +Node 1 +Devices in +Node 4 +Node 2 +Devices in +Devices in +Figure 3: Device Distributions of Different Device Map and De- +vice Matrix +partitions within a node, and they all need to communicate +with other nodes. Thus, the inter-communication times ctδW +is 4 in this case. The tensor δW’s inter-communication times +ctδW are 4, 0, and 2 in Figure 3(a)(b)(c), respectively. +Using the result of ct, we can compute the effective band- +width Be: +Be = +� +Bintra +ctT = 0, +Binter/ctT +ctT > 0, +(11) +where Bintra is the intra-node communication bandwidth, and +Binter is the inter-node communication bandwidth. +Finally, The communication cost of an AllReduce operator +is: +CAR = VAR/Be. +(12) +4.2.2. AllGather Different from AllReduce, AllGather uses +Algorithm 3 to compute the inter-communication times of +AllGather. Algorithm3 can compute the repetitive degree r of +AllGather in a device node and infer how many devices of a +communication group are within a node. Using this informa- +tion, it then outputs the ct values of corresponding AllGather. +TAPS also uses Eq.11 to compute the Be for AllGather. The +communication of an AllGather operator is: +CAG = VAG/Be. +(13) +4.2.3. AllToAll Unlike AllReduce and AllGather, which uti- +lize ring topology to communicate, AllToAll uses peer-to-peer +(P2P) communication to exchange data within a communica- +tion group. While each node in AllReduce and AllGather has +only one send link and receive link, each node in AllToAll es- +tablishes p−1 send and receive links that connect other nodes +in the communication group, where p is the device number of +the AllToAll communication group. This may influence the +communication volume we use to compute communication +costs. Therefore, we need to recompute the communication +6 + +Algorithm 3: Infer number of inter-node communica- +tion groups within a node for AllGather +Data: Device Matrix D = (dh−1,dh−2,...,d0), Tensor +map M = (m0,m1,...,ms−1), Gather axis g, +device number in a node local_device_num. +Result: Inter communication times ct +1 remain_devices ← local_device_num; +2 parallel_degree ← dmg; +3 temp_device_num ← 1; +4 repeat_num ← 1; +5 for k ← 0 to g−1 do +6 +temp_device_num ← temp_device_num×dk; +7 +if not M.find(k) then +8 +repeat_num ← repeat_num×dk; +9 end +10 if repeat_num > local_device_num then +11 +repeat_num ← local_device_num; +12 if temp_device_num ≥ local_device_num then +13 +ct ← local_device_num÷repeat_num; +14 else +15 +remain_devices ← +local_device_num÷temp_device_num; +16 +if remain_devices ≥ parallel_degree then +17 +ct ← 0; +18 +else +19 +ct ← temp_device_num÷repeat_num; +volume for AllToAll. Suppose among p devices, k devices are +within a node, and the tensor size is TS. Then for any device +in a node, there are k −1 intra-node communication with vol- +ume TS/p, and p−k inter-node communication with volume +TS/p. These k devices accumulate established (p−k)k con- +nections to other nodes, each transport Ts/p volume of data. +Then for this AllToAll communication group, the inter-node +communication volume via inter-node link is k(p − k)TS/p. +Suppose there are l devices in a node. Then there are l/k +different AllToAll communication groups within a node. We +additionally suppose the repeat degree of them is r. The repet- +itive tensor slices could share the same communication results +by synchronizing within a node using high-bandwidth links. +Then in a node, ct = l/(kr) groups simultaneously uses the +inter-node bandwidth to communication, unless p equals k. +The values k and r can also be inferred using Algorithm 3. +Effective bandwidth Be is computed using Eq.11. Thus, the +communication cost of AllToAll is: +CA2A = +� +� +� +VA2A +Be +p = k, +k(p−k) +p−1 +VA2A +Be +p > k, +(14) +5. Auxiliary Graph +Auxiliary graph GA = (VA,EA) is an extension of computa- +tion graph G where each node va ∈ VA indicates a unique +Algorithm 4: GenerateAuxiliaryGraph +Data: Computation graph G = (V,E), device graph +D = (VD,ED) +Result: Auxiliary graph GA = (VA,EA) +1 VA ← /0,EA ← /0; +2 device_num ← |VD|; +3 for (u,w) ∈ E do +4 +UA = GenerateStrategySet(u, device_num); +5 +WA = GenerateStrategySet(w, device_num); +6 +VA = VA ∪UA ∪WA ; +7 +for ua ∈ UA do +8 +for wa ∈ WA do +9 +EA = EA ∪{(ua,wa)} +10 +end +11 +end +12 end +strategy of its original vertex v ∈ V. We use Algorithm 4 to +generate auxiliary graph as Figure 4(a)(b) shows. For each +va ∈ VA, we label it with the original operator and a unique +strategy. The function GenerateStrategySet in algorithm 4 +enumerates all possible strategies of the input operator and +creates corresponding auxiliary nodes for them. More specifi- +cally, GenerateStrategySet will generate ∑ +min(p,n) +i=1 +i! +�p +i +��n−1 +i−1 +� +different parallelism strategies when there is p different parti- +tionable axes in the operator and the operator is held by N = 2n +devices. A parallelism strategy of an operator consists of the +parallelism degree and mapping of each axis. We can use +the parallelism degrees to determine the partitions of each in- +volved tensor of the operator and place them to corresponding +devices according to the mappings. For example, suppose a +matrix multiplication (MatMul) operator that does computa- +tion Y = XW can be partitioned along three axes: b axis, in +axis, and out axis; The unpartitioned shapes of X, W, and Y +are (b,in), (in,out), (b,out), respectively. Table 3 shows the +strategy set GenerateStrategySet generates when it takes the +above MatMul operator and a device number of 4 as inputs. +Taking u2 in Table 3 for illustration, number 2 in out axis +represent tensor W and Y are sliced along out axis into 2 parts; +device map (−1,1,0) indicates the mapping value of b, in, +and out axis are -1, 1, and 0, respectively. -1 here represents +tensors replicated along the b axis. 0 here indicates that the +out axis is partitioned most-innerly in clusters, which may +have a high bandwidth when communicating. Device matrix +(1,2,2) is calculated by the parallelism degree of each axis +and the device map, and it is a hierarchically logical topology +of devices. +After creating the vertices and edges of the auxiliary +graph, we then compute the communication cost Cuawa of +all ea = (ua,wa) ∈ EA using our topology-aware cost model. +The weight of ea equals the intra-operator cost of wa plus +inter-operator cost between ua and wa. +Then, we can search strategies by selecting vertices and +7 + +Input +A +B +C +Output +（a) Original Computation Graph +Input +Output +A1 A2 +AKa +B1 B2 +BKb +C1 C2 +CKc +（b) Auxiliary Computation Graph +GenerateAuxiliaryGraph +Input +Output +A1 A2 +AKa +B1 B2 +BKb +C1 C2 +CKc +（c) Strategy Decision Example +Searching Strategies +Figure 4: Auxiliary Computation Graph +edges in the auxiliary graph. Figure 4 shows an example of +the search result, where blue vertices are selected strategies. +6. Searching Strategies by ILP +We formalize the strategy searching problem as an ILP prob- +lem as below shows: +min +∑ +(i,j)∈EA +BijCi j +(15) +s.t. ∑ +va∈VA +Xva = 1, +∀v ∈ V +(16) +∑ +(i,va)∈EA +Biva = Xva ×in_degree(v) +∑ +(va,k)∈EA +Bvak = Xva ×out_degree(v),∀va ∈ VA +(17) +∑ +(i,j)∈EA +BijMi j < Device_Memory, +(18) +Bij,Xk ∈ {0,1}, +∀(i, j) ∈ EA,∀k ∈ VA +(19) +where Xva, Bij are to-be-solved bool values that indicates the +selection of the vertex va ∈VA and edge (i, j) ∈ EA. Cij and Mij +are the communication and memory costs of edge (i, j) ∈ EA. +Equation 16 informs the solver that we only select one strategy +for all v ∈ V. Equation 17 limits any va ∈ VA to have the same +indegree and outdegree as their original vertex v ∈V. To avoid +selecting multiple strategies for v ∈ V, we set the indegree and +outdegree of va ∈ VA to zero if it is not selected. Equation +18 limits the solver to produce overall strategies that do not +exceed device memory. +Instead of dynamic programming, we use integer linear +programming for two reasons. First, dynamic programming +methods like [7, 21] cannot capture the overall memory cost +during processing, which might generate strategies that exceed +memory constraints. Although methods like [4] maintain a +communication-memory-cost bound to avoid this drawback, +its computation complexity is unacceptable while generating +strategies for large-scale models. Second, we can directly +use a high-performance third-party solver to solve the ILP +problem, which saves our time from optimizing the solver +runtime. +7. Evaluation +We evaluate TAPS by comparing the communication costs of +strategies generated by volume-based searching and topology- +aware searching. In our evaluation, we assume the intra- +bandwidth equals 60GB/s and the inter-bandwidth equals +6GB/s. These two results are the peak bandwidth we get after +testing on two 8-V100 nodes using nccl-tests[1]. Addition- +ally, we assume that all the communication can fully utilize +the bandwidth and that we are running in a homogeneous +environment. +7.1. Searching Runtime +We test the searching runtime on searching strategies for +AlexNet[8] and Megatron-LM[14, 17]. Note that the main +body of transformer-based networks consists of several lay- +ers with the same structures. Given that the same structures +always have the same strategies when the devices they use are +homogeneous, we only search strategies for one transformer +layer of the networks. The applied solver can solve strategies +within a few seconds using a 16-core 3.2GHz Intel i9-12900K +CPU. In our searching runtime experiment, we suppose each +node has 8 devices. +Table 4 shows some examples of running time of solving +strategies, where VD is the total number of devices, and |EA| is +the number of auxiliary edges. The search time is irrelevant to +the number of a model’s parameters. Instead, it is relevant to +the number of a model’s operators and the total device number. +For example, the transformer layer of Megatron-LM 1.7B and +3.6B has the same structure but different parameter numbers. +We follow the configurations in [14], searching intra-operator +strategies for 1.7B, 3.6B both on overall 32 devices. As Table +4 shows, the |EA| and their time remain in the same order of +magnitude. +8 + +Table 3: Auxiliary nodes of a MatMul operator (Y = XW) partitioned on 4 devices generated by GenerateStrategySet +Node +b axis +in axis +out axis +X shape +W shape +Y shape +device map +device matrix +u1 +1 +1 +4 +(b,in) +(in,out/4) +(b,out/4) +(-1, -1, 0) +(1, 1, 4) +u2 +1 +2 +2 +(b,in/2) +(in/2,out/2) +(b,out/2) +(-1, 1, 0) +(1, 2, 2) +u3 +1 +2 +2 +(b,in/2) +(in/2,out/2) +(b,out/2) +(-1, 0, 1) +(1, 2, 2) +u4 +1 +4 +1 +(b,in/4) +(in,out/4) +(b,out) +(-1, 0, -1) +(1, 4, 1) +u5 +2 +1 +2 +(b/2,in) +(in,out/2) +(b/2,out/2) +(-1, 1, 0) +(2, 1, 2) +u6 +2 +1 +2 +(b/2,in) +(in,out/2) +(b/2,out/2) +(-1, 0, 1) +(2, 1, 2) +u7 +2 +2 +1 +(b/2,in/2) +(in/2,out) +(b/2,out) +(1, 0, -1) +(2, 2, 1) +u8 +2 +2 +1 +(b/2,in/2) +(in/2,out) +(b/2,out) +(0, 1, -1) +(2, 2, 1) +u9 +4 +1 +1 +(b/4,in) +(in,out) +(b/4,out) +(0, -1, -1) +(4, 1, 1) +#nodes x #local_devices +Communication Cost / ms +#nodes x #local_devices +#nodes x #local_devices +(a) AlexNet +(b) Megatron-LM 1.7B +(c) Megatron-LM 3.6B +Figure 5: Comparison of Topology-Aware and Volume-Based Searching on Different Models +Figure 6: Ratios of the Topology-Aware and Volume-Based +Searching +Table 4: Strategy Searching Time of Solver +Model +|VD| +|EA| +Time +AlexNet +8 +> 3×103 +< 0.1s +AlexNet +16 +> 104 +< 0.2s +AlexNet +64 +> 5×104 +< 0.8s +Megatron-LM 1.7B +32 +> 3×104 +< 0.4s +Megatron-LM 3.6B +32 +> 3×104 +< 0.4s +Megatron-LM 1T +8 +> 4×103 +< 0.1s +7.2. Comparison with Volume-Based Searching +We compare the communication cost between strategies that +volume-based searching and topology-aware searching solve. +In our experiment, we search strategies for the convolution net- +work AlexNet, and transformer-based networks Megatron-LM. +We do the volume-based searching by replacing the commu- +nication costs of the auxiliary edges with their corresponding +communication volumes, after which we use the same solver +to search for the strategies. Then we compute the communica- +tion costs of generated volume-based searching results using +our topology-aware cost model. The comparison results are +9 + +2.5 - +Volume-based +Topology-aware +2.0 +1.5 +1.0 +0.23) +0.5 +0.34x +0.15x +1.0x +0.0 +2×4 +1×8 +2×8 +4×8Volume-based +600 +Topology-aware +400 +200 +0.55x +0.26x +0.44x +0.45x +0 +2×8 +8×4 +4×8 +8×8Volume-based +600 +Topology-aware +400 +0.42x +0.53x +200 +0.62x +0.6x +0 +2×8 +8×4 +4×8 +8×8shown in Figure 5, where blue bars are the communication +costs of volume-based searching results, and orange bars are +the communication costs of topology-aware searching results. +We take AlexNet, Megatron-LM 1.7B, and 3.6B as examples. +As we can see in 5(a), when there is only one node, the com- +munication cost of two different search results will be the +same. This is intuitive since no inter-node communication ex- +ists in this case. Our experiments show that TAPS can always +find strategies that outperform those volume-based searching +solve out. In the case of searching strategies for AlexNet on +two 8-device nodes, it even reduces the communication cost +by 85%. Additionally, we merge all experiments we run into +Figure 6, where each point represents a search of a model +under a specific device topology. The x-axis represents the +ratio of topology-aware communication volume and volume- +based communication cost; The y-axis represents the ratio of +topology-aware communication cost and volume-based com- +munication cost. As we can see, all points in the graph lie on or +below the line y = 1, which means that topology-aware search- +ing can always find strategies with smaller communication +costs than volume-based searching. Moreover, topology-aware +searching reduces the communication cost by more than 20% +in most cases. +8. Related Work and Discussion +Pipeline Parallelism. auto parallleism methods like Alpa, +Chimera[12] and PipeDream[13] can generate pipeline paral- +lelism strategies that balance the stages on different devices. +The searching space of TAPS is orthogonal to pipeline par- +allelism. Thus we can use TAPS to search intra-operator +parallelism strategies for each stage of the pipeline. +Multi-dimensional Tensor Parallelism. 2D-TP[22], 3D- +TP[2] from Colossal-AI[11] generate intra-operator strategy +heuristically. TAPS currently does not support 2D-TP because +2D-TP uses Broadcast and Reduce to finish the communi- +cation, while we use AllGather, AllToAll, and Slice instead. +TAPS naturally includes strategies of 3D-TP because +Overlapping Communication and Computation. In our +implementation, we assume that the communication cannot +overlap with computation; thus, we can ignore the computa- +tion costs. However, in actual training cases, researchers[6] +delegate to overlap the computation and communication. It +is hard for us to be aware of the overlap degree. A trade-off +solution is manually setting the overlap degree for communica- +tions of different dimensions. For example, in some cases, the +communication of data parallelism can be fully overlapped, +then we can set the overlap degree to 1. +Estimating the Costs using regression models. Although +we assume the bandwidth can be fully utilized, we notice that +the effective bandwidth is very low when the size of transferred +data is small. This is because, during communication, there are +overheads like creating connections and computing average +values. Using regression models to simulate the variations of +effective bandwidth is a good choice to improve TAPS further. +9. Conclusion +We present TAPS, a topology-aware intra-operator parallelism +strategy searching algorithm that generates fine-grained intra- +operator strategies for multi-node environments. TAPS can +generate tensor redistribution operations with fewer communi- +cation costs heuristically. TAPS calculates the communication +costs of each strategy according to communication volume and +effective bandwidth, thus producing more reasonable strate- +gies compared to methods that only consider communication +volume. Based on the communication costs, TAPS formal- +izes the searching problem as an integer linear programming +problem by creating and utilizing an auxiliary graph and then +solving the result within a few seconds. Compared to volume- +based searching algorithms, TAPS can generate strategies with +up to 85% fewer communication costs for cases in multi-node +environment. The source code of TAPS will be publicly avail- +able. +References +[1] Accessed:2022-10-20. 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April 2021. +arXiv:2104.05343. +[23] Lianmin Zheng, Zhuohan Li, Hao Zhang, Yonghao Zhuang, Zhifeng +Chen, Yanping Huang, Yida Wang, Yuanzhong Xu, Danyang Zhuo, +Joseph E. Gonzalez, Ion Stoica, and Eric P. Xing. Alpa: Automat- +ing inter- and intra-operator parallelism for distributed deep learning. +January 2022. arXiv:2201.12023. +11 + diff --git a/RtE3T4oBgHgl3EQfDQmV/content/tmp_files/load_file.txt b/RtE3T4oBgHgl3EQfDQmV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..02424dc17930a8f21a9300ffbaa6c6ede7bd05f1 --- /dev/null +++ b/RtE3T4oBgHgl3EQfDQmV/content/tmp_files/load_file.txt @@ -0,0 +1,875 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf,len=874 +page_content='TAPS: Topology-Aware Intra-Operator Parallelism Strategy Searching Algorithm for Deep Neural Networks Peng Liang, Hao Zheng, *Teng Su, Linbo Qiao, Dongsheng Li National University of Defense Technology, *Huawei Technologies Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Ltd {peng_leung, zhengh, qiao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='linbo, dsli}@nudt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='cn suteng@huawei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='com * Abstract TAPS is a Topology-Aware intra-operator Parallelism strat- egy Searching algorithm that generates intra-operator paral- lelism strategies by considering both intra-node and inter-node bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Most of the existing auto-parallelism works use the communication volume as the communication cost directly when generating strategies, which we prove to be sub-optimal in multi-nodes cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' We design a topology-aware cost model for multi-node intra-operator parallelism strategy searching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Numerical experiments demonstrate that TAPS can generate strategies with up to 85% fewer communication costs, which outperform the latest baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Introduction Large-scale Deep Learning (DL) models have been a huge hot topic in recent years for their great performance improvements in fields like [3, 9, 16], which is a result of scaling up model sizes and dataset sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' For example, PaLM with 540 billion parameters is trained with a corpus of 780 billion tokens that represent a wide range of natural language use cases [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' As the model size significantly increases, training mod- els with a single device or even within a node is no longer practical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Thus, researchers use distributed deep learning to train these models [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Manual strategies like [17] have been widely used in training transformer-based models for their good performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' However, it is often not optimal because optimal parallelism strategies vary when the model or training environment changes, in which case researchers and engineers may need to redesign strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' To relieve us from the parallelism design procedure, re- searchers propose auto-parallelism algorithms [4, 7, 23] that can find decent strategies given a specific model and envi- ronment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' These algorithms first model parallelism strategies’ communication costs and then use a dynamic programming or an integer linear programming (ILP) method to find the optimal strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' As model size grows larger, a single node can no longer hold an entire large-scale model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Thus, using multi-nodes to train a model becomes necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Our key observation is that in a multi-node environment, the bandwidth within a node (intra-node bandwidth) and across nodes (inter-node band- width) are different, and the intra-node bandwidth is much A preprint version, change at any time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' higher than inter-node bandwidth in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' However, existing searching algorithms model the communication cost using the communication volume directly, ignoring the dif- ference between the bandwidths and resulting in sub-optimal strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Based on this observation, we propose a topology- aware parallelism strategy searching algorithm called TAPS, which can capture the difference between intra-node and inter- node communication and thus generates better parallelism strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' We first construct a topology-aware cost model, which can determine the inter-node communication times as well as the topology-aware communication cost given a communication axis of a tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Then we formalize the strategy searching prob- lem as an integer linear programming problem, after which we use a third-party solver to solve the final strategy decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' In summary, we make the following contributions: We prove that the volume-based communication cost model is insufficient to generate optimal intra-operator parallelism strategy in multi-nodes cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' We provide a heuristic solution in optimizing tensor redistri- bution sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' We analyze the communication in multi-node environments and propose a topology-aware communication cost model, which can calculate more accurate communication costs of a parallelism strategy of an operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' We design and implement TAPS, a strategy-searching algo- rithm that works for distributed DL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' We numerically evaluate TAPS on several models of differ- ent configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' We compare TAPS with volume-based searching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Our experiments show that TAPS can find strate- gies with up to 85% fewer communication costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Background 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Existing Parallelism Methods Since Hinton [8] trained AlexNet using two GPUs in 2012, re- searchers have proposed many parallelism methods, including data parallelism (DP), model parallelism (MP), and pipeline parallelism(PP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Data Parallelism Data parallelism partition and dis- tribute the data across devices that has a replicated model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Each device computes the gradients using the split data and uses communication like AllReduce or Broadcast to synchro- nize the gradients or model parameters with other devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' So arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='04285v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='DC] 11 Jan 2023 that after every iteration, the models on all workers are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Model Parallelism Model parallelism partition the model parameters across devices and make devices process the same data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Model parallelism produces partial-sum or sliced results when the parameter matrix is partitioned row- wisely and column-wisely, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Row-wise MP (Row- MP) requires synchronization to unify the operator’s results on different devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Column-wise MP (Column-MP) does synchronization only in backward propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Pipeline Parallelism Pipeline parallelism partition op- erators in a model into several stages and let devices hold only one or a few of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Meanwhile, PP splits a mini-batch of data into several micro-batches and feeds them one by one into the first stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' When a stage finishes its computation, it sends the result to its next stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Different stages can be handled simultaneously;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' thus, PP forms a pipeline that can improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Intra- and Inter-Operator Paralleism Alpa [23] catalog existing parallelism methods into two orthog- onal categories: intra-operator and inter-operator parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Intra-operator parallelisms are parallelism schemes that par- tition an operator’s involved tensors along some dimensions, assign the resulting partitioned computation to multiple de- vices, and let them execute different parts of the computation simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' From this view, we can treat data parallelism as a scheme that partitions an operator’s input and output tensor along the batch-size axis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' we can treat Row-MP as a scheme that partitions an operator’s input and weight ten- sor along the channel-in axis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' we can treat Column-MP as a scheme that partitions weight tensor and output tensor along the channel-out axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Inter-operator parallelism, including pipeline parallelism, partitions models into several stages with multiple operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' This paper focuses on generating multi-dimensional intra- operator parallelism strategies in multi-node environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Strategy Searching Algorithm Researchers have proposed methods to search parallelism strategies automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' ToFu[21], TensorOpt[4], and Alpa[23] generate intra-operator parallelism strategies by mini- mizing the overall communication cost of a computation graph under the observation that all different strategies of an oper- ator have the same computation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' ToFu and TensorOpt adapt the dynamic programming algorithm that OptCNN[7] propose to produce better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Alpa formalizes the search- ing problem as an integer programming problem and uses a solver to handle the solution progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' However, they assume the bandwidths of clusters are equal everywhere, ignoring the difference between the intra-node bandwidth and inter-node bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' This assumption may limit the searching algorithm to find the optimal strategies, as, in large-scale clusters, intra- node bandwidth is much higher than inter-node bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' In this paper, we propose a topology-aware communication cost model aware of the intra-node and inter-node bandwidth, which helps generate more fine-grained strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Overview TAPS is an algorithm that generates intra-operator parallelism strategies by minimizing the communication cost of the com- putation graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' TAPS takes a computation graph G = (V,E) and device graph D = (VD,ED) as inputs, and output a partition set P, which consists of strategy decisions of every operator vi ∈ V in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' The computation graph contains operator infor- mation, like shapes and operator types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' The device graph indicates the device types and the bandwidth between devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' TAPS gives a solution in two steps: First, TAPS creates an auxiliary graph where each node indicates an operator with a specific strategy and computes the weights for each edge (u,v) in the auxiliary graph, which equals the intra-operator com- munication cost of v plus tensor redistribution communication cost between u and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Then, TAPS formalizes the searching problem as an integer linear programming problem using the information in the auxiliary graph and uses a third-party solver to solve the optimal strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Communication Cost Model In this section, we give the details of our topology-aware com- munication cost model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' We first illustrate the details of the volume-based cost model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Based on the volume-based cost model, we calculate the corresponding topology-aware com- munication cost using the volumes and effective bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Volume-based cost model Previous works [4, 18, 20] model the communication cost of each strategy by symbolically computing the their communi- cation volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' The communication volume of an operator consists of intra-operator communication and inter-operator communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Intra-operator communication reduces the partial sums generated in computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Inter-operator commu- nication transforms tensor to fit the succeeding operator’s strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Intra-operator communication Taking MatMul as an example, its forward computation is shown as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='1, and its backward computation is shown as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='2 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Y = XW (1) δW = XTEy (2) Ex = EyW T (3) Let d, r, c denote the data parallelism (DP)[10], Row-MP, and Column-MP [17] degrees of a MatMul operator, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' p = drc denotes the total device number and is the power of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 2 Then we split the X and W matrices like: X = � ����� X11 X12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' X1r X21 X22 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Xd1 Xd2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Xdr � ����� , W = � ����� W11 W12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' W1c W21 W22 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Wr1 Wr2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Wrc � ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' After splitting the matrices X and W, we distribute their sub-blocks to corresponding devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' As Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' (a) shows, where each cube represents a device, each sub-block of X is replicated along axis c, and W is replicated along axis d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' As Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' (b)(c) shows, we then compute the local results of Y on each device and communicate them to form the final matrix Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' The communication is a reduction operation of local results and is mathematically equivalent to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Yi j = r ∑ k=1 XikWk j (4) Final matrix Y is split like: Y = � ����� Y11 Y12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Y1c Y21 Y22 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Yd1 Yd2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Ydc � ����� , where each sub-block Yi j is replicated along d axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Suppose we are using a bandwidth optimal Ring-AllReduce algorithm [15], the communication volume of a MatMul oper- ator accumulating results of Y on each device (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=', the volume of Row-MP) is: VY AR = 2(device_num−1) device_num data_size = 2(r −1)b out drc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' (5) Similarly, we give the communication volume of DP and Column-MP in a Matmul operator by computing the com- munication volume of acuumulating results of δW and EX, respectively, which are: V δW AR = 2(d −1)in out drc , (6) V EX AR = 2(c−1)b in drc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' (7) Finally, the overall communication volume of a MatMul oper- ator is: Volume = 2((d −1)in out +(r −1)b out +(c−1)b in) drc (8) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Inter-operator communication Inter-operator com- munication happens when there are tensor redistributions be- tween two operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Tensor redistributions are sequences that consist of several redistribution operators like All-Gather, Slice, and All-To-All.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' In this subsection, we give our solution for generating proper redistribution operator sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Let Oout,Oin denote two operators and T denote the output n-dimensional tensor of Oout and the input tensor of Oin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' ST = [s0,s1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=',sn−1] is the shape of T before partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Suppose the depths of device matrix of Oout and Oin is hout and hin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' The device matrix in Oout and Oin are Dout = [dout,hout−1,dout,hout−2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=',dout,0] and Din = [din,hin−1,din,hin−2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=',din,0], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' The tensor maps of T in Oout and Oin are Min = [mout,0,mout,1,mout,n−1] and Mout = [min,0,min,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=',min,n−1], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' To do the tensor redistribution, the device matrices and tensor shapes of Oin and Oout must be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' We unify them by two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' In step 1, we unify device matrices by factorizing some dimen- sions in two device matrices, which may result in a shape inconsistency of T in two operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Thus in step 2, we need to unify the tensor shape under the unified device matrix addi- tionally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Note that the two-step unification does not change the physical distribution of a Tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Table 1 shows an example of unifying a 2-dimensional tensor between Oout and Oin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' In step 1, we factorize "8" in two device matrices and replace them by the factorizing results [4,2] and [2,4] for Oout and Oin, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Meanwhile, we must change the tensor maps and shapes as we modify device matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Since the tensor shapes change in step 1, we need to unify it again before we infer tensor redistribution operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' In step 2, we reshape the tensor in Oin and Oout to make them have the same shape and modify tensor maps simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' After unifying the device matrix and tensor shape, we can infer the redistribution operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' A naive way to do the re- distribution is to AllGather along all the workers and then partition along axes that are not repetitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' To reduce the communication cost, we use a heuristical algorithm 1 to gen- erate tensor redistribution operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Our algorithm contains three optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' First, we only AllGather along the nec- essary axes of the tensor, which are partitioned in Oout and replicated in Oin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Second, we rearrange the redistribution se- quence, putting dependent Slice before AllGather to reduce the communication volume that AllGather produces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Third, we re- place the implicit permutations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=', AllGather and Slice along the same axis in the device matrix) with AllToAll operators, thus further reducing the communication volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' In Algo- rithm 1, InferSlice finds all necessary Slice-Op and appends them to the operator sequence S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' If there is no more SliceOp, InferSlice sets S_ flag to False.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Similarly, InferAll2All and InferAllGather do the same things for AllToAllOp and All- GatherOp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Table 3 shows an example of using above men- tioned three optimizations to fine-tune the redistribution se- quence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Finally, we obtain the inter-operator communication volume of such tensor redistribution by accumulating the communi- cation volumes of redistribution operators within sequence S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Suppose we are using bandwidth optimal Ring-AllGather ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='X11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='X12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='X21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='X22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='X21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='X22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='X11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='X12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='Y11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='Y21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='Y11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='Y21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='X21W11 X22W21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='X11W11 X12W21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='Y11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='Y21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='Y11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='Y21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='(a) Distribution of Tensor X and W ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='(b) Local computation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='(c) Communicate along the r axis ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='W12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='W11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='W22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='W12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='W21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='X21W11 X22W21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='Y21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='Y21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='Y11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='Y11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='on each device ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='to form Tensor Y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='Figure 1: Multi dimensional Intra-P of a MatMul operator on 8 devices where d = r = c = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='Table 1: Unifying Device Matrix and Tensor Shape of Tensor T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='Step ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='Operator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='Device Matrix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='Tensor Map ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='Tensor Shape ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='0: Initial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='Oout ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='[2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='8] [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='0] [s0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='s1] Oin [8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='2] [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='0] [s0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='s1] 1: Unifying device matrix Oout [2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='2] [2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='0] [s0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='s1/4] Oin [2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='0] [2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='s0/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='s1] 2: Unifying tensor shape Oout [2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='2] [2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='0] [2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='s0/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='s1/4] Oin [2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1] Table 2: Tensor Redistribution Between Mfrom = [−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='3] and Mto = [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='3] Step Operation Tensor Map Communication Volume 0: Initial AllGather(d1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='1) [−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='3] d1d2d3−1 d1d2d3 Size(T) AllGather(d2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='2) [−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='3] AllGather(d3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='4) [−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1] Slice(d1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='0) [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1] Slice(d0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='3) [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1] Slice(d3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='4) [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='3] 1: Remove AllGather(d1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='1) [−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='3] d1d2−1 d1d2d3 Size(T) AllGather(d2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='2) [−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='3] Slice(d1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='0) [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='3] Slice(d0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='3) [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='3] 2: Rearrange Slice(d0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='3) [−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='3] d1+d2−2 d0d1d2d3 Size(T) AllGather(d1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='1) [−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='3] Slice(d1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='0) [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='3] AllGather(d2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='2) [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='3] 3: Replace Slice(d0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='3) [−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='3] d1d2−1 d0d2 1d2d3 Size(T) AllToAll(d1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='1 → 0) [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='3] AllGather(d2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='2) [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='3] 4 algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' the communication volume of AllGather is: VAG = (device_num−1)data_size;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' (9) For AllToAll operators, each device only needs to send 1 device_num different data to each other in the communication group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Thus, the communication volume of AllToAll is: VA2A = (device_num−1)data_size device_num .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' (10) As we can see in Table 2, the communication volume of the operator sequence that Algorithm 1 generates is much smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' We then blend the bandwidth difference into the volume- based cost model to form our topology-aware cost model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Algorithm 1: Optimized Tensor Redistribution Data: D = [dh−1,dh−2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=',d0];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Mfrom = [mfrom,0,mfrom,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=',m from,n−1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Mto = [mto,0,mto,1,mto,n−1] Result: Redistribution operator sequence S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 1 while Mfrom ̸= Mto do 2 S_ flag ← True;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 3 while S_flag do 4 S_ flag ← InferSlice(Mfrom,Mto,S);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 5 A2A_ flag ← True;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 6 while A2A_flag do 7 A2A_ flag ← InferAll2All(Mfrom,Mto,S);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 8 end 9 end 10 AG_flag ← InferAllGather(Mfrom,Mto,S);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 11 if AG_ flag == False then 12 AllGatherFirstUndoneDim(Mfrom,Mto,S);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 13 end 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Topology-aware cost model Based on volume-based cost model, we develop a topology- aware cost model that can additionally consider the bandwidth difference when calculating communicaiton costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' TAPS uses this topology-aware cost model to generate more fine-grained strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Our observation is that in multi-node environment, we can do multiple intra-node communications of different commu- nication groups simultaneously, and they can all fully utilize the bandwidth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' But for inter-node communications, they need to share the links between nodes, thus lowering the effec- tive bandwidth of each communication group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' (a) shows a 2 DGX-V100 nodes environment, where intra-node communication uses high-bandwidth NVLink and inter-node communication uses 100GBps InfiniBand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' In Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' (b), we do communication along the axis 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' There are 8 communica- tion groups, which are (GPU0, GPU1), (GPU2, GPU3) and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Each of them has an individual NVLink to use and thus the effective bandwidth equals the bandwidth of NVLink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' The case in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' (c) also has 8 communication groups, which are (GPU0, GPU8), (GPU1, GPU9) and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' However, all of them need to transport data via the only inter-node link (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=', red line in the figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Since they are communicating simultaneously, we need to divide the bandwidth by 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Thus, the effective bandwidth is 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='5/8 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='5625GB/s in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Based on this observation, TAPS computes the number the inter-node communication groups within a node for AllRe- duce, AllGather, and AllToAll operators to obtain the effective bandwidth for every communication group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' TAPS computes the communication costs by dividing communication volumes by effective bandwidths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Algorithm 2: Infer number of inter-node communica- tion groups within a node for AllReduce Data: Device Matrix D = (dh−1,dh−2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=',d0), Tensor Map M = (m0,m1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=',ms−1), device number in a node local_device_num Result: Number of inter-node communication group ct 1 remain_devices ← local_device_num;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 2 Total_devices ← product(D);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 3 parallel_degree ← Total_devices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 4 device_in ← 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 5 for k ← 0 to s−1 do parallel_degree ← parallel_degree÷dmk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 6 for k ← 0 to h−1 do 7 if not M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='find(k) and remain_devices > 1 then 8 if remain_devices > dk then 9 device_in ← device_in×dk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 10 else 11 device_in ← remain_devices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 12 remain_devices ← remain_devices÷dk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 13 end 14 if device_in ≥ parallel_degree then 15 ct ← 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 16 else if device_in > 1 then 17 ct ← local_device_num÷device_in;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 18 else 19 ct ← local_device_num;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' AllReduce For an arbitrary AllReduce operator, we first compute the inter-communication times of its input tensor using Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Algorithm 2 takes the device matrix, tensor map of the communicated tensor, and the number of devices in a node as inputs, then infer the number of communication groups that need to do inter-node communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' The inter-communication times indicate how many com- munication groups do inter-node communications for a tensor simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' For example, suppose we are executing a Mat- Mul operator with strategy (d,r,c) = (8,2,2) with different device maps as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' The same-color cubes are devices within a node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' In Figure 3(a), there are 4 different δW 5 GPU3 GPU2 GPU0 GPU1 GPU4 GPU5 GPU7 GPU6 GPU11 GPU10 GPU8 GPU9 GPU12 GPU13 GPU15 GPU14 50GB/s 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='5GB/s GPU3 GPU2 GPU0 GPU1 GPU4 GPU5 GPU7 GPU6 GPU11 GPU10 GPU8 GPU9 GPU12 GPU13 GPU15 GPU14 GPU3 GPU2 GPU0 GPU1 GPU4 GPU5 GPU7 GPU6 GPU11 GPU10 GPU8 GPU9 GPU12 GPU13 GPU15 GPU14 (a) Topology of 2 DGX-V100 nodes (b) Device Matrix(8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Tensor Map(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Communication axis: 0 (c) Device Matrix(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 8),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Tensor Map(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Communication axis: 1 25GB/s Figure 2: Intra-communication and Inter-communication on 2 DGX-V100 nodes d c r Strategy: (d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' c) = (8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 2) d c r Strategy: (d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' c) = (8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 2) d c r Strategy: (d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' c) = (8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 2) Device Map: (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Device Matrix: (8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 2) Device Map: (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Device Matrix: (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 8) Device Map: (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Device Matrix: (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 2) (a) (b) (c) Devices in Node 3 Node 1 Devices in Node 4 Node 2 Devices in Devices in Figure 3: Device Distributions of Different Device Map and De- vice Matrix partitions within a node,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' and they all need to communicate with other nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Thus, the inter-communication times ctδW is 4 in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' The tensor δW’s inter-communication times ctδW are 4, 0, and 2 in Figure 3(a)(b)(c), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Using the result of ct, we can compute the effective band- width Be: Be = � Bintra ctT = 0, Binter/ctT ctT > 0, (11) where Bintra is the intra-node communication bandwidth, and Binter is the inter-node communication bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Finally, The communication cost of an AllReduce operator is: CAR = VAR/Be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' (12) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' AllGather Different from AllReduce, AllGather uses Algorithm 3 to compute the inter-communication times of AllGather.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Algorithm3 can compute the repetitive degree r of AllGather in a device node and infer how many devices of a communication group are within a node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Using this informa- tion, it then outputs the ct values of corresponding AllGather.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' TAPS also uses Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='11 to compute the Be for AllGather.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' The communication of an AllGather operator is: CAG = VAG/Be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' (13) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' AllToAll Unlike AllReduce and AllGather, which uti- lize ring topology to communicate, AllToAll uses peer-to-peer (P2P) communication to exchange data within a communica- tion group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' While each node in AllReduce and AllGather has only one send link and receive link, each node in AllToAll es- tablishes p−1 send and receive links that connect other nodes in the communication group, where p is the device number of the AllToAll communication group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' This may influence the communication volume we use to compute communication costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Therefore, we need to recompute the communication 6 Algorithm 3: Infer number of inter-node communica- tion groups within a node for AllGather Data: Device Matrix D = (dh−1,dh−2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=',d0), Tensor map M = (m0,m1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=',ms−1), Gather axis g, device number in a node local_device_num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Result: Inter communication times ct 1 remain_devices ← local_device_num;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 2 parallel_degree ← dmg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 3 temp_device_num ← 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 4 repeat_num ← 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 5 for k ← 0 to g−1 do 6 temp_device_num ← temp_device_num×dk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 7 if not M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='find(k) then 8 repeat_num ← repeat_num×dk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 9 end 10 if repeat_num > local_device_num then 11 repeat_num ← local_device_num;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 12 if temp_device_num ≥ local_device_num then 13 ct ← local_device_num÷repeat_num;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 14 else 15 remain_devices ← local_device_num÷temp_device_num;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 16 if remain_devices ≥ parallel_degree then 17 ct ← 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 18 else 19 ct ← temp_device_num÷repeat_num;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' volume for AllToAll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Suppose among p devices, k devices are within a node, and the tensor size is TS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Then for any device in a node, there are k −1 intra-node communication with vol- ume TS/p, and p−k inter-node communication with volume TS/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' These k devices accumulate established (p−k)k con- nections to other nodes, each transport Ts/p volume of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Then for this AllToAll communication group, the inter-node communication volume via inter-node link is k(p − k)TS/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Suppose there are l devices in a node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Then there are l/k different AllToAll communication groups within a node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' We additionally suppose the repeat degree of them is r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' The repet- itive tensor slices could share the same communication results by synchronizing within a node using high-bandwidth links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Then in a node, ct = l/(kr) groups simultaneously uses the inter-node bandwidth to communication, unless p equals k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' The values k and r can also be inferred using Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Effective bandwidth Be is computed using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Thus, the communication cost of AllToAll is: CA2A = � � � VA2A Be p = k, k(p−k) p−1 VA2A Be p > k, (14) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Auxiliary Graph Auxiliary graph GA = (VA,EA) is an extension of computa- tion graph G where each node va ∈ VA indicates a unique Algorithm 4: GenerateAuxiliaryGraph Data: Computation graph G = (V,E), device graph D = (VD,ED) Result: Auxiliary graph GA = (VA,EA) 1 VA ← /0,EA ← /0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 2 device_num ← |VD|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 3 for (u,w) ∈ E do 4 UA = GenerateStrategySet(u, device_num);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 5 WA = GenerateStrategySet(w, device_num);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 6 VA = VA ∪UA ∪WA ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 7 for ua ∈ UA do 8 for wa ∈ WA do 9 EA = EA ∪{(ua,wa)} 10 end 11 end 12 end strategy of its original vertex v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' We use Algorithm 4 to generate auxiliary graph as Figure 4(a)(b) shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' For each va ∈ VA, we label it with the original operator and a unique strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' The function GenerateStrategySet in algorithm 4 enumerates all possible strategies of the input operator and creates corresponding auxiliary nodes for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' More specifi- cally, GenerateStrategySet will generate ∑ min(p,n) i=1 i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' �p i ��n−1 i−1 � different parallelism strategies when there is p different parti- tionable axes in the operator and the operator is held by N = 2n devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' A parallelism strategy of an operator consists of the parallelism degree and mapping of each axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' We can use the parallelism degrees to determine the partitions of each in- volved tensor of the operator and place them to corresponding devices according to the mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' For example, suppose a matrix multiplication (MatMul) operator that does computa- tion Y = XW can be partitioned along three axes: b axis, in axis, and out axis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' The unpartitioned shapes of X, W, and Y are (b,in), (in,out), (b,out), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Table 3 shows the strategy set GenerateStrategySet generates when it takes the above MatMul operator and a device number of 4 as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Taking u2 in Table 3 for illustration, number 2 in out axis represent tensor W and Y are sliced along out axis into 2 parts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' device map (−1,1,0) indicates the mapping value of b, in, and out axis are -1, 1, and 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' -1 here represents tensors replicated along the b axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 0 here indicates that the out axis is partitioned most-innerly in clusters, which may have a high bandwidth when communicating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Device matrix (1,2,2) is calculated by the parallelism degree of each axis and the device map, and it is a hierarchically logical topology of devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' After creating the vertices and edges of the auxiliary graph, we then compute the communication cost Cuawa of all ea = (ua,wa) ∈ EA using our topology-aware cost model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' The weight of ea equals the intra-operator cost of wa plus inter-operator cost between ua and wa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Then, we can search strategies by selecting vertices and 7 Input A B C Output （a) Original Computation Graph Input Output A1 A2 AKa B1 B2 BKb C1 C2 CKc （b) Auxiliary Computation Graph GenerateAuxiliaryGraph Input Output A1 A2 AKa B1 B2 BKb C1 C2 CKc （c) Strategy Decision Example Searching Strategies Figure 4: Auxiliary Computation Graph edges in the auxiliary graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Figure 4 shows an example of the search result, where blue vertices are selected strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Searching Strategies by ILP We formalize the strategy searching problem as an ILP prob- lem as below shows: min ∑ (i,j)∈EA BijCi j (15) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' ∑ va∈VA Xva = 1, ∀v ∈ V (16) ∑ (i,va)∈EA Biva = Xva ×in_degree(v) ∑ (va,k)∈EA Bvak = Xva ×out_degree(v),∀va ∈ VA (17) ∑ (i,j)∈EA BijMi j < Device_Memory, (18) Bij,Xk ∈ {0,1}, ∀(i, j) ∈ EA,∀k ∈ VA (19) where Xva, Bij are to-be-solved bool values that indicates the selection of the vertex va ∈VA and edge (i, j) ∈ EA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Cij and Mij are the communication and memory costs of edge (i, j) ∈ EA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Equation 16 informs the solver that we only select one strategy for all v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Equation 17 limits any va ∈ VA to have the same indegree and outdegree as their original vertex v ∈V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' To avoid selecting multiple strategies for v ∈ V, we set the indegree and outdegree of va ∈ VA to zero if it is not selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Equation 18 limits the solver to produce overall strategies that do not exceed device memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Instead of dynamic programming, we use integer linear programming for two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' First, dynamic programming methods like [7, 21] cannot capture the overall memory cost during processing, which might generate strategies that exceed memory constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Although methods like [4] maintain a communication-memory-cost bound to avoid this drawback, its computation complexity is unacceptable while generating strategies for large-scale models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Second, we can directly use a high-performance third-party solver to solve the ILP problem, which saves our time from optimizing the solver runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Evaluation We evaluate TAPS by comparing the communication costs of strategies generated by volume-based searching and topology- aware searching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' In our evaluation, we assume the intra- bandwidth equals 60GB/s and the inter-bandwidth equals 6GB/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' These two results are the peak bandwidth we get after testing on two 8-V100 nodes using nccl-tests[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Addition- ally, we assume that all the communication can fully utilize the bandwidth and that we are running in a homogeneous environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Searching Runtime We test the searching runtime on searching strategies for AlexNet[8] and Megatron-LM[14, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Note that the main body of transformer-based networks consists of several lay- ers with the same structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Given that the same structures always have the same strategies when the devices they use are homogeneous, we only search strategies for one transformer layer of the networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' The applied solver can solve strategies within a few seconds using a 16-core 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='2GHz Intel i9-12900K CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' In our searching runtime experiment, we suppose each node has 8 devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Table 4 shows some examples of running time of solving strategies, where VD is the total number of devices, and |EA| is the number of auxiliary edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' The search time is irrelevant to the number of a model’s parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Instead, it is relevant to the number of a model’s operators and the total device number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' For example, the transformer layer of Megatron-LM 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='7B and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='6B has the same structure but different parameter numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' We follow the configurations in [14], searching intra-operator strategies for 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='7B, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='6B both on overall 32 devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' As Table 4 shows, the |EA| and their time remain in the same order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 8 Table 3: Auxiliary nodes of a MatMul operator (Y = XW) partitioned on 4 devices generated by GenerateStrategySet Node b axis in axis out axis X shape W shape Y shape device map device matrix u1 1 1 4 (b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='in) (in,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='out/4) (b,' 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(b/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='out) (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' -1) (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 1) u9 4 1 1 (b/4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='in) (in,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='out) (b/4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='out) (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' -1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' -1) (4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 1) #nodes x #local_devices Communication Cost / ms #nodes x #local_devices #nodes x #local_devices (a) AlexNet (b) Megatron-LM 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='7B (c) Megatron-LM 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='6B Figure 5: Comparison of Topology-Aware and Volume-Based Searching on Different Models Figure 6: Ratios of the Topology-Aware and Volume-Based Searching Table 4: Strategy Searching Time of Solver Model |VD| |EA| Time AlexNet 8 > 3×103 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='1s AlexNet 16 > 104 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='2s AlexNet 64 > 5×104 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='8s Megatron-LM 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='7B 32 > 3×104 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='4s Megatron-LM 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='6B 32 > 3×104 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='4s Megatron-LM 1T 8 > 4×103 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='1s 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Comparison with Volume-Based Searching We compare the communication cost between strategies that volume-based searching and topology-aware searching solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' In our experiment, we search strategies for the convolution net- work AlexNet, and transformer-based networks Megatron-LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' We do the volume-based searching by replacing the commu- nication costs of the auxiliary edges with their corresponding communication volumes, after which we use the same solver to search for the strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Then we compute the communica- tion costs of generated volume-based searching results using our topology-aware cost model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' The comparison results are 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='5 - Volume-based Topology-aware 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='23) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='34x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='15x 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='0x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='0 2×4 1×8 2×8 4×8Volume-based 600 Topology-aware 400 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='55x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='26x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='44x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='45x 0 2×8 8×4 4×8 8×8Volume-based 600 Topology-aware 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='42x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='53x 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='62x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='6x 0 2×8 8×4 4×8 8×8shown in Figure 5, where blue bars are the communication costs of volume-based searching results, and orange bars are the communication costs of topology-aware searching results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' We take AlexNet, Megatron-LM 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='7B, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='6B as examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' As we can see in 5(a), when there is only one node, the com- munication cost of two different search results will be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' This is intuitive since no inter-node communication ex- ists in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Our experiments show that TAPS can always find strategies that outperform those volume-based searching solve out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' In the case of searching strategies for AlexNet on two 8-device nodes, it even reduces the communication cost by 85%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Additionally, we merge all experiments we run into Figure 6, where each point represents a search of a model under a specific device topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' The x-axis represents the ratio of topology-aware communication volume and volume- based communication cost;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' The y-axis represents the ratio of topology-aware communication cost and volume-based com- munication cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' As we can see, all points in the graph lie on or below the line y = 1, which means that topology-aware search- ing can always find strategies with smaller communication costs than volume-based searching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Moreover, topology-aware searching reduces the communication cost by more than 20% in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Related Work and Discussion Pipeline Parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' auto parallleism methods like Alpa, Chimera[12] and PipeDream[13] can generate pipeline paral- lelism strategies that balance the stages on different devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' The searching space of TAPS is orthogonal to pipeline par- allelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Thus we can use TAPS to search intra-operator parallelism strategies for each stage of the pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Multi-dimensional Tensor Parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 2D-TP[22], 3D- TP[2] from Colossal-AI[11] generate intra-operator strategy heuristically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' TAPS currently does not support 2D-TP because 2D-TP uses Broadcast and Reduce to finish the communi- cation, while we use AllGather, AllToAll, and Slice instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' TAPS naturally includes strategies of 3D-TP because Overlapping Communication and Computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' In our implementation, we assume that the communication cannot overlap with computation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' thus, we can ignore the computa- tion costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' However, in actual training cases, researchers[6] delegate to overlap the computation and communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' It is hard for us to be aware of the overlap degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' A trade-off solution is manually setting the overlap degree for communica- tions of different dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' For example, in some cases, the communication of data parallelism can be fully overlapped, then we can set the overlap degree to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Estimating the Costs using regression models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Although we assume the bandwidth can be fully utilized, we notice that the effective bandwidth is very low when the size of transferred data is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' This is because, during communication, there are overheads like creating connections and computing average values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Using regression models to simulate the variations of effective bandwidth is a good choice to improve TAPS further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Conclusion We present TAPS, a topology-aware intra-operator parallelism strategy searching algorithm that generates fine-grained intra- operator strategies for multi-node environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' TAPS can generate tensor redistribution operations with fewer communi- cation costs heuristically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' TAPS calculates the communication costs of each strategy according to communication volume and effective bandwidth, thus producing more reasonable strate- gies compared to methods that only consider communication volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Based on the communication costs, TAPS formal- izes the searching problem as an integer linear programming problem by creating and utilizing an auxiliary graph and then solving the result within a few seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Compared to volume- based searching algorithms, TAPS can generate strategies with up to 85% fewer communication costs for cases in multi-node environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' The source code of TAPS will be publicly avail- able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' References [1] Accessed:2022-10-20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' NVIDIA nccl-tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='com/ NVIDIA/nccl-tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' [2] Zhengda Bian, Qifan Xu, Boxiang Wang, and Yang You.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Sharan Narang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Jacob Devlin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Maarten Bosma,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Gaurav Mishra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Adam Roberts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Paul Barham,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Hyung Won Chung,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Charles Sutton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Sebastian Gehrmann,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Parker Schuh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Kensen Shi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Sasha Tsvyashchenko,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Joshua Maynez,' metadata={'source': 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of large and deep neural networks for parallelization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' In Leonel Sousa, Nuno Roma, and Pedro Tomás, editors, Euro-Par 2021: Parallel Processing, pages 201–216, Cham, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Springer Interna- tional Publishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' [21] Minjie Wang, Chien-chin Huang, and Jinyang Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' Supporting very large models using automatic dataflow graph partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' In Proceedings of the Fourteenth EuroSys Conference 2019, pages 1–17, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' [22] Qifan Xu, Shenggui Li, Chaoyu Gong, and Yang You.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' An efficient 2d method for training super-large deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' April 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='05343.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content='12023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} +page_content=' 11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE3T4oBgHgl3EQfDQmV/content/2301.04285v1.pdf'} diff --git a/S9FAT4oBgHgl3EQf2B5J/content/tmp_files/2301.08712v1.pdf.txt b/S9FAT4oBgHgl3EQf2B5J/content/tmp_files/2301.08712v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fdf4619ec6355fb8099cc8743f8572f2afde2dc4 --- /dev/null +++ b/S9FAT4oBgHgl3EQf2B5J/content/tmp_files/2301.08712v1.pdf.txt @@ -0,0 +1,680 @@ +1 + +An efficient and low-cost method to create high-density nitrogen-vacancy centers in CVD +diamond for sensing applications. +Prem Bahadur Karki,1 Rupak Timalsina,2 Mohammadjavad Dowran,2 Ayodimeji E. Aregbesola,1 +Abdelghani Laraoui,2,3* and Kapildeb Ambal1† +1Department of Mathematics, Statistics, and Physics, Wichita State University, 1845 Fairmount +St. Wichita, KS 67260, United States of America +2Department of Mechanical & Materials Engineering, University of Nebraska-Lincoln, 900 N +16th St. W342 NH. Lincoln, NE 68588, United States of America +3Department of Physics and Astronomy and the Nebraska Center for Materials and Nanoscience, +University of Nebraska-Lincoln, 855 N 16th St, Lincoln, Nebraska 68588, USA +*†Author to whom correspondence should be addressed, email: + *alaraoui2@unl.edu, †Kapildeb.ambal@wichita.edu + +I. +Abstract +The negatively charged Nitrogen-Vacancy (NV-) center in diamond is one of the most versatile +and robust quantum sensors suitable for quantum technologies, including magnetic field and +temperature sensors. For precision sensing applications, densely packed NV- centers within a small +volume are preferable due to benefiting from 1/√𝑁 sensitivity enhancement (N is the number of +sensing NV centers) and efficient excitation of NV centers. However, methods for quickly and +efficiently forming high concentrations of NV- centers are in development stage. We report an +efficient, low-cost method for creating high-density NV- centers production from a relatively low +nitrogen concentration based on high-energy photons from Ar+ plasma. This study was done on +type-IIa, single crystal, CVD-grown diamond substrates with an as-grown nitrogen concentration +of 1 ppm. We estimate an NV- density of ~ 0.57 ppm (57%) distributed homogeneously over 200 +µm deep from the diamond surface facing the plasma source based on optically detected magnetic +resonance and fluorescence confocal microscopy measurements. The created NV-s have a spin- +lattice relaxation time (T1) of 5 ms and a spin-spin coherence time (T2) of 4 µs. We measure a DC +magnetic field sensitivity of ~ 104 nT Hz-1/2, an AC magnetic field sensitivity of ~ 0.12 pT Hz-1/2, +and demonstrate real-time magnetic field sensing at a rate over 10 mT s-1 using an active sample +volume of 0.2 µm3. + +II. +Introduction +The negatively charged nitrogen-vacancy (NV-) centers are one of the leading solid state-based +quantum platforms [1], enabling diverse quantum applications, including spin-qubit for quantum +information processing [2–5], quantum memory [6–8], and quantum sensing [9] for many +physical +quantities +including +magnetic +fields [10–17], +electric +fields [18–21], +and +temperature [22–24]. For potential commercial sensing applications where nanoscale spatial +resolution is not required, ensemble NV-s are preferred. The key requirements for high-sensitive +ensemble-based sensing applications are high NV- concentrations within small volumes while +preserving long spin coherence time. The high concentrations are favored for precision +magnetometry due to improved signal-to-noise and sensitivity from 1/√N enhancement [9,25]. +For example, the sensitivity to detect a constant magnetic field with NV- centers scales with +1/√𝑁𝑁𝑉 where NNV is the number of NV- centers from which the fluorescence signal is + +2 + +collected [26]. The densely packed or small sample volume is necessary to efficiently excite NV- +centers using on-chip laser and microwave excitation [27]. + +Several existing techniques are available for creating high-density NV- centers > 10 ppm (1 +ppm = 1.76 × 1017 cm−3 in diamond) [28]. The most popular ones are; (1) implanting high doses +(~100 ppm) of nitrogen (N) with subsequent annealing [29–31] and (2) electron irradiation or +implantation of He+ ions on a diamond substrate containing high-concentration of nitrogen (N) +impurities (~100 ppm) followed by subsequent annealing [32,33]. Nevertheless, the final +concentration of NV-s still depends on many factors like annealing temperature, cleaning +procedures, substrate characteristics, and the N to NV- conversion yield (typically <10%) [34]. +Recently, progress was made to increase the production yield by more than 30% by optimizing the +process (e.g., the electron irradiation dose) [35]. However, >70% of the implanted/doped nitrogen +atoms are present in the substrate as substitutional nitrogen, known as P1 paramagnetic +centers [36,37] or neutral nitrogen-vacancy (NV0) centers [1]. These excesses of nitrogen +unavoidably lead to decreased NV- center spin coherence properties by orders of magnitude [36– +38], thus degrading the sensitivity [25]. Also, a large implantation or electron irradiation dose +creates undesired defects and local graphitization of the diamond crystal, which could be beyond +repair via the thermal annealing process [39,40]. These unwanted defects decrease spin coherence +times 𝑇2 +∗ and T2 that impact the overall sensitivity. Therefore, developing low-cost and highly +efficient methods in creating high-density NV- centers from low-concentration nitrogen in the +diamond crystal is essential to increase the number of NV- centers for many scientific and +commercial applications such as the fabrication of brighter nanodiamonds for biomedical +applications [12,22,33,41]. + +This work focuses on a quick and cost-effective method of creating high-density NV- centers +in type-IIa chemical vapor deposition (CVD)-grown diamond substrates with as-grown nitrogen +concentration of 1 ppm by using high-energy photons from Ar+ plasma. Based on optically +detected magnetic resonance (ODMR) and fluorescence confocal microscopy measurements, we +estimate the density of newly created NV- ~ 0.57 ppm (> 50% of conversion yield), distributed +homogeneously over 200 µm deep from the diamond surface facing the plasma source. The created +NV- centers exhibit a spin-lattice relaxation time (T1) of 5±0.2 ms and a spin-spin coherence time +(T2) of 4±0.5 µs. We estimate a shot noise-limited DC magnetic field sensitivity of ~104 nT Hz-1/2 +an AC magnetic field sensitivity of ~ 0.12 pT Hz-1/2 respectively over an active sample volume of +0.2 µm3, and demonstrate a real-time AC magnetic field sensing with a frequency of up to 90 Hz +using the same active sample volume. +III. +Experimental methods and discussion +The NV center is a solid-state spin sensor in a diamond crystal formed by substituting a carbon +atom with a nitrogen atom and a vacancy adjacent to it [1] (inset of Fig. 1b). This type of nitrogen- +vacancy center is known as the neutral NV0 center. If the neutral NV center captures an extra +electron, it forms a negatively charged NV- center that can be photoionized to NV0 with laser or +voltage excitations [42,43]. The NV- center, a spin-1 system, is used for emerging quantum +applications, including quantum sensing [9]. The diamond substrates used for this work are single +crystal CVD-grown type-IIa diamonds (element six part# 145-500-0055) doped with ~ 1ppm (1.76 +× 1017 cm−3) of nitrogen (N) throughout the substrate during the growth process. The substrates +(labeled 1 and 2) were first cleaned in a tri-acid mixture (1 H2SO4: 1 HNO3: 1 HCLO4) [44] for +two hours at 200 oC to remove polishing-related graphitization, followed by rinsing with deionized +(DI) water or Isopropyl Alcohol (IPA) and finally drying using compressed nitrogen gas. The dry + +3 + +substrates were then exposed to Argon (Ar+) plasma by using Trion Minilock-Phantom III +Inductively Coupled Plasma (ICP) Reactive Ion Etching (RIE) system for 30 s (Fig. 1(a)). The +plasma process was performed with ICP power at 200 W, RIE power at 50 W, Ar gas flow at 5 +sccm, and the ICP-RIE chamber pressure at 10 mTorr. After the plasma exposure, the diamonds +were cleaned with the tri-acid mixture for two hours at 200 oC. The dry substrates were then +annealed under vacuum (10-8 Torr) at 1100 oC for two hours with another subsequent triacid +cleaning [31,41]. The clean and dry substrates were then used for optical and spin characterization. +We discuss below the mechanisms of NV- creation using high-energy photons from Ar+ plasma. + + +Figure 1. (a) Representation of the process for making a high-density NV- layer using Ar+ plasma. (b) +schematic of the custom-made confocal optical microscope used to characterize the high-density NV- +centers. Inset of (b) an NV center in a diamond lattice. The distribution of the created NV- centers as a +function of the depth from the side facing the plasma measured on 0.25 mm (c, substrate 1) and 0.5 mm (d, +substrate 2) thick diamonds prepared with the same conditions. + + +The NV- center is a deep bandgap defect located near the mid-gap within the 5.6 eV bandgap +of the diamond crystal [1]. NV- formation needs three processes: substitutional nitrogen atom, +adjutant vacancy, and electron capture. There are two well-established NV- creation methods: (1) +implantation of nitrogen (14N or 15N) ions followed by high-temperature annealing [45], and (2) +doping the diamond crystal with N during the growth process and create vacancies by helium ion +(He+) implantation [32,33] or electron irradiation [34,46] followed by subsequent high- + +(a) +Fluorescence (m V) +40 +Background +Plasma +Side +diamond +0 +0.0 +0.2 +Fiber +Depth (mm) +APD +(b) +(d) +Lens +White Light +20 +Source +Laser +Fluorescence +Back +round +Plasma +10 +Side +Camera +C +Objective +0.0 +0.2 +C +Depth (mm) +Diamond4 + +temperature annealing. In both methods, the implantation/irradiation of N/He ions creates a high +density of vacancies due to the broken bonds/dangling bonds. During the high-temperature +annealing process, these vacancies position themselves next to substitutional nitrogen atoms +forming the NV0 center [1]. When NV0 captures an extra electron, they become NV-. Since the +diamond has a large bandgap and NV centers are located near the middle of the bandgap, the +thermal energy is insufficient to excite electrons from the valence band to the conduction band to +be captured. These difficulties lead to poor yield in N-to-NV- formation. However, electron +irradiation creates broken bonds along with extra electrons, improving NV- formation +efficiency [34,45]. + +High energy and above bandgap photons generated from the Ar+ plasma were used to create +dangling bonds in SiO2 [47–49]. The energy of Ar+ in the plasma is very low, and the depth of the +created NV- center is more than 200 µm from the surface facing the plasma. Therefore, it is +conclusive that the enhancement of NV- formation is primarily due to high-energy photons from +Ar+ plasma [50–52]. We hypothesize that the high energy and above bandgap photons from Ar+ +plasma could have two effects: (i) Direct absorption of UV photons inducing vacancies such as +broken bonds/dangling bonds [53]. (ii) When the diamond absorbs a UV photon, it creates a +shower of electrons (because the photon energy is much larger than the diamond band gap) [54]. +These electrons could travel through the conduction band of the diamond and recapture. Therefore, +we assume that the high-energy photons would have similar effects as electron irradiation, creating +vacancies and supplying electrons for recapturing. The formation of dangling bonds will create +more NV centers from doped N0/P1 centers, similar to electron irradiation [34,46]. The excited +extra electrons in the conduction band could be captured to form NV- centers [54]. +Optical Characterization. +The fluorescence properties of the diamond substrates were investigated using a custom-built +confocal fluorescence microscope [14,55] (Fig. 1b) consisting of a green laser (532 nm) for optical +excitation of NV- centers, a permanent magnet (up to 100 mT) to apply a magnetic field for ODMR +measurements, a 100x microscope objective with a numerical aperture of 0.8 NA to focus light on +the diamond substrate, and fluorescence collection optics including 650 nm edge pass filter, +focusing lens, 9/125 single-mode fiber, single photon counter modules (SPCM) used for confocal +imaging and spin measurements in Fig. 2 and Fig. 3, and avalanche photodetector (APD) used for +measurements in Fig. 1 and Fig. 4. + +The diamond substrate is excited with a green (532 nm) laser, and the fluorescence is +measured at different depths from the surface facing the plasma, Figs. 1c, 1d. We repeat the +process to several locations and different substrates thicknesses. The measurement results from +two of the representative CVD diamond substrates of thicknesses, 0.25 mm (substrate 1) and 0.5 +mm (substrate 2) are shown in Fig. 1c and Fig. 1d. at a green laser power of 20 mW and 5 mW, +respectively. The fluorescence intensity in both diamonds is an order of magnitude higher for the +surface facing the Ar+ plasma than the other side, which has only background fluorescence coming +from the as-grown NV- centers (< 0.1 ppm). The newly created NV-s are uniformly distributed +over a depth of 150-200 µm depending on the substrate thickness, as shown in Fig. 1c and Fig. +1d. + +To estimate the spatial distribution of the Ar+ plasma-created NV- centers, we performed +fluorescence imaging on the surfaces of substrate 2 facing plasma (Fig. 2a) at a green laser power +of 0.5 mW. Fig. 2b shows the line cut profile (dashed line) taken on Fig. 2a. It is conclusive that +the Ar+ plasma created NV- centers are uniformly distributed. We compare the fluorescence +intensity from the newly created ensemble NV- centers in substrate 1 with the one collected from + +5 + +an electronic grade (EL) diamond with single NV- centers (created by ion beam implantation of +15N ions at a dose of 5×109 cm-2 and energy of 30 keV) by using the confocal microscope under +similar experimental conditions (at saturation). Since our SPCM saturates at a count rate of 10 +Mc/s corresponding to a laser power of 3 mW, we used an APD to detect the fluorescence of the +plasma-created NV-s. In Fig. 2c, we plot the fluorescence of substrate 1 as a function of green laser +power by converting the APD detected voltage (mV) to counts per second (c/s) and found a photon +count rate of 400 Mc/s at saturation (> 30 mW). Based on single NV- measurements (maximum +count rate of 0.02 Mc/s at saturation) on the EL implanted diamond (inset of Fig. 2d), we estimate +that the side facing Ar+ in substrate 1 contains > 20,000 NV- centers over the active volume of 0.2 +µm3, which translates to an NV- density of ~1017 cm-3 (= 0.57 ppm). To confirm that the detected +fluorescence comes from NV- centers, we performed fluorescence vs wavelength measurements +on the side of substrate 2 facing Ar+ plasma by using spectrometer TRIAX 320 (Horiba), Fig. 2d. +The observed spectrum consists of a typical NV- curve with high fluorescence in the wavelength +range of 650- 760 nm with zero phonon line (ZPL) peaks for NV0 and NV- at 574 and 637, nm +respectively [1]. + + + +Figure 2. Optical characterization of the Ar+ plasma photon irradiated diamond substrates. Fluorescence +image of the surface facing the Ar+ plasma (a). (b) Vertical line cut of fluorescence spatial profile taken +across the dashed lines in (a). (c) The fluorescence of substrate 1 was detected by APD as a function of +green laser power. Inset of (c) fluorescence intensity of EL diamond with single NV- centers as a function + +400 +(a) +(c) +Fluorescence (Mc/s) +200 +0.02 +Single +0.01 +NV- +0 +0 +1 +2 +Fluo. (Mc/s) +0 +10 +20 +30 +0.3 +2 +Laser Power (m W) +9 +(q) +(d) +Fluorescence (kc/s) +ZPL +2- +NV +Fluo. (Mc/s) +9 +ZPL +NVo +3 +0 +0 +5 +10 +570 +665 +760 +Y (μm) +Wavelenght (nm)6 + +of green laser power (20 Kc/s at a laser power of 2 mW). (d) Fluorescence spectrum acquired on the surface +facing the plasma of substrate 2. Scale bar in (a) is 2 m. +Spin-characterization. +The NV- center-based sensing application requires narrow resonance linewidth and sufficiently +long coherence and relaxation time. The spin-resonance properties of the Ar+ plasma created high- +density NV- are scrutinized using well-established electron spin resonance measurement +methods [14,15,56]. Fig. 3 shows the outcome of the measurements on substrate 1 performed using +the custom-built confocal microscope with SPCM modules and at an applied magnetic field of 8.6 +mT applied along [111] orientation of the (100) diamond and at laser power of 1 mW. We discuss +the ODMR measurements in detail below. + + +Figure 3. (a) CW-ODMR spectrum measured (scattered open circles) at an applied field of 8.6 mT. The +solid line curve in (a) is Lorentzian fits for the NV ODMR peaks. Insert in (a) is the cw-ODMR spectra +acquired at 2.3 mT field with optimized laser and MW parameters. (b) Rabi-nutation measurement which +shows the fluorescence intensity (scattered open circles) vs the duration of applied MW pulse at MW +frequency of 2629 MHz and a magnetic field of 8.6 mT. The solid line curve is a fit to the experimental +data (see the main text). (c) Normalized measured (scattered open circles) fluorescence intensity vs echo +time between p pulses fitted (solid line) with an exponential decay function with a decay = T2. (d) +Normalized measured (scattered open circles) fluorescence intensity vs time, fitted with an exponential +decay function with a decay = T1. + +A. cw-ODMR: The ODMR spectrum of NV- centers is measured by alternating the microwave +(MW) power between OFF and ON states. In the OFF state, NV- centers are continuously (10- + +a +() 1.00- +.00 +Exp +Norm. Fluo. (a.u.) +Norm. Fluo. (a.u.) +Fit +0.98- +-0-Exp +0.950.9 +Lorentz Fit +2920 +2960 +2600 +2800 +3000 +3200 +0 +5 +10 +(b) +Frequency (MHz) +(d) +Time (μs) +1.00- +1.00- +C +O Exp. +Exp +Norm. Fluo. (a.u.) +Fit. +Fit +0.95- +0.95- +0 +1 +2 +0 +5 +10 +Time (us) +Time (ms)7 + +100 ms pulses) pumped into the bright |0⟩ state, while in the ON state, fluorescence is reduced +as spins are driven into | ∓ 1⟩ states through the intersystem crossing to metastable singlet +states [1]. The normalized fluorescence intensity is recorded as a function of the MW +frequency at the applied magnetic field oriented along [111] direction of the (100) diamond +substrate (Fig. 3a). The resonance full-width-at-half-maximum linewidth Γ is 8.6 MHz, similar +to the ensemble NV- centers created by 14N substitution [36]. The applied magnetic field +breaks the degeneracy of the | ∓ 1⟩ state and leads to a pair of transitions for each NV- +orientation whose frequencies depend on the field projection along the NV symmetry axis. +There are four sub-ensembles of NV centers with different symmetry axes; thus, a full ODMR +spectrum contains 8 peaks (4 for |0⬌1⟩ transition and 4 for |0⬌ − 1⟩ transition) [57]. To +estimate the DC magnetic field sensitivity, we acquired the high-field (|0⬌1⟩) cw-ODMR +spectrum at an applied field of 2.3 mT with optimized measurement parameters such as the +laser power and MW power (inset of Fig. 3a) [45]. The minimum detected DC magnetic field +within the shot noise is given by [14,25,58]: CW ≅ 4 Γ (3√3 𝛾NV C)-1 (R)-1/2, where 𝛾𝑁𝑉 = +28 GHz/T is the gyromagnetic ratio of the electron spin, C is the ODMR peak contrast, R is +the NV photon-detection rate. By using the parameters of the measurements (inset of Fig. 3a, +R = 200 mV = 400 M counts/s measured by using Thorlabs APD (APD440A) at laser power +of 20 mW, Γ = 8.6 MHz, C = 0.1123) we estimate a DC sensitivity of ~104 nT/Hz1/2 over 0.2 +µm3 active volume. +B. Rabi Nutation: We performed Rabi nutation measurements to check the T2,Rabi decay [45] and +know the  pulse length required to measure NV- spin coherence lifetimes T2 and T1 of the +plasma created NV- centers (see below). We applied an MW frequency of 2629 MHz at the +|0⬌ − 1⟩ peak along [111] orientation (the left ODMR peak in Fig. 3a) and recorded the NV +fluorescence intensity vs the MW duration time t. Fig. 3b displays the measured Rabi +oscillations fitted with a function sin(Rabi t) exp (-t/ T2,Rabi) (Rabi is the Rabi frequency ~ 4 +MHz). The  pulse is 124 ns and T2,Rabi decay is 1.74 s that can be extended to > 50 s upon +a periodic reversal of the  phase [59]. +C. Transverse spin relaxation time (T2): Sensing weak dynamic (magnetic, electrical, or +thermal) signals requires both high density of NV- centers and longer T2 relaxation times. We +used a standard three-pulse Hahn-echo protocol to measure the T2 of the Ar+ plasma created +NV- centers [11,14,15]. The pulse sequence consists of a +𝜋 +2 − 𝜏 − 𝜋 − 𝜏 − +𝜋 +2 applied on the +OMDR resonance peak at 2629 MHz in Fig. 3a, and the integrated NV fluorescence intensity +is recorded as a function of the total evolution time 2𝜏 ( is the time between  and  pulses). +Fig. 3c shows a fast exponential decay of the Hahn-echo envelope (4 s), explained by the +strong dipolar interactions between the NV-s and N paramagnetic centers (1 ppm in our CVD +diamond). The contribution of 13C spins in the NV echo signal at an applied magnetic field of +8.6 mT is negligible since the signal decays before the first 13C revival (20 s). The estimated +sensitivity for dynamic (AC) magnetic fields is [25]: ηAC ≅ (𝛾NV D)-1 1 / (Rmeas)]½ Exp( /T2). + is the full field interrogation time, D is the spin echo contrast, and meas is the measurement +averaging time. By using the parameters of our measurements (T2 = 4 µs, R = 200 mV = 400 +M counts/s, D = 0.02, meas = 1 s, T = 2 µs) the sensitivity to AC magnetic fields is 0.12 pT +Hz-1/2. By using dynamical decoupling pulse sequence [25,31,41,60] T2 can be extended to > +100 s for NV- ensemble spins, and the AC sensitivity can be pushed to a few fT Hz-1/2 [61]. + +8 + +D. Longitudinal spin relaxation time (T1): We measured the longitudinal spin relaxation time +(T1) of the Ar+ plasma-created NV- centers using standard T1 measurements [15,62]. We +applied a sequence consisting of two laser pulses for NV initialization/readout (10 s & 0.3 s +in duration) with and without  pulse and recorded the subtracted fluorescence intensity as a +function of the time t between the initialization and readout pulses. We plot the result of the T1 +measurements in Fig. 3d and found a T1 ~ 5 ms, similar to the values measured in CVD and +EL diamonds. T1 measurements are useful when using spin-correlation pulse protocols to +bypass T2 and improve AC sensitivity [15,31]. + +IV. +Applications +High densities of NV-s are favored for precision magnetometry due to improved signal-to-noise +and sensitivity from 1/√𝑁 enhancement [25]. High densities also provide high fluorescence +intensity, which can be measured by regular cheap photodetectors reducing the complexity of +using SPCMs. To demonstrate magnetic field sensing applicability using the Ar+ plasma-created +NV- centers, we implemented a magnetic field tracking method described by Welter et al. [17]. A +0.15 mT oscillatory magnetic field of various frequencies is applied to the NV- sensors, which +track the changes in the applied magnetic field in real-time, as shown in Fig. 4. We connected the +tracking output voltage with an oscilloscope and recorded the time transient of the tracking signal +along with applied magnetic field signals. As shown in Fig. 4a, the applied magnetic field changes +0.3 mT (peak-to-peak) over 20 ms time, and the tracking system can follow those changes in real- +time. Fig. 4b shows the fast Fourier transform (FFT) of the measured tracking signals for all the +applied oscillatory magnetic fields. + + +Figure 4. Real-time magnetic field measured using the NV- centers created by high-energy +photons from Ar+ plasma over an active volume of 0.2 µm3. +V. +Conclusion +In summary, we demonstrated a low-cost and highly efficient method to create high-density NV- +centers on CVD-diamond crystal containing as-grown nitrogen concentrations of 1 ppm. The + +Frequency (Hz) +0 +100 +200 +300 +400 +500 +Mag. (a. u.) +(b) +FFT +0 +2 +App. Field (mT) +(a +Tracking (V) +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +time (s)9 + +measurements show that the high-density NV- layer is distributed over 150-200 µm from the +surface facing the photons created by Ar+ plasma. It demonstrates that the described process +creates an NV- density of 0.57 ppm, yielding more than 50% N-to-NV- conversion. We measured +a DC magnetic field sensitivity of ~ 104 nT Hz-1/2 and an AC magnetic sensitivity of ~ 0.12 pT Hz- +1/2 measured over 0.2 µm3 active sample volume. The measured relaxation properties are T1 = 5 +ms and T2 = 4 µs. We found that the fluorescence intensity from the NV- centers within 0.2 µm3 +volume is detectable using regular photodetectors (e.g., APD), which is beneficial for on-chip and +commercial adaptation [27]. The measurements show that using the Ar+ plasma-created high- +density NV- can be used to measure magnetic fields quickly and efficiently at a rate over 10 mT/s +with an active sample volume of 0.2 µm3. Our methodology could find applications in +magnetometry where thick NV layers (150-200 um) are needed to measure magnetic fields +generated from big cells labeled with magnetic nanoparticles [63] or planetary bodies for +paleomagnetic analysis of complex rocks such as meteorites that have heterogeneous +magnetizations ≤ 200 m [64]. + +VI. +Acknowledgments +K.A. would like to acknowledge the support of the National Science Foundation/EPSCoR RII +Track-4 Award OIA-2033210. The research was partially performed in the Nebraska Nanoscale +Facility, which is supported by the National Science Foundation under Award ECCS: 2025298 +and the Nebraska Research Initiative. P.B.K. would like to acknowledge the support of the Wichita +State University Convergence Science Initiative Program. A. L. would like to acknowledge the +support of the National Science Foundation/EPSCoR RII Track-1: Emergent Quantum Materials +and Technologies (EQUATE), Award OIA-2044049. + +VII. +AUTHOR DECLARATIONS + +a. Conflict of Interest +The authors have no conflicts to disclose. +b. Author Contributions +K.A. conceived the experiment and supervised the overall study. P.B.K., R.T., M.D., A.L., +and K.A. performed the measurements and analyzed the data. A. E. A. helped in data analysis +and figure preparation for the manuscript. A.L. and K.A. wrote the manuscript with help from +all authors. P.B.K. and R.T. contribute equally. + +VIII. +DATA AVAILABILITY +The data supporting this study's findings are available from the corresponding author upon +reasonable request. + + + + +10 + +References +[1] M. W. Doherty, N. B. Manson, P. Delaney, F. Jelezko, J. Wrachtrup, and L. C. L. Hollenberg, +The Nitrogen-Vacancy Colour Centre in Diamond, Physics Reports 528, 1 (2013). +[2] N. Y. Yao, L. Jiang, A. V. Gorshkov, P. C. Maurer, G. Giedke, J. I. Cirac, and M. D. Lukin, +Scalable Architecture for a Room Temperature Solid-State Quantum Information Processor, +Nat Commun 3, 800 (2012). +[3] T. van der Sar, Z. 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Trubko, High‐Sensitivity Moment Magnetometry +With the Quantum Diamond Microscope, Geochem Geophys Geosyst 21, (2020). + + diff --git a/S9FAT4oBgHgl3EQf2B5J/content/tmp_files/load_file.txt b/S9FAT4oBgHgl3EQf2B5J/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a97e20339a011696caa12a0e6e0ab219cc7551ce --- /dev/null +++ b/S9FAT4oBgHgl3EQf2B5J/content/tmp_files/load_file.txt @@ -0,0 +1,932 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf,len=931 +page_content='1 An efficient and low-cost method to create high-density nitrogen-vacancy centers in CVD diamond for sensing applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Prem Bahadur Karki,1 Rupak Timalsina,2 Mohammadjavad Dowran,2 Ayodimeji E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Aregbesola,1 Abdelghani Laraoui,2,3* and Kapildeb Ambal1† 1Department of Mathematics, Statistics, and Physics, Wichita State University, 1845 Fairmount St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Wichita, KS 67260, United States of America 2Department of Mechanical & Materials Engineering, University of Nebraska-Lincoln, 900 N 16th St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' W342 NH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Lincoln, NE 68588, United States of America 3Department of Physics and Astronomy and the Nebraska Center for Materials and Nanoscience, University of Nebraska-Lincoln, 855 N 16th St, Lincoln, Nebraska 68588, USA †Author to whom correspondence should be addressed, email: alaraoui2@unl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='edu, †Kapildeb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='ambal@wichita.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='edu I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Abstract The negatively charged Nitrogen-Vacancy (NV-) center in diamond is one of the most versatile and robust quantum sensors suitable for quantum technologies, including magnetic field and temperature sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' For precision sensing applications, densely packed NV- centers within a small volume are preferable due to benefiting from 1/√𝑁 sensitivity enhancement (N is the number of sensing NV centers) and efficient excitation of NV centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' However, methods for quickly and efficiently forming high concentrations of NV- centers are in development stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' We report an efficient, low-cost method for creating high-density NV- centers production from a relatively low nitrogen concentration based on high-energy photons from Ar+ plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' This study was done on type-IIa, single crystal, CVD-grown diamond substrates with an as-grown nitrogen concentration of 1 ppm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' We estimate an NV- density of ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='57 ppm (57%) distributed homogeneously over 200 µm deep from the diamond surface facing the plasma source based on optically detected magnetic resonance and fluorescence confocal microscopy measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The created NV-s have a spin- lattice relaxation time (T1) of 5 ms and a spin-spin coherence time (T2) of 4 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' We measure a DC magnetic field sensitivity of ~ 104 nT Hz-1/2, an AC magnetic field sensitivity of ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='12 pT Hz-1/2, and demonstrate real-time magnetic field sensing at a rate over 10 mT s-1 using an active sample volume of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='2 µm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Introduction The negatively charged nitrogen-vacancy (NV-) centers are one of the leading solid state-based quantum platforms [1], enabling diverse quantum applications, including spin-qubit for quantum information processing [2–5], quantum memory [6–8], and quantum sensing [9] for many physical quantities including magnetic fields [10–17], electric fields [18–21], and temperature [22–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' For potential commercial sensing applications where nanoscale spatial resolution is not required, ensemble NV-s are preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The key requirements for high-sensitive ensemble-based sensing applications are high NV- concentrations within small volumes while preserving long spin coherence time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The high concentrations are favored for precision magnetometry due to improved signal-to-noise and sensitivity from 1/√N enhancement [9,25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' For example, the sensitivity to detect a constant magnetic field with NV- centers scales with 1/√𝑁𝑁𝑉 where NNV is the number of NV- centers from which the fluorescence signal is 2 collected [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The densely packed or small sample volume is necessary to efficiently excite NV- centers using on-chip laser and microwave excitation [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Several existing techniques are available for creating high-density NV- centers > 10 ppm (1 ppm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='76 × 1017 cm−3 in diamond) [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The most popular ones are;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' (1) implanting high doses (~100 ppm) of nitrogen (N) with subsequent annealing [29–31] and (2) electron irradiation or implantation of He+ ions on a diamond substrate containing high-concentration of nitrogen (N) impurities (~100 ppm) followed by subsequent annealing [32,33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Nevertheless, the final concentration of NV-s still depends on many factors like annealing temperature, cleaning procedures, substrate characteristics, and the N to NV- conversion yield (typically <10%) [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Recently, progress was made to increase the production yield by more than 30% by optimizing the process (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=', the electron irradiation dose) [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' However, >70% of the implanted/doped nitrogen atoms are present in the substrate as substitutional nitrogen, known as P1 paramagnetic centers [36,37] or neutral nitrogen-vacancy (NV0) centers [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' These excesses of nitrogen unavoidably lead to decreased NV- center spin coherence properties by orders of magnitude [36– 38], thus degrading the sensitivity [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Also, a large implantation or electron irradiation dose creates undesired defects and local graphitization of the diamond crystal, which could be beyond repair via the thermal annealing process [39,40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' These unwanted defects decrease spin coherence times 𝑇2 ∗ and T2 that impact the overall sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Therefore, developing low-cost and highly efficient methods in creating high-density NV- centers from low-concentration nitrogen in the diamond crystal is essential to increase the number of NV- centers for many scientific and commercial applications such as the fabrication of brighter nanodiamonds for biomedical applications [12,22,33,41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' This work focuses on a quick and cost-effective method of creating high-density NV- centers in type-IIa chemical vapor deposition (CVD)-grown diamond substrates with as-grown nitrogen concentration of 1 ppm by using high-energy photons from Ar+ plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Based on optically detected magnetic resonance (ODMR) and fluorescence confocal microscopy measurements, we estimate the density of newly created NV- ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='57 ppm (> 50% of conversion yield), distributed homogeneously over 200 µm deep from the diamond surface facing the plasma source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The created NV- centers exhibit a spin-lattice relaxation time (T1) of 5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='2 ms and a spin-spin coherence time (T2) of 4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='5 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' We estimate a shot noise-limited DC magnetic field sensitivity of ~104 nT Hz-1/2 an AC magnetic field sensitivity of ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='12 pT Hz-1/2 respectively over an active sample volume of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='2 µm3, and demonstrate a real-time AC magnetic field sensing with a frequency of up to 90 Hz using the same active sample volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Experimental methods and discussion The NV center is a solid-state spin sensor in a diamond crystal formed by substituting a carbon atom with a nitrogen atom and a vacancy adjacent to it [1] (inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' This type of nitrogen- vacancy center is known as the neutral NV0 center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' If the neutral NV center captures an extra electron, it forms a negatively charged NV- center that can be photoionized to NV0 with laser or voltage excitations [42,43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The NV- center, a spin-1 system, is used for emerging quantum applications, including quantum sensing [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The diamond substrates used for this work are single crystal CVD-grown type-IIa diamonds (element six part# 145-500-0055) doped with ~ 1ppm (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='76 × 1017 cm−3) of nitrogen (N) throughout the substrate during the growth process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The substrates (labeled 1 and 2) were first cleaned in a tri-acid mixture (1 H2SO4: 1 HNO3: 1 HCLO4) [44] for two hours at 200 oC to remove polishing-related graphitization, followed by rinsing with deionized (DI) water or Isopropyl Alcohol (IPA) and finally drying using compressed nitrogen gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The dry 3 substrates were then exposed to Argon (Ar+) plasma by using Trion Minilock-Phantom III Inductively Coupled Plasma (ICP) Reactive Ion Etching (RIE) system for 30 s (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' 1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The plasma process was performed with ICP power at 200 W, RIE power at 50 W, Ar gas flow at 5 sccm, and the ICP-RIE chamber pressure at 10 mTorr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' After the plasma exposure, the diamonds were cleaned with the tri-acid mixture for two hours at 200 oC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The dry substrates were then annealed under vacuum (10-8 Torr) at 1100 oC for two hours with another subsequent triacid cleaning [31,41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The clean and dry substrates were then used for optical and spin characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' We discuss below the mechanisms of NV- creation using high-energy photons from Ar+ plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' (a) Representation of the process for making a high-density NV- layer using Ar+ plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' (b) schematic of the custom-made confocal optical microscope used to characterize the high-density NV- centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Inset of (b) an NV center in a diamond lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The distribution of the created NV- centers as a function of the depth from the side facing the plasma measured on 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='25 mm (c, substrate 1) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='5 mm (d, substrate 2) thick diamonds prepared with the same conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The NV- center is a deep bandgap defect located near the mid-gap within the 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='6 eV bandgap of the diamond crystal [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' NV- formation needs three processes: substitutional nitrogen atom, adjutant vacancy, and electron capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' There are two well-established NV- creation methods: (1) implantation of nitrogen (14N or 15N) ions followed by high-temperature annealing [45], and (2) doping the diamond crystal with N during the growth process and create vacancies by helium ion (He+) implantation [32,33] or electron irradiation [34,46] followed by subsequent high- (a) Fluorescence (m V) 40 Background Plasma Side diamond 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='2 Fiber Depth (mm) APD (b) (d) Lens White Light 20 Source Laser Fluorescence Back round Plasma 10 Side Camera C Objective 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='2 C Depth (mm) Diamond4 temperature annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' In both methods, the implantation/irradiation of N/He ions creates a high density of vacancies due to the broken bonds/dangling bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' During the high-temperature annealing process, these vacancies position themselves next to substitutional nitrogen atoms forming the NV0 center [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' When NV0 captures an extra electron, they become NV-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Since the diamond has a large bandgap and NV centers are located near the middle of the bandgap, the thermal energy is insufficient to excite electrons from the valence band to the conduction band to be captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' These difficulties lead to poor yield in N-to-NV- formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' However, electron irradiation creates broken bonds along with extra electrons, improving NV- formation efficiency [34,45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' High energy and above bandgap photons generated from the Ar+ plasma were used to create dangling bonds in SiO2 [47–49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The energy of Ar+ in the plasma is very low, and the depth of the created NV- center is more than 200 µm from the surface facing the plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Therefore, it is conclusive that the enhancement of NV- formation is primarily due to high-energy photons from Ar+ plasma [50–52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' We hypothesize that the high energy and above bandgap photons from Ar+ plasma could have two effects: (i) Direct absorption of UV photons inducing vacancies such as broken bonds/dangling bonds [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' (ii) When the diamond absorbs a UV photon, it creates a shower of electrons (because the photon energy is much larger than the diamond band gap) [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' These electrons could travel through the conduction band of the diamond and recapture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Therefore, we assume that the high-energy photons would have similar effects as electron irradiation, creating vacancies and supplying electrons for recapturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The formation of dangling bonds will create more NV centers from doped N0/P1 centers, similar to electron irradiation [34,46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The excited extra electrons in the conduction band could be captured to form NV- centers [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Optical Characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The fluorescence properties of the diamond substrates were investigated using a custom-built confocal fluorescence microscope [14,55] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' 1b) consisting of a green laser (532 nm) for optical excitation of NV- centers, a permanent magnet (up to 100 mT) to apply a magnetic field for ODMR measurements, a 100x microscope objective with a numerical aperture of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='8 NA to focus light on the diamond substrate, and fluorescence collection optics including 650 nm edge pass filter, focusing lens, 9/125 single-mode fiber, single photon counter modules (SPCM) used for confocal imaging and spin measurements in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' 2 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' 3, and avalanche photodetector (APD) used for measurements in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The diamond substrate is excited with a green (532 nm) laser, and the fluorescence is measured at different depths from the surface facing the plasma, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' 1c, 1d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' We repeat the process to several locations and different substrates thicknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The measurement results from two of the representative CVD diamond substrates of thicknesses, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='25 mm (substrate 1) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='5 mm (substrate 2) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' 1c and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' 1d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' at a green laser power of 20 mW and 5 mW, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The fluorescence intensity in both diamonds is an order of magnitude higher for the surface facing the Ar+ plasma than the other side, which has only background fluorescence coming from the as-grown NV- centers (< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='1 ppm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The newly created NV-s are uniformly distributed over a depth of 150-200 µm depending on the substrate thickness, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' 1c and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' 1d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' To estimate the spatial distribution of the Ar+ plasma-created NV- centers, we performed fluorescence imaging on the surfaces of substrate 2 facing plasma (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' 2a) at a green laser power of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='5 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' 2b shows the line cut profile (dashed line) taken on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' It is conclusive that the Ar+ plasma created NV- centers are uniformly distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' We compare the fluorescence intensity from the newly created ensemble NV- centers in substrate 1 with the one collected from 5 an electronic grade (EL) diamond with single NV- centers (created by ion beam implantation of 15N ions at a dose of 5×109 cm-2 and energy of 30 keV) by using the confocal microscope under similar experimental conditions (at saturation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Since our SPCM saturates at a count rate of 10 Mc/s corresponding to a laser power of 3 mW, we used an APD to detect the fluorescence of the plasma-created NV-s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' 2c, we plot the fluorescence of substrate 1 as a function of green laser power by converting the APD detected voltage (mV) to counts per second (c/s) and found a photon count rate of 400 Mc/s at saturation (> 30 mW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Based on single NV- measurements (maximum count rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='02 Mc/s at saturation) on the EL implanted diamond (inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' 2d), we estimate that the side facing Ar+ in substrate 1 contains > 20,000 NV- centers over the active volume of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='2 µm3, which translates to an NV- density of ~1017 cm-3 (= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='57 ppm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' To confirm that the detected fluorescence comes from NV- centers, we performed fluorescence vs wavelength measurements on the side of substrate 2 facing Ar+ plasma by using spectrometer TRIAX 320 (Horiba), Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The observed spectrum consists of a typical NV- curve with high fluorescence in the wavelength range of 650- 760 nm with zero phonon line (ZPL) peaks for NV0 and NV- at 574 and 637, nm respectively [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Optical characterization of the Ar+ plasma photon irradiated diamond substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Fluorescence image of the surface facing the Ar+ plasma (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' (b) Vertical line cut of fluorescence spatial profile taken across the dashed lines in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' (c) The fluorescence of substrate 1 was detected by APD as a function of green laser power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Inset of (c) fluorescence intensity of EL diamond with single NV- centers as a function 400 (a) (c) Fluorescence (Mc/s) 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='02 Single 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='01 NV- 0 0 1 2 Fluo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' (Mc/s) 0 10 20 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='3 2 Laser Power (m W) 9 (q) (d) Fluorescence (kc/s) ZPL 2- NV Fluo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' (Mc/s) 9 ZPL NVo 3 0 0 5 10 570 665 760 Y (μm) Wavelenght (nm)6 of green laser power (20 Kc/s at a laser power of 2 mW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' (d) Fluorescence spectrum acquired on the surface facing the plasma of substrate 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Scale bar in (a) is 2 \uf06dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Spin-characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The NV- center-based sensing application requires narrow resonance linewidth and sufficiently long coherence and relaxation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The spin-resonance properties of the Ar+ plasma created high- density NV- are scrutinized using well-established electron spin resonance measurement methods [14,15,56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' 3 shows the outcome of the measurements on substrate 1 performed using the custom-built confocal microscope with SPCM modules and at an applied magnetic field of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='6 mT applied along [111] orientation of the (100) diamond and at laser power of 1 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' We discuss the ODMR measurements in detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' (a) CW-ODMR spectrum measured (scattered open circles) at an applied field of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='6 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The solid line curve in (a) is Lorentzian fits for the NV ODMR peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Insert in (a) is the cw-ODMR spectra acquired at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='3 mT field with optimized laser and MW parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' (b) Rabi-nutation measurement which shows the fluorescence intensity (scattered open circles) vs the duration of applied MW pulse at MW frequency of 2629 MHz and a magnetic field of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='6 mT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The solid line curve is a fit to the experimental data (see the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' (c) Normalized measured (scattered open circles) fluorescence intensity vs echo time between p pulses fitted (solid line) with an exponential decay function with a decay = T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' (d) Normalized measured (scattered open circles) fluorescence intensity vs time, fitted with an exponential decay function with a decay = T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' cw-ODMR: The ODMR spectrum of NV- centers is measured by alternating the microwave (MW) power between OFF and ON states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' In the OFF state, NV- centers are continuously (10- a () 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='00- .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='00 Exp Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Fluo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=') Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Fluo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=') Fit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='98- 0-Exp 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='950.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='9 Lorentz Fit 2920 2960 2600 2800 3000 3200 0 5 10 (b) Frequency (MHz) (d) Time (μs) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='00- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='00- C O Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Exp Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Fluo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=') Fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Fit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='95- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='95- 0 1 2 0 5 10 Time (us) Time (ms)7 100 ms pulses) pumped into the bright |0⟩ state, while in the ON state, fluorescence is reduced as spins are driven into | ∓ 1⟩ states through the intersystem crossing to metastable singlet states [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The normalized fluorescence intensity is recorded as a function of the MW frequency at the applied magnetic field oriented along [111] direction of the (100) diamond substrate (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The resonance full-width-at-half-maximum linewidth Γ is 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='6 MHz, similar to the ensemble NV- centers created by 14N substitution [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The applied magnetic field breaks the degeneracy of the | ∓ 1⟩ state and leads to a pair of transitions for each NV- orientation whose frequencies depend on the field projection along the NV symmetry axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' There are four sub-ensembles of NV centers with different symmetry axes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' thus, a full ODMR spectrum contains 8 peaks (4 for |0⬌1⟩ transition and 4 for |0⬌ − 1⟩ transition) [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' To estimate the DC magnetic field sensitivity, we acquired the high-field (|0⬌1⟩) cw-ODMR spectrum at an applied field of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='3 mT with optimized measurement parameters such as the laser power and MW power (inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' 3a) [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The minimum detected DC magnetic field within the shot noise is given by [14,25,58]: \uf068CW ≅ 4 Γ (3√3 𝛾NV C)-1 (R)-1/2, where 𝛾𝑁𝑉 = 28 GHz/T is the gyromagnetic ratio of the electron spin, C is the ODMR peak contrast, R is the NV photon-detection rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' By using the parameters of the measurements (inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' 3a, R = 200 mV = 400 M counts/s measured by using Thorlabs APD (APD440A) at laser power of 20 mW, Γ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='6 MHz, C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='1123) we estimate a DC sensitivity of ~104 nT/Hz1/2 over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='2 µm3 active volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Rabi Nutation: We performed Rabi nutation measurements to check the T2,Rabi decay [45] and know the \uf070 pulse length required to measure NV- spin coherence lifetimes T2 and T1 of the plasma created NV- centers (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' We applied an MW frequency of 2629 MHz at the |0⬌ − 1⟩ peak along [111] orientation (the left ODMR peak in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' 3a) and recorded the NV fluorescence intensity vs the MW duration time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' 3b displays the measured Rabi oscillations fitted with a function sin(\uf077Rabi t) exp (-t/ T2,Rabi) (\uf077Rabi is the Rabi frequency ~ 4 MHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The \uf070 pulse is 124 ns and T2,Rabi decay is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='74 \uf06ds that can be extended to > 50 \uf06ds upon a periodic reversal of the \uf070 phase [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Transverse spin relaxation time (T2): Sensing weak dynamic (magnetic, electrical, or thermal) signals requires both high density of NV- centers and longer T2 relaxation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' We used a standard three-pulse Hahn-echo protocol to measure the T2 of the Ar+ plasma created NV- centers [11,14,15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The pulse sequence consists of a 𝜋 2 − 𝜏 − 𝜋 − 𝜏 − 𝜋 2 applied on the OMDR resonance peak at 2629 MHz in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' 3a, and the integrated NV fluorescence intensity is recorded as a function of the total evolution time 2𝜏 (\uf074 is the time between \uf070\uf02f\uf032 and \uf070 pulses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' 3c shows a fast exponential decay of the Hahn-echo envelope (4 \uf06ds), explained by the strong dipolar interactions between the NV-s and N paramagnetic centers (1 ppm in our CVD diamond).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The contribution of 13C spins in the NV echo signal at an applied magnetic field of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='6 mT is negligible since the signal decays before the first 13C revival (20 \uf06ds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The estimated sensitivity for dynamic (AC) magnetic fields is [25]: ηAC ≅ (𝛾NV D)-1 1 / (R\uf054meas)]½ Exp(\uf054 /T2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' \uf054 is the full field interrogation time, D is the spin echo contrast, and \uf054meas is the measurement averaging time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' By using the parameters of our measurements (T2 = 4 µs, R = 200 mV = 400 M counts/s, D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='02, \uf054meas = 1 s, T = 2 µs) the sensitivity to AC magnetic fields is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='12 pT Hz-1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' By using dynamical decoupling pulse sequence [25,31,41,60] T2 can be extended to > 100 \uf06ds for NV- ensemble spins, and the AC sensitivity can be pushed to a few fT Hz-1/2 [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' 8 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Longitudinal spin relaxation time (T1): We measured the longitudinal spin relaxation time (T1) of the Ar+ plasma-created NV- centers using standard T1 measurements [15,62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' We applied a sequence consisting of two laser pulses for NV initialization/readout (10 \uf06ds & 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='3 \uf06ds in duration) with and without \uf070 pulse and recorded the subtracted fluorescence intensity as a function of the time t between the initialization and readout pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' We plot the result of the T1 measurements in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' 3d and found a T1 ~ 5 ms, similar to the values measured in CVD and EL diamonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' T1 measurements are useful when using spin-correlation pulse protocols to bypass T2 and improve AC sensitivity [15,31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Applications High densities of NV-s are favored for precision magnetometry due to improved signal-to-noise and sensitivity from 1/√𝑁 enhancement [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' High densities also provide high fluorescence intensity, which can be measured by regular cheap photodetectors reducing the complexity of using SPCMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' To demonstrate magnetic field sensing applicability using the Ar+ plasma-created NV- centers, we implemented a magnetic field tracking method described by Welter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='15 mT oscillatory magnetic field of various frequencies is applied to the NV- sensors, which track the changes in the applied magnetic field in real-time, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' We connected the tracking output voltage with an oscilloscope and recorded the time transient of the tracking signal along with applied magnetic field signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' 4a, the applied magnetic field changes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='3 mT (peak-to-peak) over 20 ms time, and the tracking system can follow those changes in real- time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' 4b shows the fast Fourier transform (FFT) of the measured tracking signals for all the applied oscillatory magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Real-time magnetic field measured using the NV- centers created by high-energy photons from Ar+ plasma over an active volume of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='2 µm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Conclusion In summary, we demonstrated a low-cost and highly efficient method to create high-density NV- centers on CVD-diamond crystal containing as-grown nitrogen concentrations of 1 ppm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The Frequency (Hz) 0 100 200 300 400 500 Mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=') (b) FFT 0 2 App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Field (mT) (a Tracking (V) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='25 time (s)9 measurements show that the high-density NV- layer is distributed over 150-200 µm from the surface facing the photons created by Ar+ plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' It demonstrates that the described process creates an NV- density of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='57 ppm, yielding more than 50% N-to-NV- conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' We measured a DC magnetic field sensitivity of ~ 104 nT Hz-1/2 and an AC magnetic sensitivity of ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='12 pT Hz- 1/2 measured over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='2 µm3 active sample volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The measured relaxation properties are T1 = 5 ms and T2 = 4 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' We found that the fluorescence intensity from the NV- centers within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='2 µm3 volume is detectable using regular photodetectors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=', APD), which is beneficial for on-chip and commercial adaptation [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The measurements show that using the Ar+ plasma-created high- density NV- can be used to measure magnetic fields quickly and efficiently at a rate over 10 mT/s with an active sample volume of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='2 µm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Our methodology could find applications in magnetometry where thick NV layers (150-200 um) are needed to measure magnetic fields generated from big cells labeled with magnetic nanoparticles [63] or planetary bodies for paleomagnetic analysis of complex rocks such as meteorites that have heterogeneous magnetizations ≤ 200 \uf06dm [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Acknowledgments K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' would like to acknowledge the support of the National Science Foundation/EPSCoR RII Track-4 Award OIA-2033210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' The research was partially performed in the Nebraska Nanoscale Facility, which is supported by the National Science Foundation under Award ECCS: 2025298 and the Nebraska Research Initiative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' would like to acknowledge the support of the Wichita State University Convergence Science Initiative Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' would like to acknowledge the support of the National Science Foundation/EPSCoR RII Track-1: Emergent Quantum Materials and Technologies (EQUATE), Award OIA-2044049.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' AUTHOR DECLARATIONS a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Conflict of Interest The authors have no conflicts to disclose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Author Contributions K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' conceived the experiment and supervised the overall study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=', R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=', and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' performed the measurements and analyzed the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' helped in data analysis and figure preparation for the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' wrote the manuscript with help from all authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' contribute equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=" DATA AVAILABILITY The data supporting this study's findings are available from the corresponding author upon reasonable request." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' 10 References [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/S9FAT4oBgHgl3EQf2B5J/content/2301.08712v1.pdf'} +page_content=' Doherty, N.' 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b/SNE4T4oBgHgl3EQfLAy5/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..fa44c2787ed27b147358d6cca99695ae9a06e1f7 --- /dev/null +++ b/SNE4T4oBgHgl3EQfLAy5/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:729705dfed0d13c793353339ff2aec91b0db2f5a07d3e6bf8f5e991ac9b76b45 +size 274135 diff --git a/StAzT4oBgHgl3EQfJPvm/content/tmp_files/2301.01078v1.pdf.txt b/StAzT4oBgHgl3EQfJPvm/content/tmp_files/2301.01078v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e885176a42987e045a68de0921b6c4d19ece390c --- /dev/null +++ b/StAzT4oBgHgl3EQfJPvm/content/tmp_files/2301.01078v1.pdf.txt @@ -0,0 +1,1226 @@ +arXiv:2301.01078v1 [physics.flu-dyn] 3 Jan 2023 +Under consideration for publication in J. Fluid Mech. +1 +Banner appropriate to article type will appear here in typeset article +Modal decomposition of nonlinear interactions in +wall turbulence +U. Karban1,2†, E. Martini1, A.V.G. Cavalieri3, P. Jordan1 +1Département Fluides, Thermique, Combustion, Institut Pprime, CNRS-University of Poitiers-ENSMA, +France +2Department of Aerospace Engineering, Middle East Technical University, Ankara 06800, Turkey +3Instituto Tecnológico de Aeronáutica, São José dos Campos/SP, Brazil +(Received xx; revised xx; accepted xx) +Coherent structures are found in many different turbulent flows and are known to drive self- +sustaining processes in wall turbulence. Identifying the triadic interactions which generate +coherent structures can provide insights beyond what is possible in the framework of +linearized models. There are infinite possible interactions that may generate a given structure. +Thus a method to systematically study those, ranking them in terms of their contribution +to the structure of interest, is essential. We here use the resolvent-based extended spectral +proper orthogonal decomposition (RESPOD) approach (Karban, U. et al. 2022 Self-similar +mechanisms in wall turbulence studied using resolvent analysis. Journal of Fluid Mechanics +969, A36) to rank the triadic interactions which give rise to wall-attached structures in +a minimal Couette flow at Reynolds number 400. Our analysis identifies that six triadic +interactions dominate the most-energetic wall-attached structure, revealing the capability of +the methodology to identify and rank nonlinear interactions responsible for a given coherent +structure. The approach can be used to analyse the energy exchange in turbulent flows and +may guide the construction of reduced-order models based on the interplay between different +flow modes. +Key words: +1. Introduction +Turbulent flows contain coherent structures that span large spatial and temporal scales. +These structures are responsible for many important phenomena observed in different +flows, ranging from sustaining the near-wall cycle (Hamilton et al. 1995) in wall-bounded +flows to noise generation in jets (Jordan & Colonius 2013; Cavalieri et al. 2019). It has +been shown that the linear mechanisms play a major role in the generation of coherent +structures (Ellingsen & Palm 1975; Landahl 1980; Trefethen et al. 1993; Hwang & Cossu +2010b; Brandt 2014; Schmidt et al. 2018; Pickering et al. 2020). A now popular approach +to investigate these mechanisms is the resolvent analysis, where the Navier-Stokes (N-S) +† Email address for correspondence: ukarban@metu.edu.tr +Abstract must not spill onto p.2 + +2 +U. Karban et al. +equation are arranged in input-output form in the frequency domain (Farrell & Ioannou +1993; Jovanović & Bamieh 2005; McKeon & Sharma 2010; Hwang & Cossu 2010a; +Sipp & Marquet 2012; Towne et al. 2018; Lesshafft et al. 2019). Although resolvent analysis +provides a dynamical framework, in most cases, it provides a qualitative understanding of the +coherent structures and the associated mechanisms. It has been shown for certain flows that +modelling the nonlinear fluctuations, i.e., the color of the turbulence, is essential for better +prediction of these structures (Zare et al. 2017; Martini et al. 2020; Amaral et al. 2021; +Morra et al. 2021; Nogueira et al. 2021; Karban et al. 2022), particularly when developing +flow models that can quantitatively predict coherent structures. +One way to tackle the nonlinearity is to use eddy viscosity. It has been shown for many +flows that adopting an eddy viscosity model while constructing the resolvent operator +improves the prediction of coherent structures (Hwang & Cossu 2010b; Morra et al. 2019, +2021; Pickering et al. 2021; Kuhn et al. 2021). One can consider the use of eddy viscosity +within resolvent framework as the following: it is known that the resolvent operator yields +the exact coherent structures observed in the flow if the forcing is white. The actual forcing +is not white, and inclusion of an eddy viscosity in the linear operator allows to model at least +part of the forcing colour. Given that the eddy viscosity models incoherent disturbances (cf. +Hussain & Reynolds 1970), one may conjecture that it provides a good model of the effect +of the incoherent disturbances on coherent structures. +An alternative approach to model the nonlinearity is to use quasi-linear approximation +(Malkus 1956), where the N-S equations are split into somehow-averaged quantities and the +remaining fluctuating terms. The equations for the averaged quantities are then solved directly +taking into account the coupling with the fluctuation equations, while the fluctuation equa- +tions are linearised by neglecting the nonlinear fluctuating terms (Malkus 1956) or replacing +them with a linear model (Farrell & Ioannou 2012; Thomas et al. 2014; Constantinou et al. +2014; Bretheim et al. 2015; Farrell et al. 2017; Bretheim et al. 2018). +All these approaches try modelling the nonlinear terms as a whole rather than tracing +separately the triadic interactions that add up to form them. When decomposing the flow +into Fourier modes, the quadratic nonlinearity of the incompressible N-S equations become +triadic interactions between these modes. For a high-Reynolds-number turbulent flow, the +vast number of possible interactions forming a given nonlinear term prohibits their direct +modelling. There are some studies which analytically investigate triadic interactions in +simple cases such as homogeneous turbulence (Kraichnan 1973; Waleffe 1992; Moffatt +2014). Cheung & Zaki (2014) derived the spectral N-S equations. Investigating the analytical +properties of triadic interactions in homogeneous, isotropic turbulence, they showed that the +famous -5/3 decay is embedded in the N-S equations. In a recent study, Cho et al. (2018) +employed the spectral turbulent kinetic energy equation to trace the energy transfer between +different scales in a turbulent channel via triadic interactions. Jin et al. (2021) adopted a +similar approach to study the energy transfer in cylinder wake. +Triadic interactions in turbulent flows are also investigated within the resolvent framework. +The interactions between the response modes of the resolvent operator and their effect +of the self-similar nature of these modes was first discussed in Sharma et al. (2017). +Rosenberg et al. (2019) showed that by including the effect of triadic interactions among +the dominant response modes of the resolvent operator, prediction of coherent structures can +be significantly improved in oscillatory flows. This approach was followed by Symon et al. +(2019) where they studied flow over airfoils, and then by Symon et al. (2021), where they +investigated the energy transfer in some minimal flow units. A formalism was provided by +Padovan et al. (2020) to extend the resolvent framework to oscillatory flows, taking into +account the cross-frequency interactions. Rigas et al. (2021) used the resolvent framework +together with limited triadic interactions to investigate boundary layer transition. Bae et al. + +Modal decomposition of nonlinear interactions in wall turbulence +3 +(2021) investigated critical nonlinear mechanisms in Couette flow, again using resolvent +framework, by filtering the contribution of the dominantforcing mode to response generation. +In this study, we investigate dominant nonlinear mechanisms in wall-bounded turbulence. +The complexity of all possible triadic interactions in a turbulent flow can be reduced by +focusing on a certain quantity and eliminating all the non-relevant interactions. We use the +resolvent-based extended spectral proper orthogonal decomposition (RESPOD) (Towne et al. +2015; Karban et al. 2022) for this purpose. RESPOD is used to rank the triadic interactions +in terms of their correlation and/or their contribution to a given observable. The method +is implemented using a direct numerical simulation (DNS) of minimal Couette flow with +Reynolds number 400, where the spanwise wall shear is considered the target observable. In +similar minimal channel configurations, Bae et al. (2021) investigated the triadic interactions +contributing to the (훼, 훽) = (0, 2휋/퐿푧) mode, where 훼 and 훽 are streamwise and spanwise +wavenumbers, respectively, and 퐿푧 is the domain size in 푧-direction. We investigate here the +triadic interactions systematically extracted using RESPOD for the same mode. By doing so, +we present an approach to investigate nonlinear interactions in numerical datasets, where the +effect of each triad on the observable of interest may be studied separately using the resolvent +operator. This provides a quantitative analysis of the contribution of the various triads at play. +In this approach we wish to move a step further in the analysis of turbulence using the +resolvent operator. With numerical datasets, it is possible to recover forcings and responses +and relate them through the resolvent operator, as discussed above. The set of tools we +wish to develop here aim at a further exploration of the forcing, which is first split into +constituent triads, whose role in exciting coherent structures may be quantified using the +resolvent operator. Next, once a particular triad is isolated and recognised as dynamically +relevant, we wish to extract the individual structures in the flow that form each element of the +triad; it is also our objective to propose a modal analysis for that task. A schematic depicting +the flow chart of the analysis is presented in figure 1. +The flow configuration chosen here is minimal Couette flow due to its simplicity, leading +to a lower number of non-linear interactions and a few dominant coherent structures, which +simplifies the task. The available knowledge on the dynamics of this flow allows us to +demonstrate that the tool we propose does indeed identifies the dominant flow interactions. +The methods are general and may be employed in other flows of interest. With the approaches +developed here we move beyond the analysis capabilities given by the resolvent operator, +by analysing the non-linear terms at play, which are unquestionably relevant in turbulence +dynamics. +The remainder of the paper is structured as follows: the mathematical framework to extract +triadic interactions associated with a measured quantity is explained in §2. The details about +the DNS database of the minimal Couette flow are provided in §3. The results of identifying +the relevant triadic interactions and the energy transfer via these interactions in the minimal +Couette flow are discussed in §4. An modelling approach to predict the relevant forcing using +response structures is proposed in §5. Finally, some concluding remarks are provided in §6. +2. Extracting nonlinear interactions using RESPOD +We consider the incompressible Navier-Stokes (N-S) equations in Cartesian coordinates as, +M휕푡풒(풙, 푡) = N (풒(풙, 푡)) , +(2.1) +where 풒 = [푢 푣 푤 푝]⊤ is the state vector, N denotes the nonlinear N-S operator for +incompressible flows and the matrix M is zero for the continuity equation and identity +matrix for the remaining equations. Discretisation in space and linearisation around the + +4 +U. Karban et al. +Figure 1: Schematic depicting different stages of the analysis conducted in the study. +mean, 풒(풙), yields +M휕푡풒′(풙, 푡) − A(풙)풒′(풙, 푡) = B 풇 (풙, 푡), +(2.2) +where A(풙) = 휕푞N|풒 is the linear operator obtained from the Jacobian of N and 풇 (풙, 푡) +denotes all the remaining nonlinear terms, interpreted as a forcing term in the momentum +equations; B imposes zero forcing at the continuity equation. Full expressions for the +operators are given in Nogueira et al. (2021). We focus on parallel flow, i.e., a flow that +is homogeneous in two directions, for instance, in 푥 and 푧, with the mean flow varying only +in 푦. We can modify (2.2) to cast it in the resolvent form by applying Fourier transforms in +all homogeneous dimensions and rearranging as, +ˆ풒( ˜훼, 푦, ˜훽, ˜휔) = R( ˜훼, 푦, ˜훽, ˜휔) ˆ풇 ( ˜훼, 푦, ˜훽, ˜휔), +(2.3) +where ˜훼 and ˜훽 are the streamwise and spanwise wavenumbers, respectively, and ˜휔 is the +angular frequency, the hat indicates a Fourier transformed quantity and R( ˜훼, 푦, ˜훽, ˜휔) ≜ +(−푖 ˜휔M −A( ˜훼, 푦, ˜훽))−1B is the resolvent operator. For brevity, we drop the notation showing +dependence on wavenumber, wall-normal coordinate and frequency in what follows. +One can restrict and/or transform the response to a given set of observables ˆ풚풌, using a +measurement matrix, C as, +ˆ풚풌 = C ˆ풒풌, +(2.4) +which yields +ˆ풚풌 = ˜R풌 ˆ풇풌, +(2.5) +Focus on Fluids articles must not exceed this page length + +→ TwallForcing:RESPODmode +Response:ESPODmode +2mpose +into coherent +ctures +Vpk-i +2 +Pk-i +V +-2 +0 +2-2 +0 +2 +-2 +0 +2 +Decompose +RESPODmode +into individual +Forcing: +Interaction map: +triads +Deco +RESPODmode +Role ofeachtriad +Tik-i +triad forcing +in building the +stru +Xk +RESPODmode +-2 +0 +2 +2 +2 +2 +2 +1 +-2 +0 +-2 +0 +2 +-2 +0 +2 +-2 +0 +2 +-2 +0 +2 +2 +0 +2Modal decomposition of nonlinear interactions in wall turbulence +5 +where ˜R풌 ≜ C(−푖 ˜휔M − A)−1B is called the modified resolvent operator. +For the incompressible N-S equations, the forcing term in (2.2) is given as 풇 = 풖′⊤ · +∇풖′ − 풖′⊤ · ∇풖′, where (·) and ()⊤ denote dot product and transpose, respectively, and the +overbar denotes averaging in time and homogeneous directions 푥 and 푧. The forcing in the +wavenumber-frequency space, ˆ풇풌, is then obtained via a convolution, +ˆ풇풌 = +� +풊 +ˆ풖⊤ +풊 · ∇ ˆ풖풌−풊, +(2.6) +where 풊 = (훼푖, 훽푖, 휔푖), and 풌 = (훼푘, 훽푘, 휔푘) denote wavenumber-frequency combinations, +and summation over 풊 implies a nested summation over 훼푖, 훽푖 and 휔푖. Here, we consider +that 휔 is discretised. Note that (2.6) is valid assuming that the triplet 풌 contains at least one +non-zero element, such that the averaged term in 풇 has no contribution. +The RESPOD method, adapted from extended proper orthogonal decomposition (Borée +2003; Hoarau et al. 2006), finds, for a given observable, all structures in a ‘target’ event that +are correlated to the SPOD modes of the observable. Here we choose the target event to be +the nonlinear interactions, which give rise to the forcing terms in the resolvent framework, as +in Towne et al. (2015) and Karban et al. (2022). The goal is to map the triadic interactions +underpinning the dominant coherent structures of the flow. +The SPOD modes of an observable, ˆ풚푘, can be estimated using the ensemble matrix of +realisations, through the eigendecomposition, +ˆY 퐻 +풌 W ˆY풌 = ˆ휣풌휦풌 ˆ휣 +퐻 +풌 , +(2.7) +and the SPOD modes are obtained from ˆ휣풌 as, +휳풌 = ˆY풌 ˆ휣풌휦−1/2 +풌 +, +(2.8) +where ˆY풌 ≜ [ ˆ풚풌 (1) ˆ풚풌 (2) · · · ˆ풚풌 (푃)] denotes the ensemble matrix for different realisations of +ˆ풚풌 with 푃 being the total number of realisations, 휳풌 and 휦풌 are SPOD modes and their +associated eigenvalues, respectively (see Towne et al. (2018)), and W is a positive-definite +matrix of quadrature gains along 푦, which is discretised. The SPOD modes in the columns +of 휳풌 are the optimal orthonormal basis for the realisations of the observable ˆ풚풌. +In Karban et al. (2022), it was shown that the coefficient matrix ˆ휣풌 can be used to extract +the part in the forcing that is correlated with the observed SPOD mode as +흌풌 = ˆF풌 ˆ휣풌휦−1/2 +풌 +, +(2.9) +where, ˆF풌 is the ensemble matrix of ˆ풇풌. The RESPOD forcing mode 흌풌 satisfies +휳풌 = ˜R풌 흌풌, +(2.10) +i.e., the RESPOD forcing mode excites precisely the SPOD mode via the resolvent operator. +As discussed in Karban et al. (2022), the RESPOD mode includes the part of the forcing +that is correlated to the SPOD mode of interest. This comprises a “silent” part 흌풌,푠 , which +generates no response ( ˜R풌 흌풌,푠 = 0) but is nonetheless present in the dataset and correlated +to the SPOD mode. +Substituting into (2.9) the expansion in (2.6), which shows the triadic interactions summing +up to yield the forcing ˆ풇풌, one can compute the triadic interactions correlated with the +observable as +ˆ풇풌 ˆ휣풌휦−1/2 +풌 += +� +풊 +� +ˆU⊤ +풊 · ∇ ˆU풌−풊 +�⊤ ˆ휣풌휦−1/2 +풌 +, +(2.11) + +6 +U. Karban et al. +where ˆU denotes the ensemble matrix of ˆ풖. Defining +휞풊,풌−풊 ≜ +� +ˆU⊤ +풊 · ∇ ˆU풌−풊 +�⊤ ˆ휣풌휦−1/2 +풌 +, +(2.12) +the correlated forcing 흌풌 can be decomposed as, +흌풌 = +� +풊 +휞풊,풌−풊 = +� +풊 +� +ˆU⊤ +풊 · ∇ ˆU풌−풊 +�⊤ ˆ휣풌휦−1/2 +풌 +. +(2.13) +We define the energy as, +∥(·)∥2 = 휀{(·)퐻W (·)}, +(2.14) +where the superscript 퐻 indicates Hermitian transpose, and 휀{·} denotes the expectation +operator. In what follows, 휀{·} corresponds to time-averaging for time-dependent structures, +and to ensemble averaging for Fourier realisations in the frequency space. The energy of +휞풊,풌−풊, denoted by ∥휞풊,풌−풊∥2, for all 풊 shows the correlation map of the nonlinear interactions +related to the observed SPOD mode, 휳풌. One can instead investigate ∥휼풊,풌−풊∥2, where +휼풊,풌−풊 ≜ R풌휞풊,풌−풊, which provides the contribution of a triadic interaction to a given SPOD +mode of the measured state, as suggested by (2.10) and (2.13). By removing or including +terms in the sum in equation (2.13), one is able to inspect the contributions of each triad 풊 to +the observable. +3. Database of the minimal Couette flow +The use of RESPOD for detection of ‘important’ nonlinear interactions associated with +a specific measurement is tested on a minimal Couette flow (Hamilton et al. 1995), sim- +ilar to that investigated by Nogueira et al. (2021). The simulations are performed using +the ‘ChannelFlow’ code, a pseudo-spectral incompressible flow solver using a Fourier- +Chebyshev discretisation in the wall-parallel and wall-normal directions, respectively (see +www.channelflow.ch for details). The dimensions of the minimal box are (퐿푥, 퐿푦, 퐿푧) = +(1.75휋ℎ, 2ℎ, 1.2휋ℎ), where the subscripts 푥, 푦 and 푧 denote the streamwise, wall-normal and +spanwise directions, and ℎ is the channel half-height. These are the minimal dimensions to +sustain turbulence in Couette flow at low Reynolds number, as studied by Hamilton et al. +(1995). The domain was discretised as (푛푥, 푛푦, 푛푧) = (32, 65, 32) with a dealiasing factor of +3/2 in the wall-parallel directions. The channel walls move with wall velocity, ±푈푤 yielding +a Reynolds number, 푅푒 = 400 based on 푈푤 and ℎ, corresponding to a friction Reynolds +number, 푅푒휏 ≈ 34. Once the initial transients disappeared, the flow data was stored for +7000 convective units with a sampling rate, Δ푡 = 0.25. Temporal data is transformed into +frequency space using blocks of 2048 time steps with 50% overlapping and using a second- +order exponential windowing function given in Martini et al. (2019). While computing the +forcing data, the correction due to using windowing functions is implemented as described +in Martini et al. (2019) and Nogueira et al. (2021). We verified that the forcing acting on the +resolvent operator accurately yields the response, however, the comparison is not shown here +for brevity. +Figure 2 presents the profiles for the mean and the root-mean-square (RMS) of the velocity +components, 푢, 푣 and 푤 in the streamwise, wall-normal and spanwise directions, respectively, +along the wall-normal direction, 푦. We see that the mean flow deviates from the laminar +solution given by (푦 − 1) due to nonlinear interactions between turbulent fluctuations. The +RMS plots indicate that the fluctuations in 푢 peak around 푦 = 1.5 and 푦 = 0.5. A similar but +smaller double-peak structure is seen in the RMS of 푤 with the peaks occurring at the same + +Modal decomposition of nonlinear interactions in wall turbulence +7 +Figure 2: Mean (a) and the RMS (b) profiles of the velocity components, 푢 (black solid), 푣 +(red dashed) and 푤 (blue dash-dotted) along the wall-normal direction. +wall-normal positions. The RMS of 푣 peaks around the centre at an amplitude slightly lower +than that of 푤. +We choose wall shear fluctuations in the spanwise direction, 휏푧 ≜ 휕푧푢′|푦={0,2} at both +upper and lower walls as our observable. Spanwise wall shear was used to extract self-similar +wall-attached structures in a turbulent channel in Karban et al. (2022). We use it to have a +low-rank representation of the flow associated with this quantity in this study. Here and in +what follows, we use the term ‘wall-attached’ to define quantities that are correlated with the +wall-shear. +For simpler notation, wavenumbers will be presented in integers defined as 훼 = ˜훼퐿푥/2휋 +and 훽 = ˜훽퐿푧/2휋. Similarly, mode frequencies will be presented in integer bins denoted by +휔 = ˜휔푁퐹/ 푓푠, ranging in [−푁퐹/2, 푁퐹/2 − 1], where 푁퐹 = 2048 is the number of temporal +points used for taking the Fourier transform (FT) and 푓푠 ≜ 1/Δ푡 is the sampling rate of the +database. +Minimal Couette flow is known to be dominated most of the time by rolls and streaks +spanning the entire computational domain, corresponding to (훼, 훽) = (0, 1). Occasionally +wavy disturbanceswith 훼 = 1 appearafter streak instability and breakdown,and subsequently +non-linear interactions among such “waves” lead to the formation of new rolls, restarting +the process (Hamilton et al. 1995; Hall & Sherwin 2010). Figure 3 shows the time-averaged +energy contained in each wavenumberpair together with the ratio of the time-averaged energy +of the wall-attached structures to the total energy at each wavenumber pair. We see that the +mode pair (훼, 훽) = (0, 1), related to streaks and rolls, contains most of the fluctuation energy +(∼75%). In contrast, the modes (±1, 0), related to waves, and (0, 2), which we will refer to as +roll-streak harmonic, contain slightly less than 5% of the total energy, and all the other mode +pairs have less than 2%. The energy of the wall-attached part of the state, denoted by 풒푎, of + +8 +U. Karban et al. +Figure 3: a) Energy of flow structures at different wavenumber pairs averaged over time. b) +Ratio of the average energy of the wall-attached structures to the total energy at their +wavenumbers. +the roll-streak mode (0,1) is around 80% of its total energy. Therefore, the coherent structures +correlated with the spanwise wall-shear can constitute a good low-rank representative of the +flow at this wavenumber pair. A similar case is observed for the wave modes (±1, 0) while +for the mode (0,2), the energy ratio of the wall-attached part is around 15%. +Figure 4 shows the power spectral density (PSD), integrated along the wall-normal +direction, of the velocity field 풒 at wavenumber pairs (훼, 훽) = (0, 1), (0,2), (1,0) and +(1,1). Although the oblique-wave mode (훼, 훽) = (1, 1) is energy-wise insignificant, it plays +a critical role for transfer of energy to (훼, 훽) = (0, 1) mode, as will be shown later, and hence +is included here. We see that the streamwise-constant modes peak around the zero frequency, +which is expected due to their quasi-steady nature, while the wave modes (1,0) and (1,1) have +their peak around ˜휔 ≈ 0.1 (휔 = 8), leading to a phase speed of 푐+ ≜ ˜휔+/ ˜훼+ = ±1 in wall +units (negative values arise if frequency or wavenumber is negative) corresponding to ∼ 10% +of wall velocity. The shape of the spectra is observed to be similar for the modes that have +the same streamwise wavenumber. This trend can be more clearly seen in figure 5, where the +integrated PSDs normalised with respect to the peak value are plotted for different modes. +We see two different families of PSD distributions for the two streamwise wavenumbers, +훼 = 0 and 훼 = 1, respectively. +We now focus on the most energetic mode (훼, 훽) = (0, 1) at its peak-energy frequency, +휔 = 0. The wall-attached forcing and response modes, 흌풌 and R풌 흌풌, respectively, are shown +in figure 6. The response field, which is the velocity field correlated to the wall-shear, consists +of streaks and rolls. Given that the upper and lower walls have positive and negative mean +velocities, respectively, the phase relation between streaks and rolls is reminiscent of the +lift-up mechanism (Brandt 2014). This is further supported regarding the associated forcing +mode. At the spanwise positions where the streamwise vortices are located, the forcing is +mainly located near the walls aligned with the 푦-direction, causing a moment to generate +the streamwise vortices. These vortices then generate streaks by carrying the high- and +low-velocity structures near the upper and lower walls, respectively, towards the channel +centre. Note that the forcing component in the streamwise direction is in opposite phase to + +Modal decomposition of nonlinear interactions in wall turbulence +9 +Figure 4: PSDs of ˆ풒(0,1) (blue), ˆ풒(0,2) (orange), ˆ풒(1,0) (yellow) and ˆ풒(1,1) (violet) +integrated over the wall-normal direction. +Figure 5: PSDs of ˆ풒(0,{1,3}) (black; solid, dashed, dash-dotted, respectively), and +ˆ풒(1,{0,3}) (red; solid, dashed, dash-dotted, dotted, respectively) integrated over the +wall-normal direction and normalised with respect to the peak value of each mode. +the streaks seen in the response. This indicates that the streaks are generated by the lift-up +mechanism despite the counteracting effect of the streamwise forcing, as previously reported +by Nogueira et al. (2021). The response generation at this triplet can therefore be considered +suboptimal. +We also plot the wall-attached response fields for the modes (훼, 훽, 휔) = (0, 2, 0), (1,0,8) +and (1,1,8), respectively, in figure 7. Each mode is shown at its peak frequency (see figure +4). The response field contains streaks and rolls for the mode (0,2,0) as in the roll-streak +mode (0,1,0), but with doubled periodicity, and thus, is called roll-streak harmonic. The +mode (1,0,8) is dominated by its spanwise component, leading to a wave mode. Finally, the + +10 +U. Karban et al. +Figure 6: Wall-attached part of the velocity (a) and the associated forcing (b) reconstructed +in the 푦-푧 plane for the mode (훼, 훽, 휔) = (0, 1, 0). The color plot indicates the streamwise +component and the arrows show the spanwise and wall-normal components. +Figure 7: Wall-attached part of the velocity reconstructed in the 푦-푧 plane for the modes +(훼, 훽, 휔) = (0, 2, 0) (left), (1,0,8) (center) and (1,1,8) (right). The color plot indicates the +streamwise component and the arrows show the spanwise and wall-normal components. +response field for the mode (1,1,8) contains some oblique wave structures tilted with the +mean flow. +4. Nonlinear interactions in the minimal Couette flow +4.1. Extracting important triadic interactions +The maps showing the energy of the nonlinear interactions contributing to the dominant +mode, 풌 = (훼푘, 훽푘, 휔푘) = (0, 1, 0) are shown in figure 8. Different columns compares the +maps ∥ ˆ풖풊∇ ˆ풖풌−풊∥2, ∥휞풊,풌−풊∥2 and ∥휼풊,풌−풊∥2, which correspond respectively to energies of the +direct triadic interactions, the interactions correlated with the wall shear, and the response +to the latter obtained by the resolvent operator. Note that only the triplet 풊 = (훼푖, 훽푖, 휔푖) is +shown, where for each 풊, there exists a 풌 − 풊 such that the nonlinear interaction between 풊 +and 풌 − 풊 yields 풌 = (0, 1, 0). Starting with 휔푖 = 0, we see that the interaction between the +roll-streak mode, 풊 = (0, −1, 0) and its complementary, roll-streak harmonic 풌 −풊 = (0, 2, 0), +is dominant in all three maps, indicating that the interaction is large in amplitude, highly +correlated to the dominant mode, and generates the response with the largest amplitude. We +also observe large amplitude for the interaction (0, 2, 0) + (0, −1, 0), which involves the same +structures with the previous one, but with the gradient operator acting on the roll-streak +mode (0, −1, 0). The interactions involving wave modes (±1, 1, 0) + (∓1, 0, 0), although not +yielding a large forcing component (low amplitudes at the first two rows of figure 8), are +seen to be present in the response map (third row of figure 8), implying that these modes +efficiently drive the observable. For non-zero frequencies 휔푖 = 4 and 8, we observe that the +contribution of streamwise-constantmodes with 훼푖 = 0 decreases with increasing 휔, whereas +Rapids articles must not exceed this page length + +=Modal decomposition of nonlinear interactions in wall turbulence +11 +Figure 8: Amplitude maps of ∥풖⊤ +풊 · ∇풖풌−풊∥2 (top), ∥휞풊,풌−풊 ∥2 (middle), and ∥ ˜R풌휞풊,풌−풊 ∥2 +(bottom) obtained at 휔푖 = 0 (left), 휔푖 = 4 (center) and 휔푖 = 8 (right), for the mode +풌 = (훼푘, 훽푘, 휔푘) = (0, 1, 0). Only the modes 풊 are shown while the complementary +modes 풌 − 풊 are selected to yield 풌 = (0, 1, 0). +wave modes with 훼푖 = 1 drive an increasingly strongerresponse for higher frequencies, which +may be attributed to the different frequency content of streamwise-constant and wavy modes, +explored in figures 4 and 5. +To investigate the overall contribution to the dominant mode (0, 1, 0) via a given wavenum- +ber pair (훼푖, 훽푖) and its complementary, we define the forcing mode, ˇ휞풊,풌−풊, obtained by +summing 휞풊,풌−풊 over the frequency index, 휔푖, i.e., adding the nonlinear interactions between +all different frequency combinations, and compute its response via ˜R풌 ˇ휞풊,풌−풊. Similar to +the energy maps shown in figure 8, the map of ∥ ˜R풌 ˇ휞풊,풌−풊∥2 is plotted in figure 9-a, which +shows that the response generation is dominated by six interactions: two streamwise-constant, +which are (0, {−1, 2})+(0, {2, −1}) involving the roll-streak and roll-streak harmonic modes, +and four streamwise-periodic over 퐿푥, which are (±1, {0, 1}) + (∓1, {1, 0}) involving wave +modes. Note that here and in what follows, we use curly brackets for short hand notation of +multiple modes. For instance, (0, {−1, 2}) denotes the modes (0, −1) and (0, 2). +Besides the magnitude of the response to a given triadic interaction, it is important to +evaluate how it contributes to the overall response. As shown in Nogueira et al. (2021) +and Morra et al. (2021), different forcing components can interfere destructively. In what +follows we propose a measure to identify which interactions are constructive, amplifying +a given mode, or destructive, saturating or damping it. As a measure of constructive- +ness/destructiveness of a given interaction, we calculate the inner product between the + +12 +U. Karban et al. +Figure 9: a) Amplitude map of the response generated by the wall-correlated interactions +at all frequencies added together. b) The map of normalised inner product between the +overall response and the response with contribution of a single interaction masked, +computed at different wavenumber pairs used for masking. Both maps are generated for +the mode (훼푘, 훽푘, 휔푘) = (0, 1, 0). +response to a single wall-attached interaction, ˆ풒풊,풌−풊 푎, and the wall-attached velocity field, +ˆ풒풌 푎, +⟨ ˆ풒풊,풌−풊 푎, ˆ풒풌 푎⟩ ≜ 휀{ ˆ풒퐻 +풊,풌−풊 푎W ˆ풒풌 푎}. +(4.1) +An interaction map is obtained by calculating (4.1) for each wavenumber pair and normalising +the result with ∥ ˆ풒풌 푎∥2, which shows a normalised projection, and thus the construc- +tive/destructive role of each triadic interaction in generating the wall-attached response. The +resulting map is shown in figure 9-b. Note that the interaction map should sum up to 1, i.e., the +sum of all destructive and constructive interactions lead to the mode observed in the system. +The analysis reveals that the contributions from the interactions (0, {−1, 2}) + (0, {2, −1}) +decrease the response energy, implying a destructive interference between these interactions +and the remaining ones. The interactions involving wave modes (±1, {0, 1}) + (∓1, {1, 0}) , +on the other hand, cause the response energy to increase, implying a constructive effect. +The effect of these interactions on the response field is shown in figure 10 by masking +these interactions, i.e., subtracting the contributions from the designated interactions from +the overall response computed by the resolvent. We see that masking the interactions +(0, {−1, 2}) + (0, {2, −1}) mainly affects the streaks causing an increase in their amplitude, +while the roll remains nearly unchanged. Masking the interactions (±1, {0, 1}) + (∓1, {1, 0}) +almost completely eliminates the streamwise vortices, which also causes the lift-up effect +to be eliminated. This results in streaks with smaller amplitude and reversed phase. This +result is consistent with models of self-sustaining process in wall turbulence, where rolls +are excited by non-linear interactions involving waves with non-zero 훼 (Hamilton et al. +1995; Hall & Sherwin 2010). Remember that in the RESPOD forcing mode shown in figure +6, the streamwise component counteracts the lift-up mechanism forced by the spanwise +components. These results, when combined, imply that the streamwise and spanwise +components in the RESPOD forcing mode, 흌풌, are mainly constructed by the nonlinear +interaction groups (0, {−1, 2}) + (0, {2, −1}) and (±1, {0, 1}) + (∓1, {1, 0}), respectively. +Masking (±1, {0, 1}) + (∓1, {1, 0}) causes the lift-up mechanism, which is an efficient +means to generate streaks via streamwise vortices, to disappear. The remaining streamwise +component in 흌풌 is mainly constructed by (0, {−1, 2}) + (0, {2, −1}) and generates streaks +with negative phase, reducing the amplitude ofthe streaks generated by the lift-up mechanism. + +Modal decomposition of nonlinear interactions in wall turbulence +13 +Figure 10: Velocity field corresponding to the wall-attached structure +(훼푘, 훽푘, 휔푘) = (0, 1, 0) in the 푦-푧 plane. Top-left: the entire response; top-right: the +response obtained by masking the interactions between the modes (훼푖, 훽푖) = (0, {−1, 2}) +and their complementary modes; bottom-left: the response obtained by masking the +interactions between the modes (훼푖, 훽푖) = (±1, {0, 1}) and their complementary modes; +bottom-right: the response obtained by masking the interactions between the modes +(훼푖, 훽푖) = (0, {−1, 2}) and their complementary modes as well as the interactions between +the modes (훼푖, 훽푖) = (±1, {0, 1}) and their complementary modes. +This elucidates the destructive interference among components observed by Nogueira et al. +(2021). The present results show that such destructive interference occurs among different +triadic interactions. Masking all six modes almost entirely eliminates the response as seen in +figure 10. +4.2. Energy transfer via triadic interactions +The interaction map shown in figure 9-b can also be interpreted in terms of energy exchange +between different modes via nonlinear interactions. Symon et al. (2021) investigated, by +employing the spectral form of the transport equation of turbulent kinetic energy (TKE), +the overall relation between production, dissipation and the transfer of energy for individual +wavenumber pairs in parallel, stationary turbulent flows. The spectral TKE equation is given, +using indicial notation for the last two terms for convenience, as +휕 ˆ퐸 +휕푡 = ℜ +� +− +∫ 2 +0 +휕푢 +휕푦 ˆ푢∗ˆ푣푑푦 − 1 +푅푒 +∫ 2 +0 +휕 ˆ푢푚 +휕푥푛 +휕 ˆ푢∗푚 +휕푥푛 +푑푦 − +∫ 2 +0 +ˆ푢∗푚 ˆ푓푚푑푦 +� +, +(4.2) +where ℜ{·} indicates the real part, the hat in this equation denotes, by abuse of notation, +Fourier transformed quantities in the stremwise and spanwise directions, ˆ퐸 is the spectral +TKE of a given wavenumber pair, the superscript * denotes complex conjugate, 푚 and 푛 +denote that the vector indices, ˆ푓푚 is the 푚th componentof the forcing vector (see Symon et al. + +14 +U. Karban et al. +(2021) for derivation of (4.2)). Here, we assume that the Couette flow is stationary in the +time interval we investigate, which renders +휕 ˆ퐸 +휕푡 = 0. +(4.3) +The three terms on the right-hand side of (4.2) correspond to the production, dissipation and +nonlinear transfer of the turbulent kinetic energy, respectively, which, thanks to (4.3), sum up +to zero for a given wavenumber pair. One can write (4.2) in the frequency domain by Fourier +transforming in the time domain each term on the right-hand side and Welch averaging +(Jin et al. 2021), which still satisfies the energy balance for any wavenumber-frequency +triplet 풌 as, +0 = ℜ +� +− +� 휕푢 +휕푦 ˆ푢풌, ˆ푣풌 +� +− 1 +푅푒 +� 휕 ˆ푢푚풌 +휕푥푛 +, 휕 ˆ푢푚풌 +휕푥푛 +� +− +� +ˆ푢푚풌, ˆ푓푚풌 +�� +. +(4.4) +The contributions of production, dissipation and nonlinear transfer to the energy balance +for different wavenumber pairs at zero frequency are illustrated in figure 11. We see that the +roll-streak mode (0,1) draws the most energy from the mean flow to produce TKE, and is +the only mode to transfer this energy to other modes via nonlinear transfer. All the modes +are seen to lose energy via dissipation as expected. Note that the energy balance map is +symmetric in the 훽푖 axis, which is not shown for better readability of the plot. +Cho et al. (2018) computed the nonlinear energy transfer via each triadic interaction by +expanding the convolution in the third term on the right-hand side of (4.2) as in (2.6). Here, +we do a similar analysis for the nonlinear transfer term in the frequency-domain energy +balance equation given in (4.4). The resulting energy transfer map for (0,1,0) mode is shown +in figure 12-a, where blue and red colors indicate losing and gaining energy, respectively, via +the corresponding interaction. We see that roll-streak mode (0,1,0) mostly transfer energy via +the interactions (0, {−1, 1}, 휔)+(0, {2, 0}, −휔) and (±1, 0, 휔)+(∓1, 1, −휔) with (0, −1, 휔)+ +(0, 2, −휔) being the dominant one. It is also seen to gain energy via a number of interactions +but the incoming energy rate is negligible compared to outgoing rate, and hence, is not +discussed here. For the triadic interaction (0, −1, 휔) + (0, 2, −휔), we calculated that the +transfer from the interaction (0, −1, 0) + (0, 2, 0) is -0.06, which constitutes half of the +transfer from all the frequencies. Since the flow is symmetric in the spanwise direction, +the modes (0, 1, 0) and (0, −1, 0) are complex conjugates of each other. Thus, we can say +that the mode (0,1,0) is transferring energy to roll-streak harmonic mode (0,2,0) via a triadic +interaction involving its conjugate mode, making the (훼푘, 훽푘) = (0, 2) mode the second-most +energetic with a peak at zero frequency (see figure 4). This sort of transfer can be associated +to the energy cascade in turbulence from large to small scales observed in high-푅푒 flows. +Combining the results obtained inspecting the energy transfer and the findings of the +previous subsection indicate that the mode (0,1,0) gains almost all of its energy from the +mean flow via the lift-up mechanism and transfers some of this energy via nonlinear transfer to +streamwise-constant modes as the onset of energy cascade. This nonlinear transfer appears as +a streak with opposite phase (see bottom left plot in figure 10). This will be further discussed +below. The response seen in (0,1,0) mode is a result of the destructive relation between +the lift-up mechanism, excited by the wave modes, and the nonlinear transfer associated to +roll-streak modes. +We have seen that (0,2,0) mode receives its energy via a nonlinear transfer mechanism +according to the energy budget plot given in figure 11. The nonlinear transfer via each +triadic interaction contributing to (0,2,0) mode is shown in figure 12-b. The map reveals +that (0,2,0) mode receives energy via the interactions (0, 1, 휔) + (0, 1, −휔) at a rate similar + +Modal decomposition of nonlinear interactions in wall turbulence +15 +Figure 11: Production (left), dissipation (center), and nonlinear transfer (right) of the +spectral turbulent kinetic energy for different wavenumber pairs at 휔 = 0. Both the size +and the color intensity of the markers indicate amplitude. +Figure 12: Nonlinear energy transfer to the modes (a) 풌 = (0, 1, 0), (b) (0,2,0) and (c) +(0,3,0) from different triadic interactions. Transfer from different frequencies for a given +wavenumber pair is integrated. +to the energy transfer in (0,1,0) mode via the interactions (0, −1, 휔) + (0, 2, −휔). Some of +the energy that (0,2,0) mode receives is transferred to the next harmonic via the interaction +(0, −1, 휔) + (0, 3, 휔). Looking at the nonlinear transfer map for (0,3,0) mode in figure 12-c, +we see that energy is transferred via the interaction (0, 1, 0) + (0, 2, 0), once again, at a rate +similar to the transfer from (0,2,0) shown in figure 12-b. One can trace the energy cascade +for the negative 훽 modes in the same way, which yields the same transfer maps mirrored in +the 훼푖 and 훽푖 axes. This suggests that the modes (훼, 훽) = (0, ±1), once extracting energy +from the mean and transferring it to harmonics (0, ±2), plays a role to transfer energy via +the triadic interactions associated to higher 훽, i.e., they provide a medium for the nonlinear +transfer to higher 훽 modes without losing noticeable energy. +One can further analyse the nonlinear energy transfer by dissecting the contributions to + +16 +U. Karban et al. +Figure 13: Nonlinear energy transfer to the mode 풌 = (0, 1, 0) from the 푥-, (left), 푦- +(center) and 푧- (right) components of different triadic interactions. Transfer from different +frequencies for a given wavenumber pair is integrated. +Figure 14: The same map with figure 9-b obtained for the mode (훼푘, 훽푘, 휔푘) = (0, 2, 0) +(right) and the corresponding response in the 푦-푧 plane (left). +the nonlinear energy transfer from 푥-, 푦- and 푧-components, i.e., the terms ˆ푢풌 ˆ푓푥풌, ˆ푣풌 ˆ푓푦풌 +and ˆ푤풌 ˆ푓푧풌. In figure 13, we show the map of nonlinear energy transfer to the roll-streak +mode (0,1,0) via each spatial component. The three maps shown in this figure sum up +to yield the nonlinear transfer map shown in figure 12-a. The energy in (0,1,0) mode is +transferred to other modes mostly via the 푥-component, i.e., the streaks. This is aligned with +the results shown in 10, where it was shown that the response generated by the interactions +responsible for nonlinear transfer yielded streaky structures. The roll-streak mode is seen +to receive energy via the spanwise component of the triadic interactions involving wave +modes (±1, {0, 1}) + (∓1, {1, 0}), which can be associated to the amplification by the lift- +up mechanism, consistent with the analysis done with 10. This positive energy transfer is +nonetheless lower than the loss of energy via the streaks. No significant energy transfer +occurs via the wall-normal component. +We now investigate whether a similar destructive relation takes place between the nonlinear +transferand the lift-up mechanism forroll-streak harmonic mode (0,2,0)as in roll-streak mode +(0,1,0). In figure 14, we show the same interaction map given in figure 9-b for the mode +(훼푘, 훽푘, 휔푘) = (0, 2, 0) together with the reconstruction of the mode in the 푦-푧 plane. Similar +to the (0,1,0) mode, we see streamwise vortices and streaks in (0,2,0) with a halved period in +the spanwise direction. The rolls and the streaks are in opposite phase compared to the (0,1,0), + +Modal decomposition of nonlinear interactions in wall turbulence +17 +Figure 15: Velocity field corresponding to the wall-attached structure +(훼푘, 훽푘, 휔푘) = (0, 2, 0) in the 푦-푧 plane. Top-left: the entire response; top-right: response +obtained by masking the interactions (훼푖, 훽푖) = (0, ±1)+complementary and +(±1, 0)+complementary; bottom-left: response obtained by masking the interactions +(±1, 2)+complementary; bottom-right: the response obtained by masking the six +interactions. +which implies that they are not directly associated to the lift-up mechanism. The interaction +map indicates strong contribution to response generation from the interactions associated +with the nonlinear energy transfer, i.e., the ones that appear in figure 12-b as well. Besides +these interactions, we see some positive contributions from the interactions (±1, 2) + (∓1, 0). +To see the effect of these interactions on the response, we mask these interactions and +observe the change in the response in figure 15. Masking the interactions associated to +nonlinear transfer, we obtain a response field reminiscent of the lift-up mechanism. This +partial response is due to the interactions (±1, 2) + (∓1, 0). Masking these interactions, on +the other hand, yields a response field with inverted streaks and vortices. Similar to the case +of (0,1,0) mode, there exists a destructive interference between the nonlinear energy transfer +and the lift-up mechanism. Masking both groups of interactions causes the response to be +almost zero, indicating that these six interactions are the active ones for response generation. +5. Modelling coherent structures in triadic interactions +RESPOD provides the forcing mode that drives a desired SPOD mode of a given observable, +and the individual triadic interactions that sum up to yield this forcing mode. However, a +RESPOD forcing mode for a triadic interaction, +휞풊,풌−풊 = � ˆ풖⊤ +풊 · ∇ ˆ풖풌−풊 +�⊤ ˆ휣풌휦−1/2 +풌 +, +(5.1) +does not provide the modes associated to ˆ풖풊 and ˆ풖풌−풊 that generate the observable-correlated +forcing component 휞풊,풌−풊. Predicting the structures that form the forcing modes that drive + +18 +U. Karban et al. +Figure 16: The wall-shear-correlated forcing mode, 휞풊,풌−풊 (left) in comparison to the +forcing mode predictions, 휳풊∇휳풌−풊 and 휻풊∇휻풌−풊, which are based on the SPOD modes +(center) and ESPOD modes (right), respectively. The triplet 풊 = (훼, 훽, 휔) is set to be +(0,2,1), (0, −1, 1), (1,0,1) and (1,1,1) (from top to bottom) and the complementary triplet +풌 − 풊 is chosen to yield 풌 = (0, 1, 1). +the most energetic structures in the observable is important, as it shows which structures +are actually involved in nonlinear interactions, and may also help designing reduced-order +models for turbulent flows. Note that these response modes are not unique as it is always +possible to multiply one of them with a dummy vector, 휼, and and the other by the inverse +for elementwise multiplication, 1/휼, respectively, which eventually yields the same forcing +mode assuming that 1/휼 is finite. Among potential candidates to yield the forcing mode +given in (5.1) are the optimal SPOD modes, 휳풊 and 휳풌−풊 of 풖풊 and 풖풌−풊, respectively, and +the extended SPOD (ESPOD) mode defined as +휻풊 ≜ ˆ풖풊 ˆ휣 +(1) +풌 λ(1) +풌 +−1/2, +(5.2) +휻풌−풊 ≜ ˆ풖풌−풊 ˆ휣 +(1) +풌 λ(1) +풌 +−1/2, +(5.3) +where λ(1) +풌 +is the largest eigenvalue of 휦풌, and ˆ휣 +(1) +풌 +is the associated eigenvector. +In figure 16, we compare the forcing modes for the triplet 풌 = (0, 1, 1) using the triadic +interactions 휳풊∇휳풌−풊 and 휻풊∇휻풌−풊, respectively, for different 풊’s. The triplets 풌 and 풊 in +this figure are chosen such that the difference between the two forcing predictions manifests +clearly. All the modes are normalised to have unit magnitude with respect to ∥ · ∥2 norm. +The phase in the forcing modes obtained using SPOD modes is random, while the phases of +the RESPOD forcing modes 휞풊,풌−풊 and the forcing predictions based on the ESPOD modes, + +Sk-iTik-i +wiVk-i +20 +2 +0 +2 +0 +20 +0 +-2 +0 +2 +-2 +0 +2 +-2 +2 +2 +91 +0 +0 +-2 +2 +-2 +2 +0 +0 +-2 +2 +2 +2 +91 +0 +0 +0 +-2 +0 +2 +-2 +0 +2 +-2 +2 +2 +2 +1 +1 +1 +0 +0 +0 +-2 +0 +2 +-2 +2 +-2 +2 +2Modal decomposition of nonlinear interactions in wall turbulence +19 +휻풊∇휻풌−풊, are both associated with the phase of the wall shear, and therefore, not random with +respect to each other. In the following, we only compare the mode shapes. +Using the optimal SPOD modes yields forcing modes similar to the RESPOD forcing mode +for the interactions among wave modes (1, 0, 1) + (−1, 1, 0) and (1, 1, 1) + (−1, 0, 0), while +the mode shapes significantly differ for the interactions (0, 2, 1) + (0, −1, 0) and (0, −1, 1) + +(0, 2, 0). Using the ESPOD modes, on the other hand, yields forcing modes that are similar +to the RESPOD forcing modes for the latter two interactions, while it is not the case for the +former two. This suggests that one may blend the two modes as +흋풊 = 푎휻풊 + (1 − 푎)휳풊, +흋풌−풊 = 푏휻풌−풊 + (1 − 푏)휳풌−풊, +(5.4) +to predict the response modes generating the RESPOD forcing mode, where 푎 and 푏 denote +the blending coefficients. To be able to quantify the accuracy of the prediction, we define a +similarity measure, +훾휼 = +흌퐻 +풌 W (휼⊤ +풊 · ∇휼풌−풊)⊤ +� +∥ 흌풌 ∥2∥휼⊤ +풊 · ∇휼풌−풊∥2 +, +(5.5) +for a given mode pair 휼풊 and 휼풌−풊. We propose deciding the coefficients 푎 and 푏 regarding +the correlation levels between the observable ˆ풚풌 and the response and forcing structures, ˆ풖풊, +ˆ풖풌−풊 and ˆ풇풌 that are in a triadic interaction. Since the ESPOD and RESPOD modes yield the +correlated parts of the response and forcing to the wall shear, respectively, the correlation +levels can be computed as +푐휻 +풊 = λ(1) +풌 ∥휻풊∥2/∥ ˆ풖풊∥2, +(5.6) +푐휞 = λ(1) +풌 ∥휞풊,풌−풊∥2/∥ ˆ풖⊤ +풊 · ∇ ˆ풖풌−풊∥2. +(5.7) +Figure 17 shows the values of 훾휳, 훾휻, max{푐휻 +풊 , 푐휻 +풌−풊} and 푐휞 for a number of triplets 풌 +and 풊. We observe that the similarity measure 훾휻 is high when 푐휞 is not very small (> 0.1) +and either of 푐휻 +풊 or 푐휻 +풌−풊 is close to 푐휞, i.e., the forcing structures and at least one of the +response structures in the triadic interaction are correlated to the observed quantity. On the +other hand, the similarity measure 훾휳 is high when 푐휞 is again not very small and 푐휻 +풊 and +푐휻 +풌−풊 are close to zero, i.e., the forcing is correlated to the observable ˆY풌 while the response +structures in the triadic interaction are not. Note that neither of 훾휻 or 훾휳 is close to 1 when the +correlation between the forcing and the observable, 푐휞, is low as in the case of 풊 = (1, 2, 1) +and (−1, −1, 1) shown in figure 17. However, this usually constitutes a case where accurate +prediction of the forcing mode shape is not important due to its small contribution to response +generation. +Based on these observations, we present the following empirical relation for the blending +coefficients +푎 ≜ min{푐휻 +풊 /푐휞, 1} and 푏 ≜ min{푐휻 +풌−풊/푐휞, 1}. +(5.8) +The model in (5.4) provides a smooth blending of the ESPOD modes and SPOD modes +depending on the maximum correlation level between the ESPOD modes and the observed +quantity. The resulting modes are normalized afterwards using ∥ · ∥2 norm. We compare the +similarity measures 훾휳, 훾휻 and 훾흋 in figure 18. We see that the model improves the alignment +everywhere compared to the alignment levels obtained using the SPOD and ESPOD modes +separately. The improvement is more evident when comparing 훾휻 and 훾흋 for the interactions + +20 +U. Karban et al. +Figure 17: Comparison of the similarity values, 훾휳 (a) and 훾휻 (b) against the correlation +levels max{푐휻 +풊 , 푐휻 +풌−풊} (c) and 푐휞 (d) for 풌 = (0, 1, 1) and 풊 = ([−3, 3], [−3, 3], 1). +Figure 18: Comparison of the similarity values, 훾휳 (left), 훾휻 (center) and 훾흋 (right) for +풌 = (0, 1, 1) and 풊 = ([−3, 3], [−3, 3], 1). +involving streamwise-constant modes, while the change is only marginal between 훾휳 and 훾흋 +for the streamwise-periodic modes. +We present in figure 19 the predicted flow structures 흋풊 and 흋풌−풊 which yield the triadic +interactions shown in figure 16, together with the resulting forcing modes 흋풊∇흋풌−풊, which +are compared to the RESPOD forcing modes 휞풊,풌−풊. The predicted flow structures are seen to +yield forcing predictions that are in good alignment with the actual RESPOD forcing modes +present in the flow. For the streamwise-constant modes both the mode shape and phase are +predicted accurately while a phase shift is observed in streamwise-periodic modes although +the mode shapes are well predicted. + +Modal decomposition of nonlinear interactions in wall turbulence +21 +Figure 19: The flow structures 흋풊 and 흋풌−풊 (first and second columns, respectively) +predicted by (5.4) and the resulting forcing prediction 흋풊∇흋풌−풊 (third column) in +comparison to the RESPOD forcing mode 휞풊,풌−풊 (fourth column). The triplet +풊 = (훼, 훽, 휔) are set to (0,2,1), (0, −1, 1), (1,0,1) and (1,1,1) (from top to bottom) and the +complementary triplet 풌 − 풊 is chosen to yield 풌 = (0, 1, 1). +To understand the cumulative effect of the small imperfections in the predicted forcing +modes obtained using (5.4), we replace the RESPOD forcing modes with these forcing +predictions and investigate the change in the resulting response mode. The comparison of the +resulting response structure against the response to the RESPOD forcing modes, is shown +in figure 20 for the tuples shown in figure 4 at their peak-energy frequency. Note that the +model is used to obtain the forcing mode shapes only, while the phase and the norm of the +resulting modes are calibrated using the actual forcing data. The comparison reveals that the +response generated by the modeled forcing matches the ESPOD mode with good accuracy +for all the wavenumber-frequency pairs except the streamwise velocity component 푢 in the +triplet, 풌 = (0, 2, 0). +6. Conclusions +We have discussed a method to investigate the triadic interactions that underpinthe generation +of flow structures associated with a given observable. The method is based on the resolvent- +based extended spectral proper orthogonal decomposition (RESPOD), used in Karban et al. +(2022) to identify self-similar structures in a turbulent channel flow. A minimal Couette +flow is here chosen as the test case, where the triadic interactions associated with spanwise +wall-shear are investigated. +We identify the forcing modes correlated to the SPOD modes of an observable via +RESPOD. These forcing modes generates the associated SPOD modes when applied to +the resolvent operator. We show in this study that using RESPOD it is also possible to +identify individual triadic interactions that are correlated to the observable. Summation of + +Tik-iPk-i +PiVPk-i +310 +2 +0 +2 +0 +2 +0 +2 +20 +0 +0 +-2 +0 +2 +-2 +0 +2 +-2 +0 +2 +-2 +2 +2 +2 +3.1 +- +- +0 +0 +-2 +-2 +0 +2 +2 +0 +2 +0 +2 +2 +2 +2 +2 +31 +1 +0 +0 +0 +0 +-2 +0 +2 +-2 +0 +2 +-2 +0 +2 +-2 +2 +2 +2 +2 +21 +- +1 +1 +0 +0 +0 +0 +-2 +0 +2 +-2 +0 +2 +-2 +0 +2 +-2 +2 +222 +U. Karban et al. +Figure 20: Comparison of the response to the forcing predictions obtained using equation +(5.4)(dashed) against the ESPOD modes (solid) at 풌 = (0, 1, 0) (top-left), (0,2,0) +(top-right), (1,0,8) (bottom-left) and (1,1,8) (bottom-right). Blue, orange and yellow lines +indicate the velocity components, 푢, 푣 and 푤, respectively. +the correlated triadic interactions is by definition equal to the RESPOD forcing mode. This +procedure allows identifying interactions that dominate generating the observable. For each +of these interactions, we propose a model for the structures that from the triad, using a linear +combination of the SPOD and ESPOD modes. The propose model shows higher coherence +than when each of these modes is used individually. +The analysis reveals that the most energetic mode, (훼, 훽) = 0, at its peak-energy +frequency, 휔 = 0, was mainly driven by six triadic interactions: four interactions involving +modes periodic over 퐿푥 in the streamwise direction, that generate small-in-amplitude but +efficient forcing, and two interactions involving streamwise-constant modes that, although +being less efficient, generate forcing structures with large amplitudes. The streamwise- +periodic interactions generate a combined streak-streamwise vortex structure via the lift-up +mechanism, while the streamwise-constant interactions counteract the streak generation by +generating a streamwise forcing componentin phaseopposition to the lift-up mechanism. This +explains in physical terms the destructive interference of forcing observed by Nogueira et al. +(2021): forcing is composed of different triadic interactions with opposing effects in exciting +streamwise vortices and streaks. +Our framework also allows us to investigate energy transfer between different modes via +triadic interactions. We observe that the triadic interactions involving the (0,1) mode provide +a constructive contribution to all the modes investigated. This is an expected result since it +is the only mode with a negative nonlinear transfer rate of turbulent kinetic energy, as shown + +Modal decomposition of nonlinear interactions in wall turbulence +23 +by the energy balance analysis we conducted following Symon et al. (2021). Investigating +the nonlinear transfer for different modes via a range of triadic interactions, we observe the +energy cascade mechanism transferring energy from (0,1) to (0,2) and then from (0,2) to +(0,3). Comparing the interaction map and the map of nonlinear energy transfer revealed that +the triadic interactions associated to the lift-up mechanism and the nonlinear transfer are in +destructive interference for the modes (0,1,0) and (0,2,0). +The method we discuss provides a systematic means by which to understand mechanisms +responsible for the generation of a given observable in a turbulent flow. It does not however +provide the flow structures that form the forcing modes active in response generation. To +predict these flow structures, we proposed an empirical model blending the SPOD and +ESPOD modes based on the correlation between the ESPOD modes and the measured +observable. We showed that the forcing structures predicted via this empirical approach +generates a response very similar to the actual response, pointing towards its potential in the +construction of reduced-order models. +Funding. This work has received funding from the Clean Sky 2 Joint Undertaking under the European +Union’s Horizon 2020 research and innovation programme under grant agreement No 785303. U.K. has +received funding from TUBITAK 2236 Co-funded Brain Circulation Scheme 2 (Project No: 121C061). +Declaration of interests. The authors report no conflict of interest. +REFERENCES +Amaral, F. R., Cavalieri, A. V., Martini, E., Jordan, P. & Towne, A. 2021 Resolvent-based estimation +of turbulent channel flow using wall measurements. Journal of Fluid Mechanics 927, A17. +Bae, H. J., Lozano-Durán, A. & McKeon, B. J. 2021 Nonlinear mechanism of the self-sustaining process +in the buffer and logarithmic layer of wall-bounded flows. Journal of Fluid Mechanics 914, A3. +Borée, J. 2003 Extended proper orthogonal decomposition: a tool to analyse correlated events in turbulent +flows. 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Journal of Fluid Mechanics +812, 636–680. + +This figure "jfm_graphical_abstract.jpg" is available in "jpg"� format from: +http://arxiv.org/ps/2301.01078v1 + diff --git a/StAzT4oBgHgl3EQfJPvm/content/tmp_files/load_file.txt b/StAzT4oBgHgl3EQfJPvm/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5cc4c07d8bf10f3f1459ea7c71161352932e08d5 --- /dev/null +++ b/StAzT4oBgHgl3EQfJPvm/content/tmp_files/load_file.txt @@ -0,0 +1,996 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf,len=995 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='01078v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='flu-dyn] 3 Jan 2023 Under consideration for publication in J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 1 Banner appropriate to article type will appear here in typeset article Modal decomposition of nonlinear interactions in wall turbulence U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Karban1,2†, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Martini1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Cavalieri3, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Jordan1 1Département Fluides, Thermique, Combustion, Institut Pprime, CNRS-University of Poitiers-ENSMA, France 2Department of Aerospace Engineering, Middle East Technical University, Ankara 06800, Turkey 3Instituto Tecnológico de Aeronáutica, São José dos Campos/SP, Brazil (Received xx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' revised xx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' accepted xx) Coherent structures are found in many different turbulent flows and are known to drive self- sustaining processes in wall turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Identifying the triadic interactions which generate coherent structures can provide insights beyond what is possible in the framework of linearized models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' There are infinite possible interactions that may generate a given structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Thus a method to systematically study those, ranking them in terms of their contribution to the structure of interest, is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' We here use the resolvent-based extended spectral proper orthogonal decomposition (RESPOD) approach (Karban, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 2022 Self-similar mechanisms in wall turbulence studied using resolvent analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Journal of Fluid Mechanics 969, A36) to rank the triadic interactions which give rise to wall-attached structures in a minimal Couette flow at Reynolds number 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Our analysis identifies that six triadic interactions dominate the most-energetic wall-attached structure, revealing the capability of the methodology to identify and rank nonlinear interactions responsible for a given coherent structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The approach can be used to analyse the energy exchange in turbulent flows and may guide the construction of reduced-order models based on the interplay between different flow modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Key words: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Introduction Turbulent flows contain coherent structures that span large spatial and temporal scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' These structures are responsible for many important phenomena observed in different flows, ranging from sustaining the near-wall cycle (Hamilton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 1995) in wall-bounded flows to noise generation in jets (Jordan & Colonius 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Cavalieri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' It has been shown that the linear mechanisms play a major role in the generation of coherent structures (Ellingsen & Palm 1975;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Landahl 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Trefethen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Hwang & Cossu 2010b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Brandt 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Schmidt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Pickering et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' A now popular approach to investigate these mechanisms is the resolvent analysis, where the Navier-Stokes (N-S) † Email address for correspondence: ukarban@metu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='tr Abstract must not spill onto p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='2 2 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Karban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' equation are arranged in input-output form in the frequency domain (Farrell & Ioannou 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Jovanović & Bamieh 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' McKeon & Sharma 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Hwang & Cossu 2010a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Sipp & Marquet 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Towne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Lesshafft et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Although resolvent analysis provides a dynamical framework, in most cases, it provides a qualitative understanding of the coherent structures and the associated mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' It has been shown for certain flows that modelling the nonlinear fluctuations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=', the color of the turbulence, is essential for better prediction of these structures (Zare et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Martini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Amaral et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Morra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Nogueira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Karban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 2022), particularly when developing flow models that can quantitatively predict coherent structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' One way to tackle the nonlinearity is to use eddy viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' It has been shown for many flows that adopting an eddy viscosity model while constructing the resolvent operator improves the prediction of coherent structures (Hwang & Cossu 2010b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Morra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 2019, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Pickering et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Kuhn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' One can consider the use of eddy viscosity within resolvent framework as the following: it is known that the resolvent operator yields the exact coherent structures observed in the flow if the forcing is white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The actual forcing is not white, and inclusion of an eddy viscosity in the linear operator allows to model at least part of the forcing colour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Given that the eddy viscosity models incoherent disturbances (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Hussain & Reynolds 1970), one may conjecture that it provides a good model of the effect of the incoherent disturbances on coherent structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' An alternative approach to model the nonlinearity is to use quasi-linear approximation (Malkus 1956), where the N-S equations are split into somehow-averaged quantities and the remaining fluctuating terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The equations for the averaged quantities are then solved directly taking into account the coupling with the fluctuation equations, while the fluctuation equa- tions are linearised by neglecting the nonlinear fluctuating terms (Malkus 1956) or replacing them with a linear model (Farrell & Ioannou 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Thomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Constantinou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Bretheim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Farrell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Bretheim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' All these approaches try modelling the nonlinear terms as a whole rather than tracing separately the triadic interactions that add up to form them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' When decomposing the flow into Fourier modes, the quadratic nonlinearity of the incompressible N-S equations become triadic interactions between these modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' For a high-Reynolds-number turbulent flow, the vast number of possible interactions forming a given nonlinear term prohibits their direct modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' There are some studies which analytically investigate triadic interactions in simple cases such as homogeneous turbulence (Kraichnan 1973;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Waleffe 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Moffatt 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Cheung & Zaki (2014) derived the spectral N-S equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Investigating the analytical properties of triadic interactions in homogeneous, isotropic turbulence, they showed that the famous -5/3 decay is embedded in the N-S equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' In a recent study, Cho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' (2018) employed the spectral turbulent kinetic energy equation to trace the energy transfer between different scales in a turbulent channel via triadic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' (2021) adopted a similar approach to study the energy transfer in cylinder wake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Triadic interactions in turbulent flows are also investigated within the resolvent framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The interactions between the response modes of the resolvent operator and their effect of the self-similar nature of these modes was first discussed in Sharma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Rosenberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' (2019) showed that by including the effect of triadic interactions among the dominant response modes of the resolvent operator, prediction of coherent structures can be significantly improved in oscillatory flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' This approach was followed by Symon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' (2019) where they studied flow over airfoils, and then by Symon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' (2021), where they investigated the energy transfer in some minimal flow units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' A formalism was provided by Padovan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' (2020) to extend the resolvent framework to oscillatory flows, taking into account the cross-frequency interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Rigas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' (2021) used the resolvent framework together with limited triadic interactions to investigate boundary layer transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Bae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Modal decomposition of nonlinear interactions in wall turbulence 3 (2021) investigated critical nonlinear mechanisms in Couette flow, again using resolvent framework, by filtering the contribution of the dominantforcing mode to response generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' In this study, we investigate dominant nonlinear mechanisms in wall-bounded turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The complexity of all possible triadic interactions in a turbulent flow can be reduced by focusing on a certain quantity and eliminating all the non-relevant interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' We use the resolvent-based extended spectral proper orthogonal decomposition (RESPOD) (Towne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Karban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 2022) for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' RESPOD is used to rank the triadic interactions in terms of their correlation and/or their contribution to a given observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The method is implemented using a direct numerical simulation (DNS) of minimal Couette flow with Reynolds number 400, where the spanwise wall shear is considered the target observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' In similar minimal channel configurations, Bae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' (2021) investigated the triadic interactions contributing to the (훼, 훽) = (0, 2휋/퐿푧) mode, where 훼 and 훽 are streamwise and spanwise wavenumbers, respectively, and 퐿푧 is the domain size in 푧-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' We investigate here the triadic interactions systematically extracted using RESPOD for the same mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' By doing so, we present an approach to investigate nonlinear interactions in numerical datasets, where the effect of each triad on the observable of interest may be studied separately using the resolvent operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' This provides a quantitative analysis of the contribution of the various triads at play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' In this approach we wish to move a step further in the analysis of turbulence using the resolvent operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' With numerical datasets, it is possible to recover forcings and responses and relate them through the resolvent operator, as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The set of tools we wish to develop here aim at a further exploration of the forcing, which is first split into constituent triads, whose role in exciting coherent structures may be quantified using the resolvent operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Next, once a particular triad is isolated and recognised as dynamically relevant, we wish to extract the individual structures in the flow that form each element of the triad;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' it is also our objective to propose a modal analysis for that task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' A schematic depicting the flow chart of the analysis is presented in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The flow configuration chosen here is minimal Couette flow due to its simplicity, leading to a lower number of non-linear interactions and a few dominant coherent structures, which simplifies the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The available knowledge on the dynamics of this flow allows us to demonstrate that the tool we propose does indeed identifies the dominant flow interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The methods are general and may be employed in other flows of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' With the approaches developed here we move beyond the analysis capabilities given by the resolvent operator, by analysing the non-linear terms at play, which are unquestionably relevant in turbulence dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The remainder of the paper is structured as follows: the mathematical framework to extract triadic interactions associated with a measured quantity is explained in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The details about the DNS database of the minimal Couette flow are provided in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The results of identifying the relevant triadic interactions and the energy transfer via these interactions in the minimal Couette flow are discussed in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' An modelling approach to predict the relevant forcing using response structures is proposed in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Finally, some concluding remarks are provided in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Extracting nonlinear interactions using RESPOD We consider the incompressible Navier-Stokes (N-S) equations in Cartesian coordinates as, M휕푡풒(풙, 푡) = N (풒(풙, 푡)) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='1) where 풒 = [푢 푣 푤 푝]⊤ is the state vector, N denotes the nonlinear N-S operator for incompressible flows and the matrix M is zero for the continuity equation and identity matrix for the remaining equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Discretisation in space and linearisation around the 4 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Karban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Figure 1: Schematic depicting different stages of the analysis conducted in the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' mean, 풒(풙), yields M휕푡풒′(풙, 푡) − A(풙)풒′(풙, 푡) = B 풇 (풙, 푡), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='2) where A(풙) = 휕푞N|풒 is the linear operator obtained from the Jacobian of N and 풇 (풙, 푡) denotes all the remaining nonlinear terms, interpreted as a forcing term in the momentum equations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' B imposes zero forcing at the continuity equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Full expressions for the operators are given in Nogueira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' We focus on parallel flow, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=', a flow that is homogeneous in two directions, for instance, in 푥 and 푧, with the mean flow varying only in 푦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' We can modify (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='2) to cast it in the resolvent form by applying Fourier transforms in all homogeneous dimensions and rearranging as, ˆ풒( ˜훼, 푦, ˜훽, ˜휔) = R( ˜훼, 푦, ˜훽, ˜휔) ˆ풇 ( ˜훼, 푦, ˜훽, ˜휔), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='3) where ˜훼 and ˜훽 are the streamwise and spanwise wavenumbers, respectively, and ˜휔 is the angular frequency, the hat indicates a Fourier transformed quantity and R( ˜훼, 푦, ˜훽, ˜휔) ≜ (−푖 ˜휔M −A( ˜훼, 푦, ˜훽))−1B is the resolvent operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' For brevity, we drop the notation showing dependence on wavenumber, wall-normal coordinate and frequency in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' One can restrict and/or transform the response to a given set of observables ˆ풚풌, using a measurement matrix, C as, ˆ풚풌 = C ˆ풒풌, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='4) which yields ˆ풚풌 = ˜R풌 ˆ풇풌, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='5) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='Focus on Fluids articles must not exceed this page length ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='→ TwallForcing:RESPODmode ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='Response:ESPODmode ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='2mpose ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='into coherent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='ctures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='Vpk-i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='Pk-i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='2-2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='Decompose ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='RESPODmode ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='into individual ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='Forcing: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='Interaction map: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='triads ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='Deco ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='RESPODmode ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='Role ofeachtriad ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='Tik-i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='triad forcing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='in building the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='stru ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='Xk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='RESPODmode ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='1 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='2Modal decomposition of nonlinear interactions in wall turbulence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='where ˜R풌 ≜ C(−푖 ˜휔M − A)−1B is called the modified resolvent operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' For the incompressible N-S equations, the forcing term in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='2) is given as 풇 = 풖′⊤ · ∇풖′ − 풖′⊤ · ∇풖′, where (·) and ()⊤ denote dot product and transpose, respectively, and the overbar denotes averaging in time and homogeneous directions 푥 and 푧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The forcing in the wavenumber-frequency space, ˆ풇풌, is then obtained via a convolution, ˆ풇풌 = � 풊 ˆ풖⊤ 풊 · ∇ ˆ풖풌−풊, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='6) where 풊 = (훼푖, 훽푖, 휔푖), and 풌 = (훼푘, 훽푘, 휔푘) denote wavenumber-frequency combinations, and summation over 풊 implies a nested summation over 훼푖, 훽푖 and 휔푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Here, we consider that 휔 is discretised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Note that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='6) is valid assuming that the triplet 풌 contains at least one non-zero element, such that the averaged term in 풇 has no contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The RESPOD method, adapted from extended proper orthogonal decomposition (Borée 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Hoarau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 2006), finds, for a given observable, all structures in a ‘target’ event that are correlated to the SPOD modes of the observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Here we choose the target event to be the nonlinear interactions, which give rise to the forcing terms in the resolvent framework, as in Towne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' (2015) and Karban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The goal is to map the triadic interactions underpinning the dominant coherent structures of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The SPOD modes of an observable, ˆ풚푘, can be estimated using the ensemble matrix of realisations, through the eigendecomposition, ˆY 퐻 풌 W ˆY풌 = ˆ휣풌휦풌 ˆ휣 퐻 풌 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='7) and the SPOD modes are obtained from ˆ휣풌 as, 휳풌 = ˆY풌 ˆ휣풌휦−1/2 풌 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='8) where ˆY풌 ≜ [ ˆ풚풌 (1) ˆ풚풌 (2) · · · ˆ풚풌 (푃)] denotes the ensemble matrix for different realisations of ˆ풚풌 with 푃 being the total number of realisations, 휳풌 and 휦풌 are SPOD modes and their associated eigenvalues, respectively (see Towne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' (2018)), and W is a positive-definite matrix of quadrature gains along 푦, which is discretised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The SPOD modes in the columns of 휳풌 are the optimal orthonormal basis for the realisations of the observable ˆ풚풌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' In Karban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' (2022), it was shown that the coefficient matrix ˆ휣풌 can be used to extract the part in the forcing that is correlated with the observed SPOD mode as 흌풌 = ˆF풌 ˆ휣풌휦−1/2 풌 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='9) where, ˆF풌 is the ensemble matrix of ˆ풇풌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The RESPOD forcing mode 흌풌 satisfies 휳풌 = ˜R풌 흌풌, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='10) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=', the RESPOD forcing mode excites precisely the SPOD mode via the resolvent operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' As discussed in Karban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' (2022), the RESPOD mode includes the part of the forcing that is correlated to the SPOD mode of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' This comprises a “silent” part 흌풌,푠 , which generates no response ( ˜R풌 흌풌,푠 = 0) but is nonetheless present in the dataset and correlated to the SPOD mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Substituting into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='9) the expansion in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='6), which shows the triadic interactions summing up to yield the forcing ˆ풇풌, one can compute the triadic interactions correlated with the observable as ˆ풇풌 ˆ휣풌휦−1/2 풌 = � 풊 � ˆU⊤ 풊 · ∇ ˆU풌−풊 �⊤ ˆ휣풌휦−1/2 풌 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='11) 6 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Karban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' where ˆU denotes the ensemble matrix of ˆ풖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Defining 휞풊,풌−풊 ≜ � ˆU⊤ 풊 · ∇ ˆU풌−풊 �⊤ ˆ휣풌휦−1/2 풌 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='12) the correlated forcing 흌풌 can be decomposed as, 흌풌 = � 풊 휞풊,풌−풊 = � 풊 � ˆU⊤ 풊 · ∇ ˆU풌−풊 �⊤ ˆ휣풌휦−1/2 풌 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='13) We define the energy as, ∥(·)∥2 = 휀{(·)퐻W (·)}, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='14) where the superscript 퐻 indicates Hermitian transpose, and 휀{·} denotes the expectation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' In what follows, 휀{·} corresponds to time-averaging for time-dependent structures, and to ensemble averaging for Fourier realisations in the frequency space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The energy of 휞풊,풌−풊, denoted by ∥휞풊,풌−풊∥2, for all 풊 shows the correlation map of the nonlinear interactions related to the observed SPOD mode, 휳풌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' One can instead investigate ∥휼풊,풌−풊∥2, where 휼풊,풌−풊 ≜ R풌휞풊,풌−풊, which provides the contribution of a triadic interaction to a given SPOD mode of the measured state, as suggested by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='10) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' By removing or including terms in the sum in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='13), one is able to inspect the contributions of each triad 풊 to the observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Database of the minimal Couette flow The use of RESPOD for detection of ‘important’ nonlinear interactions associated with a specific measurement is tested on a minimal Couette flow (Hamilton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 1995), sim- ilar to that investigated by Nogueira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The simulations are performed using the ‘ChannelFlow’ code, a pseudo-spectral incompressible flow solver using a Fourier- Chebyshev discretisation in the wall-parallel and wall-normal directions, respectively (see www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='channelflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='ch for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The dimensions of the minimal box are (퐿푥, 퐿푦, 퐿푧) = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='75휋ℎ, 2ℎ, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='2휋ℎ), where the subscripts 푥, 푦 and 푧 denote the streamwise, wall-normal and spanwise directions, and ℎ is the channel half-height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' These are the minimal dimensions to sustain turbulence in Couette flow at low Reynolds number, as studied by Hamilton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The domain was discretised as (푛푥, 푛푦, 푛푧) = (32, 65, 32) with a dealiasing factor of 3/2 in the wall-parallel directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The channel walls move with wall velocity, ±푈푤 yielding a Reynolds number, 푅푒 = 400 based on 푈푤 and ℎ, corresponding to a friction Reynolds number, 푅푒휏 ≈ 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Once the initial transients disappeared, the flow data was stored for 7000 convective units with a sampling rate, Δ푡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Temporal data is transformed into frequency space using blocks of 2048 time steps with 50% overlapping and using a second- order exponential windowing function given in Martini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' While computing the forcing data, the correction due to using windowing functions is implemented as described in Martini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' (2019) and Nogueira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' We verified that the forcing acting on the resolvent operator accurately yields the response, however, the comparison is not shown here for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Figure 2 presents the profiles for the mean and the root-mean-square (RMS) of the velocity components, 푢, 푣 and 푤 in the streamwise, wall-normal and spanwise directions, respectively, along the wall-normal direction, 푦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' We see that the mean flow deviates from the laminar solution given by (푦 − 1) due to nonlinear interactions between turbulent fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The RMS plots indicate that the fluctuations in 푢 peak around 푦 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='5 and 푦 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' A similar but smaller double-peak structure is seen in the RMS of 푤 with the peaks occurring at the same Modal decomposition of nonlinear interactions in wall turbulence 7 Figure 2: Mean (a) and the RMS (b) profiles of the velocity components, 푢 (black solid), 푣 (red dashed) and 푤 (blue dash-dotted) along the wall-normal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' wall-normal positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The RMS of 푣 peaks around the centre at an amplitude slightly lower than that of 푤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' We choose wall shear fluctuations in the spanwise direction, 휏푧 ≜ 휕푧푢′|푦={0,2} at both upper and lower walls as our observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Spanwise wall shear was used to extract self-similar wall-attached structures in a turbulent channel in Karban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' We use it to have a low-rank representation of the flow associated with this quantity in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Here and in what follows, we use the term ‘wall-attached’ to define quantities that are correlated with the wall-shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' For simpler notation, wavenumbers will be presented in integers defined as 훼 = ˜훼퐿푥/2휋 and 훽 = ˜훽퐿푧/2휋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Similarly, mode frequencies will be presented in integer bins denoted by 휔 = ˜휔푁퐹/ 푓푠, ranging in [−푁퐹/2, 푁퐹/2 − 1], where 푁퐹 = 2048 is the number of temporal points used for taking the Fourier transform (FT) and 푓푠 ≜ 1/Δ푡 is the sampling rate of the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Minimal Couette flow is known to be dominated most of the time by rolls and streaks spanning the entire computational domain, corresponding to (훼, 훽) = (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Occasionally wavy disturbanceswith 훼 = 1 appearafter streak instability and breakdown,and subsequently non-linear interactions among such “waves” lead to the formation of new rolls, restarting the process (Hamilton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Hall & Sherwin 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Figure 3 shows the time-averaged energy contained in each wavenumberpair together with the ratio of the time-averaged energy of the wall-attached structures to the total energy at each wavenumber pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' We see that the mode pair (훼, 훽) = (0, 1), related to streaks and rolls, contains most of the fluctuation energy (∼75%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' In contrast, the modes (±1, 0), related to waves, and (0, 2), which we will refer to as roll-streak harmonic, contain slightly less than 5% of the total energy, and all the other mode pairs have less than 2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The energy of the wall-attached part of the state, denoted by 풒푎, of 8 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Karban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Figure 3: a) Energy of flow structures at different wavenumber pairs averaged over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' b) Ratio of the average energy of the wall-attached structures to the total energy at their wavenumbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' the roll-streak mode (0,1) is around 80% of its total energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Therefore, the coherent structures correlated with the spanwise wall-shear can constitute a good low-rank representative of the flow at this wavenumber pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' A similar case is observed for the wave modes (±1, 0) while for the mode (0,2), the energy ratio of the wall-attached part is around 15%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Figure 4 shows the power spectral density (PSD), integrated along the wall-normal direction, of the velocity field 풒 at wavenumber pairs (훼, 훽) = (0, 1), (0,2), (1,0) and (1,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Although the oblique-wave mode (훼, 훽) = (1, 1) is energy-wise insignificant, it plays a critical role for transfer of energy to (훼, 훽) = (0, 1) mode, as will be shown later, and hence is included here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' We see that the streamwise-constant modes peak around the zero frequency, which is expected due to their quasi-steady nature, while the wave modes (1,0) and (1,1) have their peak around ˜휔 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='1 (휔 = 8), leading to a phase speed of 푐+ ≜ ˜휔+/ ˜훼+ = ±1 in wall units (negative values arise if frequency or wavenumber is negative) corresponding to ∼ 10% of wall velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The shape of the spectra is observed to be similar for the modes that have the same streamwise wavenumber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' This trend can be more clearly seen in figure 5, where the integrated PSDs normalised with respect to the peak value are plotted for different modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' We see two different families of PSD distributions for the two streamwise wavenumbers, 훼 = 0 and 훼 = 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' We now focus on the most energetic mode (훼, 훽) = (0, 1) at its peak-energy frequency, 휔 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The wall-attached forcing and response modes, 흌풌 and R풌 흌풌, respectively, are shown in figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The response field, which is the velocity field correlated to the wall-shear, consists of streaks and rolls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Given that the upper and lower walls have positive and negative mean velocities, respectively, the phase relation between streaks and rolls is reminiscent of the lift-up mechanism (Brandt 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' This is further supported regarding the associated forcing mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' At the spanwise positions where the streamwise vortices are located, the forcing is mainly located near the walls aligned with the 푦-direction, causing a moment to generate the streamwise vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' These vortices then generate streaks by carrying the high- and low-velocity structures near the upper and lower walls, respectively, towards the channel centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Note that the forcing component in the streamwise direction is in opposite phase to Modal decomposition of nonlinear interactions in wall turbulence 9 Figure 4: PSDs of ˆ풒(0,1) (blue), ˆ풒(0,2) (orange), ˆ풒(1,0) (yellow) and ˆ풒(1,1) (violet) integrated over the wall-normal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Figure 5: PSDs of ˆ풒(0,{1,3}) (black;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' solid, dashed, dash-dotted, respectively), and ˆ풒(1,{0,3}) (red;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' solid, dashed, dash-dotted, dotted, respectively) integrated over the wall-normal direction and normalised with respect to the peak value of each mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' the streaks seen in the response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' This indicates that the streaks are generated by the lift-up mechanism despite the counteracting effect of the streamwise forcing, as previously reported by Nogueira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The response generation at this triplet can therefore be considered suboptimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' We also plot the wall-attached response fields for the modes (훼, 훽, 휔) = (0, 2, 0), (1,0,8) and (1,1,8), respectively, in figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Each mode is shown at its peak frequency (see figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The response field contains streaks and rolls for the mode (0,2,0) as in the roll-streak mode (0,1,0), but with doubled periodicity, and thus, is called roll-streak harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The mode (1,0,8) is dominated by its spanwise component, leading to a wave mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Finally, the 10 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Karban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Figure 6: Wall-attached part of the velocity (a) and the associated forcing (b) reconstructed in the 푦-푧 plane for the mode (훼, 훽, 휔) = (0, 1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The color plot indicates the streamwise component and the arrows show the spanwise and wall-normal components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Figure 7: Wall-attached part of the velocity reconstructed in the 푦-푧 plane for the modes (훼, 훽, 휔) = (0, 2, 0) (left), (1,0,8) (center) and (1,1,8) (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The color plot indicates the streamwise component and the arrows show the spanwise and wall-normal components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' response field for the mode (1,1,8) contains some oblique wave structures tilted with the mean flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Nonlinear interactions in the minimal Couette flow 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Extracting important triadic interactions The maps showing the energy of the nonlinear interactions contributing to the dominant mode, 풌 = (훼푘, 훽푘, 휔푘) = (0, 1, 0) are shown in figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Different columns compares the maps ∥ ˆ풖풊∇ ˆ풖풌−풊∥2, ∥휞풊,풌−풊∥2 and ∥휼풊,풌−풊∥2, which correspond respectively to energies of the direct triadic interactions, the interactions correlated with the wall shear, and the response to the latter obtained by the resolvent operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Note that only the triplet 풊 = (훼푖, 훽푖, 휔푖) is shown, where for each 풊, there exists a 풌 − 풊 such that the nonlinear interaction between 풊 and 풌 − 풊 yields 풌 = (0, 1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Starting with 휔푖 = 0, we see that the interaction between the roll-streak mode, 풊 = (0, −1, 0) and its complementary, roll-streak harmonic 풌 −풊 = (0, 2, 0), is dominant in all three maps, indicating that the interaction is large in amplitude, highly correlated to the dominant mode, and generates the response with the largest amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' We also observe large amplitude for the interaction (0, 2, 0) + (0, −1, 0), which involves the same structures with the previous one, but with the gradient operator acting on the roll-streak mode (0, −1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The interactions involving wave modes (±1, 1, 0) + (∓1, 0, 0), although not yielding a large forcing component (low amplitudes at the first two rows of figure 8), are seen to be present in the response map (third row of figure 8), implying that these modes efficiently drive the observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' For non-zero frequencies 휔푖 = 4 and 8, we observe that the contribution of streamwise-constantmodes with 훼푖 = 0 decreases with increasing 휔, whereas Rapids articles must not exceed this page length =Modal decomposition of nonlinear interactions in wall turbulence 11 Figure 8: Amplitude maps of ∥풖⊤ 풊 · ∇풖풌−풊∥2 (top), ∥휞풊,풌−풊 ∥2 (middle), and ∥ ˜R풌휞풊,풌−풊 ∥2 (bottom) obtained at 휔푖 = 0 (left), 휔푖 = 4 (center) and 휔푖 = 8 (right), for the mode 풌 = (훼푘, 훽푘, 휔푘) = (0, 1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Only the modes 풊 are shown while the complementary modes 풌 − 풊 are selected to yield 풌 = (0, 1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' wave modes with 훼푖 = 1 drive an increasingly strongerresponse for higher frequencies, which may be attributed to the different frequency content of streamwise-constant and wavy modes, explored in figures 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' To investigate the overall contribution to the dominant mode (0, 1, 0) via a given wavenum- ber pair (훼푖, 훽푖) and its complementary, we define the forcing mode, ˇ휞풊,풌−풊, obtained by summing 휞풊,풌−풊 over the frequency index, 휔푖, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=', adding the nonlinear interactions between all different frequency combinations, and compute its response via ˜R풌 ˇ휞풊,풌−풊.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Similar to the energy maps shown in figure 8, the map of ∥ ˜R풌 ˇ휞풊,풌−풊∥2 is plotted in figure 9-a, which shows that the response generation is dominated by six interactions: two streamwise-constant, which are (0, {−1, 2})+(0, {2, −1}) involving the roll-streak and roll-streak harmonic modes, and four streamwise-periodic over 퐿푥, which are (±1, {0, 1}) + (∓1, {1, 0}) involving wave modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Note that here and in what follows, we use curly brackets for short hand notation of multiple modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' For instance, (0, {−1, 2}) denotes the modes (0, −1) and (0, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Besides the magnitude of the response to a given triadic interaction, it is important to evaluate how it contributes to the overall response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' As shown in Nogueira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' (2021) and Morra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' (2021), different forcing components can interfere destructively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' In what follows we propose a measure to identify which interactions are constructive, amplifying a given mode, or destructive, saturating or damping it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' As a measure of constructive- ness/destructiveness of a given interaction, we calculate the inner product between the 12 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Karban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Figure 9: a) Amplitude map of the response generated by the wall-correlated interactions at all frequencies added together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' b) The map of normalised inner product between the overall response and the response with contribution of a single interaction masked, computed at different wavenumber pairs used for masking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Both maps are generated for the mode (훼푘, 훽푘, 휔푘) = (0, 1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' response to a single wall-attached interaction, ˆ풒풊,풌−풊 푎, and the wall-attached velocity field, ˆ풒풌 푎, ⟨ ˆ풒풊,풌−풊 푎, ˆ풒풌 푎⟩ ≜ 휀{ ˆ풒퐻 풊,풌−풊 푎W ˆ풒풌 푎}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='1) An interaction map is obtained by calculating (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='1) for each wavenumber pair and normalising the result with ∥ ˆ풒풌 푎∥2, which shows a normalised projection, and thus the construc- tive/destructive role of each triadic interaction in generating the wall-attached response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The resulting map is shown in figure 9-b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Note that the interaction map should sum up to 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=', the sum of all destructive and constructive interactions lead to the mode observed in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The analysis reveals that the contributions from the interactions (0, {−1, 2}) + (0, {2, −1}) decrease the response energy, implying a destructive interference between these interactions and the remaining ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The interactions involving wave modes (±1, {0, 1}) + (∓1, {1, 0}) , on the other hand, cause the response energy to increase, implying a constructive effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The effect of these interactions on the response field is shown in figure 10 by masking these interactions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=', subtracting the contributions from the designated interactions from the overall response computed by the resolvent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' We see that masking the interactions (0, {−1, 2}) + (0, {2, −1}) mainly affects the streaks causing an increase in their amplitude, while the roll remains nearly unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Masking the interactions (±1, {0, 1}) + (∓1, {1, 0}) almost completely eliminates the streamwise vortices, which also causes the lift-up effect to be eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' This results in streaks with smaller amplitude and reversed phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' This result is consistent with models of self-sustaining process in wall turbulence, where rolls are excited by non-linear interactions involving waves with non-zero 훼 (Hamilton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Hall & Sherwin 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Remember that in the RESPOD forcing mode shown in figure 6, the streamwise component counteracts the lift-up mechanism forced by the spanwise components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' These results, when combined, imply that the streamwise and spanwise components in the RESPOD forcing mode, 흌풌, are mainly constructed by the nonlinear interaction groups (0, {−1, 2}) + (0, {2, −1}) and (±1, {0, 1}) + (∓1, {1, 0}), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Masking (±1, {0, 1}) + (∓1, {1, 0}) causes the lift-up mechanism, which is an efficient means to generate streaks via streamwise vortices, to disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The remaining streamwise component in 흌풌 is mainly constructed by (0, {−1, 2}) + (0, {2, −1}) and generates streaks with negative phase, reducing the amplitude ofthe streaks generated by the lift-up mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Modal decomposition of nonlinear interactions in wall turbulence 13 Figure 10: Velocity field corresponding to the wall-attached structure (훼푘, 훽푘, 휔푘) = (0, 1, 0) in the 푦-푧 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Top-left: the entire response;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' top-right: the response obtained by masking the interactions between the modes (훼푖, 훽푖) = (0, {−1, 2}) and their complementary modes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' bottom-left: the response obtained by masking the interactions between the modes (훼푖, 훽푖) = (±1, {0, 1}) and their complementary modes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' bottom-right: the response obtained by masking the interactions between the modes (훼푖, 훽푖) = (0, {−1, 2}) and their complementary modes as well as the interactions between the modes (훼푖, 훽푖) = (±1, {0, 1}) and their complementary modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' This elucidates the destructive interference among components observed by Nogueira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The present results show that such destructive interference occurs among different triadic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Masking all six modes almost entirely eliminates the response as seen in figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Energy transfer via triadic interactions The interaction map shown in figure 9-b can also be interpreted in terms of energy exchange between different modes via nonlinear interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Symon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' (2021) investigated, by employing the spectral form of the transport equation of turbulent kinetic energy (TKE), the overall relation between production, dissipation and the transfer of energy for individual wavenumber pairs in parallel, stationary turbulent flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The spectral TKE equation is given, using indicial notation for the last two terms for convenience, as 휕 ˆ퐸 휕푡 = ℜ � − ∫ 2 0 휕푢 휕푦 ˆ푢∗ˆ푣푑푦 − 1 푅푒 ∫ 2 0 휕 ˆ푢푚 휕푥푛 휕 ˆ푢∗푚 휕푥푛 푑푦 − ∫ 2 0 ˆ푢∗푚 ˆ푓푚푑푦 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='2) where ℜ{·} indicates the real part, the hat in this equation denotes, by abuse of notation, Fourier transformed quantities in the stremwise and spanwise directions, ˆ퐸 is the spectral TKE of a given wavenumber pair, the superscript * denotes complex conjugate, 푚 and 푛 denote that the vector indices, ˆ푓푚 is the 푚th componentof the forcing vector (see Symon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 14 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Karban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' (2021) for derivation of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Here, we assume that the Couette flow is stationary in the time interval we investigate, which renders 휕 ˆ퐸 휕푡 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='3) The three terms on the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='2) correspond to the production, dissipation and nonlinear transfer of the turbulent kinetic energy, respectively, which, thanks to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='3), sum up to zero for a given wavenumber pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' One can write (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='2) in the frequency domain by Fourier transforming in the time domain each term on the right-hand side and Welch averaging (Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 2021), which still satisfies the energy balance for any wavenumber-frequency triplet 풌 as, 0 = ℜ � − � 휕푢 휕푦 ˆ푢풌, ˆ푣풌 � − 1 푅푒 � 휕 ˆ푢푚풌 휕푥푛 , 휕 ˆ푢푚풌 휕푥푛 � − � ˆ푢푚풌, ˆ푓푚풌 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='4) The contributions of production, dissipation and nonlinear transfer to the energy balance for different wavenumber pairs at zero frequency are illustrated in figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' We see that the roll-streak mode (0,1) draws the most energy from the mean flow to produce TKE, and is the only mode to transfer this energy to other modes via nonlinear transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' All the modes are seen to lose energy via dissipation as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Note that the energy balance map is symmetric in the 훽푖 axis, which is not shown for better readability of the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Cho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' (2018) computed the nonlinear energy transfer via each triadic interaction by expanding the convolution in the third term on the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='2) as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Here, we do a similar analysis for the nonlinear transfer term in the frequency-domain energy balance equation given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The resulting energy transfer map for (0,1,0) mode is shown in figure 12-a, where blue and red colors indicate losing and gaining energy, respectively, via the corresponding interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' We see that roll-streak mode (0,1,0) mostly transfer energy via the interactions (0, {−1, 1}, 휔)+(0, {2, 0}, −휔) and (±1, 0, 휔)+(∓1, 1, −휔) with (0, −1, 휔)+ (0, 2, −휔) being the dominant one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' It is also seen to gain energy via a number of interactions but the incoming energy rate is negligible compared to outgoing rate, and hence, is not discussed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' For the triadic interaction (0, −1, 휔) + (0, 2, −휔), we calculated that the transfer from the interaction (0, −1, 0) + (0, 2, 0) is -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='06, which constitutes half of the transfer from all the frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Since the flow is symmetric in the spanwise direction, the modes (0, 1, 0) and (0, −1, 0) are complex conjugates of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Thus, we can say that the mode (0,1,0) is transferring energy to roll-streak harmonic mode (0,2,0) via a triadic interaction involving its conjugate mode, making the (훼푘, 훽푘) = (0, 2) mode the second-most energetic with a peak at zero frequency (see figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' This sort of transfer can be associated to the energy cascade in turbulence from large to small scales observed in high-푅푒 flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Combining the results obtained inspecting the energy transfer and the findings of the previous subsection indicate that the mode (0,1,0) gains almost all of its energy from the mean flow via the lift-up mechanism and transfers some of this energy via nonlinear transfer to streamwise-constant modes as the onset of energy cascade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' This nonlinear transfer appears as a streak with opposite phase (see bottom left plot in figure 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' This will be further discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The response seen in (0,1,0) mode is a result of the destructive relation between the lift-up mechanism, excited by the wave modes, and the nonlinear transfer associated to roll-streak modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' We have seen that (0,2,0) mode receives its energy via a nonlinear transfer mechanism according to the energy budget plot given in figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The nonlinear transfer via each triadic interaction contributing to (0,2,0) mode is shown in figure 12-b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The map reveals that (0,2,0) mode receives energy via the interactions (0, 1, 휔) + (0, 1, −휔) at a rate similar Modal decomposition of nonlinear interactions in wall turbulence 15 Figure 11: Production (left), dissipation (center), and nonlinear transfer (right) of the spectral turbulent kinetic energy for different wavenumber pairs at 휔 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Both the size and the color intensity of the markers indicate amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Figure 12: Nonlinear energy transfer to the modes (a) 풌 = (0, 1, 0), (b) (0,2,0) and (c) (0,3,0) from different triadic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Transfer from different frequencies for a given wavenumber pair is integrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' to the energy transfer in (0,1,0) mode via the interactions (0, −1, 휔) + (0, 2, −휔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Some of the energy that (0,2,0) mode receives is transferred to the next harmonic via the interaction (0, −1, 휔) + (0, 3, 휔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Looking at the nonlinear transfer map for (0,3,0) mode in figure 12-c, we see that energy is transferred via the interaction (0, 1, 0) + (0, 2, 0), once again, at a rate similar to the transfer from (0,2,0) shown in figure 12-b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' One can trace the energy cascade for the negative 훽 modes in the same way, which yields the same transfer maps mirrored in the 훼푖 and 훽푖 axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' This suggests that the modes (훼, 훽) = (0, ±1), once extracting energy from the mean and transferring it to harmonics (0, ±2), plays a role to transfer energy via the triadic interactions associated to higher 훽, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=', they provide a medium for the nonlinear transfer to higher 훽 modes without losing noticeable energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' One can further analyse the nonlinear energy transfer by dissecting the contributions to 16 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Karban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Figure 13: Nonlinear energy transfer to the mode 풌 = (0, 1, 0) from the 푥-, (left), 푦- (center) and 푧- (right) components of different triadic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Transfer from different frequencies for a given wavenumber pair is integrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Figure 14: The same map with figure 9-b obtained for the mode (훼푘, 훽푘, 휔푘) = (0, 2, 0) (right) and the corresponding response in the 푦-푧 plane (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' the nonlinear energy transfer from 푥-, 푦- and 푧-components, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=', the terms ˆ푢풌 ˆ푓푥풌, ˆ푣풌 ˆ푓푦풌 and ˆ푤풌 ˆ푓푧풌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' In figure 13, we show the map of nonlinear energy transfer to the roll-streak mode (0,1,0) via each spatial component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The three maps shown in this figure sum up to yield the nonlinear transfer map shown in figure 12-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The energy in (0,1,0) mode is transferred to other modes mostly via the 푥-component, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=', the streaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' This is aligned with the results shown in 10, where it was shown that the response generated by the interactions responsible for nonlinear transfer yielded streaky structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The roll-streak mode is seen to receive energy via the spanwise component of the triadic interactions involving wave modes (±1, {0, 1}) + (∓1, {1, 0}), which can be associated to the amplification by the lift- up mechanism, consistent with the analysis done with 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' This positive energy transfer is nonetheless lower than the loss of energy via the streaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' No significant energy transfer occurs via the wall-normal component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' We now investigate whether a similar destructive relation takes place between the nonlinear transferand the lift-up mechanism forroll-streak harmonic mode (0,2,0)as in roll-streak mode (0,1,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' In figure 14, we show the same interaction map given in figure 9-b for the mode (훼푘, 훽푘, 휔푘) = (0, 2, 0) together with the reconstruction of the mode in the 푦-푧 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Similar to the (0,1,0) mode, we see streamwise vortices and streaks in (0,2,0) with a halved period in the spanwise direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The rolls and the streaks are in opposite phase compared to the (0,1,0), Modal decomposition of nonlinear interactions in wall turbulence 17 Figure 15: Velocity field corresponding to the wall-attached structure (훼푘, 훽푘, 휔푘) = (0, 2, 0) in the 푦-푧 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Top-left: the entire response;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' top-right: response obtained by masking the interactions (훼푖, 훽푖) = (0, ±1)+complementary and (±1, 0)+complementary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' bottom-left: response obtained by masking the interactions (±1, 2)+complementary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' bottom-right: the response obtained by masking the six interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' which implies that they are not directly associated to the lift-up mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The interaction map indicates strong contribution to response generation from the interactions associated with the nonlinear energy transfer, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=', the ones that appear in figure 12-b as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Besides these interactions, we see some positive contributions from the interactions (±1, 2) + (∓1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' To see the effect of these interactions on the response, we mask these interactions and observe the change in the response in figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Masking the interactions associated to nonlinear transfer, we obtain a response field reminiscent of the lift-up mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' This partial response is due to the interactions (±1, 2) + (∓1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Masking these interactions, on the other hand, yields a response field with inverted streaks and vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Similar to the case of (0,1,0) mode, there exists a destructive interference between the nonlinear energy transfer and the lift-up mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Masking both groups of interactions causes the response to be almost zero, indicating that these six interactions are the active ones for response generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Modelling coherent structures in triadic interactions RESPOD provides the forcing mode that drives a desired SPOD mode of a given observable, and the individual triadic interactions that sum up to yield this forcing mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' However, a RESPOD forcing mode for a triadic interaction, 휞풊,풌−풊 = � ˆ풖⊤ 풊 · ∇ ˆ풖풌−풊 �⊤ ˆ휣풌휦−1/2 풌 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='1) does not provide the modes associated to ˆ풖풊 and ˆ풖풌−풊 that generate the observable-correlated forcing component 휞풊,풌−풊.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Predicting the structures that form the forcing modes that drive 18 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Karban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Figure 16: The wall-shear-correlated forcing mode, 휞풊,풌−풊 (left) in comparison to the forcing mode predictions, 휳풊∇휳풌−풊 and 휻풊∇휻풌−풊, which are based on the SPOD modes (center) and ESPOD modes (right), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The triplet 풊 = (훼, 훽, 휔) is set to be (0,2,1), (0, −1, 1), (1,0,1) and (1,1,1) (from top to bottom) and the complementary triplet 풌 − 풊 is chosen to yield 풌 = (0, 1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' the most energetic structures in the observable is important, as it shows which structures are actually involved in nonlinear interactions, and may also help designing reduced-order models for turbulent flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Note that these response modes are not unique as it is always possible to multiply one of them with a dummy vector, 휼, and and the other by the inverse for elementwise multiplication, 1/휼, respectively, which eventually yields the same forcing mode assuming that 1/휼 is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Among potential candidates to yield the forcing mode given in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='1) are the optimal SPOD modes, 휳풊 and 휳풌−풊 of 풖풊 and 풖풌−풊, respectively, and the extended SPOD (ESPOD) mode defined as 휻풊 ≜ ˆ풖풊 ˆ휣 (1) 풌 λ(1) 풌 −1/2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='2) 휻풌−풊 ≜ ˆ풖풌−풊 ˆ휣 (1) 풌 λ(1) 풌 −1/2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='3) where λ(1) 풌 is the largest eigenvalue of 휦풌, and ˆ휣 (1) 풌 is the associated eigenvector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' In figure 16, we compare the forcing modes for the triplet 풌 = (0, 1, 1) using the triadic interactions 휳풊∇휳풌−풊 and 휻풊∇휻풌−풊, respectively, for different 풊’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The triplets 풌 and 풊 in this figure are chosen such that the difference between the two forcing predictions manifests clearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' All the modes are normalised to have unit magnitude with respect to ∥ · ∥2 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The phase in the forcing modes obtained using SPOD modes is random,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' while the phases of the RESPOD forcing modes 휞풊,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='풌−풊 and the forcing predictions based on the ESPOD modes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Sk-iTik-i wiVk-i 20 2 0 2 0 20 0 2 0 2 2 0 2 2 2 2 91 0 0 2 2 2 2 0 0 2 2 2 2 91 0 0 0 2 0 2 2 0 2 2 2 2 2 1 1 1 0 0 0 2 0 2 2 2 2 2 2Modal decomposition of nonlinear interactions in wall turbulence 19 휻풊∇휻풌−풊,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' are both associated with the phase of the wall shear,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' and therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' not random with respect to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' In the following, we only compare the mode shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Using the optimal SPOD modes yields forcing modes similar to the RESPOD forcing mode for the interactions among wave modes (1, 0, 1) + (−1, 1, 0) and (1, 1, 1) + (−1, 0, 0), while the mode shapes significantly differ for the interactions (0, 2, 1) + (0, −1, 0) and (0, −1, 1) + (0, 2, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Using the ESPOD modes, on the other hand, yields forcing modes that are similar to the RESPOD forcing modes for the latter two interactions, while it is not the case for the former two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' This suggests that one may blend the two modes as 흋풊 = 푎휻풊 + (1 − 푎)휳풊, 흋풌−풊 = 푏휻풌−풊 + (1 − 푏)휳풌−풊, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='4) to predict the response modes generating the RESPOD forcing mode, where 푎 and 푏 denote the blending coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' To be able to quantify the accuracy of the prediction, we define a similarity measure, 훾휼 = 흌퐻 풌 W (휼⊤ 풊 · ∇휼풌−풊)⊤ � ∥ 흌풌 ∥2∥휼⊤ 풊 · ∇휼풌−풊∥2 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='5) for a given mode pair 휼풊 and 휼풌−풊.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' We propose deciding the coefficients 푎 and 푏 regarding the correlation levels between the observable ˆ풚풌 and the response and forcing structures, ˆ풖풊, ˆ풖풌−풊 and ˆ풇풌 that are in a triadic interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Since the ESPOD and RESPOD modes yield the correlated parts of the response and forcing to the wall shear, respectively, the correlation levels can be computed as 푐휻 풊 = λ(1) 풌 ∥휻풊∥2/∥ ˆ풖풊∥2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='6) 푐휞 = λ(1) 풌 ∥휞풊,풌−풊∥2/∥ ˆ풖⊤ 풊 · ∇ ˆ풖풌−풊∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='7) Figure 17 shows the values of 훾휳, 훾휻, max{푐휻 풊 , 푐휻 풌−풊} and 푐휞 for a number of triplets 풌 and 풊.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' We observe that the similarity measure 훾휻 is high when 푐휞 is not very small (> 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='1) and either of 푐휻 풊 or 푐휻 풌−풊 is close to 푐휞, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=', the forcing structures and at least one of the response structures in the triadic interaction are correlated to the observed quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' On the other hand, the similarity measure 훾휳 is high when 푐휞 is again not very small and 푐휻 풊 and 푐휻 풌−풊 are close to zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=', the forcing is correlated to the observable ˆY풌 while the response structures in the triadic interaction are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Note that neither of 훾휻 or 훾휳 is close to 1 when the correlation between the forcing and the observable, 푐휞, is low as in the case of 풊 = (1, 2, 1) and (−1, −1, 1) shown in figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' However, this usually constitutes a case where accurate prediction of the forcing mode shape is not important due to its small contribution to response generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Based on these observations, we present the following empirical relation for the blending coefficients 푎 ≜ min{푐휻 풊 /푐휞, 1} and 푏 ≜ min{푐휻 풌−풊/푐휞, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='8) The model in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='4) provides a smooth blending of the ESPOD modes and SPOD modes depending on the maximum correlation level between the ESPOD modes and the observed quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The resulting modes are normalized afterwards using ∥ · ∥2 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' We compare the similarity measures 훾휳, 훾휻 and 훾흋 in figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' We see that the model improves the alignment everywhere compared to the alignment levels obtained using the SPOD and ESPOD modes separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The improvement is more evident when comparing 훾휻 and 훾흋 for the interactions 20 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Karban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Figure 17: Comparison of the similarity values, 훾휳 (a) and 훾휻 (b) against the correlation levels max{푐휻 풊 , 푐휻 풌−풊} (c) and 푐휞 (d) for 풌 = (0, 1, 1) and 풊 = ([−3, 3], [−3, 3], 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Figure 18: Comparison of the similarity values, 훾휳 (left), 훾휻 (center) and 훾흋 (right) for 풌 = (0, 1, 1) and 풊 = ([−3, 3], [−3, 3], 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' involving streamwise-constant modes, while the change is only marginal between 훾휳 and 훾흋 for the streamwise-periodic modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' We present in figure 19 the predicted flow structures 흋풊 and 흋풌−풊 which yield the triadic interactions shown in figure 16, together with the resulting forcing modes 흋풊∇흋풌−풊, which are compared to the RESPOD forcing modes 휞풊,풌−풊.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The predicted flow structures are seen to yield forcing predictions that are in good alignment with the actual RESPOD forcing modes present in the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' For the streamwise-constant modes both the mode shape and phase are predicted accurately while a phase shift is observed in streamwise-periodic modes although the mode shapes are well predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Modal decomposition of nonlinear interactions in wall turbulence 21 Figure 19: The flow structures 흋풊 and 흋풌−풊 (first and second columns, respectively) predicted by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='4) and the resulting forcing prediction 흋풊∇흋풌−풊 (third column) in comparison to the RESPOD forcing mode 휞풊,풌−풊 (fourth column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The triplet 풊 = (훼, 훽, 휔) are set to (0,2,1), (0, −1, 1), (1,0,1) and (1,1,1) (from top to bottom) and the complementary triplet 풌 − 풊 is chosen to yield 풌 = (0, 1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' To understand the cumulative effect of the small imperfections in the predicted forcing modes obtained using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='4), we replace the RESPOD forcing modes with these forcing predictions and investigate the change in the resulting response mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The comparison of the resulting response structure against the response to the RESPOD forcing modes, is shown in figure 20 for the tuples shown in figure 4 at their peak-energy frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Note that the model is used to obtain the forcing mode shapes only, while the phase and the norm of the resulting modes are calibrated using the actual forcing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The comparison reveals that the response generated by the modeled forcing matches the ESPOD mode with good accuracy for all the wavenumber-frequency pairs except the streamwise velocity component 푢 in the triplet, 풌 = (0, 2, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Conclusions We have discussed a method to investigate the triadic interactions that underpinthe generation of flow structures associated with a given observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The method is based on the resolvent- based extended spectral proper orthogonal decomposition (RESPOD), used in Karban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' (2022) to identify self-similar structures in a turbulent channel flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' A minimal Couette flow is here chosen as the test case, where the triadic interactions associated with spanwise wall-shear are investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' We identify the forcing modes correlated to the SPOD modes of an observable via RESPOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' These forcing modes generates the associated SPOD modes when applied to the resolvent operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' We show in this study that using RESPOD it is also possible to identify individual triadic interactions that are correlated to the observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Summation of Tik-iPk-i PiVPk-i 310 2 0 2 0 2 0 2 20 0 0 2 0 2 2 0 2 2 0 2 2 2 2 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='1 0 0 2 2 0 2 2 0 2 0 2 2 2 2 2 31 1 0 0 0 0 2 0 2 2 0 2 2 0 2 2 2 2 2 2 21 1 1 0 0 0 0 2 0 2 2 0 2 2 0 2 2 2 222 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Karban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Figure 20: Comparison of the response to the forcing predictions obtained using equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='4)(dashed) against the ESPOD modes (solid) at 풌 = (0, 1, 0) (top-left), (0,2,0) (top-right), (1,0,8) (bottom-left) and (1,1,8) (bottom-right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Blue, orange and yellow lines indicate the velocity components, 푢, 푣 and 푤, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' the correlated triadic interactions is by definition equal to the RESPOD forcing mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' This procedure allows identifying interactions that dominate generating the observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' For each of these interactions, we propose a model for the structures that from the triad, using a linear combination of the SPOD and ESPOD modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The propose model shows higher coherence than when each of these modes is used individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The analysis reveals that the most energetic mode, (훼, 훽) = 0, at its peak-energy frequency, 휔 = 0, was mainly driven by six triadic interactions: four interactions involving modes periodic over 퐿푥 in the streamwise direction, that generate small-in-amplitude but efficient forcing, and two interactions involving streamwise-constant modes that, although being less efficient, generate forcing structures with large amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The streamwise- periodic interactions generate a combined streak-streamwise vortex structure via the lift-up mechanism, while the streamwise-constant interactions counteract the streak generation by generating a streamwise forcing componentin phaseopposition to the lift-up mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' This explains in physical terms the destructive interference of forcing observed by Nogueira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' (2021): forcing is composed of different triadic interactions with opposing effects in exciting streamwise vortices and streaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Our framework also allows us to investigate energy transfer between different modes via triadic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' We observe that the triadic interactions involving the (0,1) mode provide a constructive contribution to all the modes investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' This is an expected result since it is the only mode with a negative nonlinear transfer rate of turbulent kinetic energy, as shown Modal decomposition of nonlinear interactions in wall turbulence 23 by the energy balance analysis we conducted following Symon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Investigating the nonlinear transfer for different modes via a range of triadic interactions, we observe the energy cascade mechanism transferring energy from (0,1) to (0,2) and then from (0,2) to (0,3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Comparing the interaction map and the map of nonlinear energy transfer revealed that the triadic interactions associated to the lift-up mechanism and the nonlinear transfer are in destructive interference for the modes (0,1,0) and (0,2,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The method we discuss provides a systematic means by which to understand mechanisms responsible for the generation of a given observable in a turbulent flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' It does not however provide the flow structures that form the forcing modes active in response generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' To predict these flow structures, we proposed an empirical model blending the SPOD and ESPOD modes based on the correlation between the ESPOD modes and the measured observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' We showed that the forcing structures predicted via this empirical approach generates a response very similar to the actual response, pointing towards its potential in the construction of reduced-order models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' This work has received funding from the Clean Sky 2 Joint Undertaking under the European Union’s Horizon 2020 research and innovation programme under grant agreement No 785303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' has received funding from TUBITAK 2236 Co-funded Brain Circulation Scheme 2 (Project No: 121C061).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Declaration of interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' The authors report no conflict of interest.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Journal of Fluid Mechanics 643, 333–348.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Hwang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' & Cossu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 2010b Linear non-normal energy amplification of harmonic and stochastic forcing in the turbulent channel flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Journal of Fluid Mechanics 664, 51–73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Jin, 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Fluids 6, 024702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Jordan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' & Colonius, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 2013 Wave packets and turbulent jet noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Annual Review of Fluid Mechanics 45 (1), 173–195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Jovanović, M.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=', Cavalieri, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=', Lesshafft, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' & Jordan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 2022 Self-similar mechanisms in wall turbulence studied using resolvent analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Journal of Fluid Mechanics 939, A36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Kraichnan, R.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Journal of Fluid Mechanics 900, A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' McKeon, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' & Sharma, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 2010 A critical-layer framework for turbulent pipe flow.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' 2021 Forcing statistics in resolvent analysis: application in minimal turbulent couette flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Journal of Fluid Mechanics 908, A32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' Padovan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=', Otto, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAzT4oBgHgl3EQfJPvm/content/2301.01078v1.pdf'} +page_content=' E.' metadata={'source': 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This study aims at improving the performance of scoring +student responses in science education automatically. BERT-based lan- +guage models have shown significant superiority over traditional NLP +models in various language-related tasks. However, science writing of +students, including argumentation and explanation, is domain-specific. +In addition, the language used by students is different from the language +in journals and Wikipedia, which are training sources of BERT and its +existing variants. All these suggest that a domain-specific model pre- +trained using science education data may improve model performance. +However, the ideal type of data to contextualize pre-trained language +model and improve the performance in automatically scoring student +written responses remains unclear. Therefore, we employ different data +in this study to contextualize both BERT and SciBERT models and +compare their performance on automatic scoring of assessment tasks for +scientific argumentation. We use three datasets to pre-train the model: +1) journal articles in science education, 2) a large dataset of students’ +written responses (sample size over 50,000), and 3) a small dataset of +students’ written responses of scientific argumentation tasks. Our exper- +imental results show that in-domain training corpora constructed from +science questions and responses improve language model performance on +a wide variety of downstream tasks. Our study confirms the effectiveness +of continual pre-training on domain-specific data in the education do- +main and demonstrates a generalizable strategy for automating science +education tasks with high accuracy. We plan to release our data and +SciEdBERT models for public use and community engagement. +1 +Introduction +Writing is critical in science learning because it is the medium for students to +express their thought processes. In classroom settings, educators have engaged +students in writing explanations of phenomena, design solutions, arguments, etc. +[10][15], with which students develop scientific knowledge and competence. How- +ever, it is time-consuming for teachers to review and evaluate natural language +arXiv:2301.12031v1 [cs.AI] 27 Jan 2023 + +2 +Zhengliang Liu, Xinyu He , Lei Liu, Tianming Liu, and Xiaoming Zhai +writing, thus preventing the timely understanding of students’ thought processes +and academic progress. Recent development in machine learning (ML), especially +natural language processing (NLP), has proved to be a promising approach to +promoting the use of writing in science teaching and learning [17]. For example, +various NLP methods have been employed in science assessment practices that +involve constructed responses, essays, simulation, or educational games [14]. In +this rapidly developing domain, the state-of-the-art Bidirectional Encoder Rep- +resentations from Transformers (BERT) model [4], a transformer-based machine +learning architecture developed by Google, demonstrates superiority over other +machine learning methods in scoring student responses to science assessment +tasks [1]. +Fig. 1. The SciEdBERT framework. A student response instance is classified based on +the latent representation of word vectors. +Studies have shown that the performance on NLP tasks can be improved by +using domain-specific data to contextualize language models [5]. Several BERT- +based language models, such as SciBERT [3], AgriBERT [12], BioBERT [8], +and ClinicalRadioBERT [11], have demonstrated significant success on domain- + ++ Score = 0 ++ Score = 1 ++ Score = 2 ++ Score = 3 +Classification +Encoding +... +！ +Embedding +！！ +！！ +. +！ +！！ +1>" +↑ +t +T +SciEdBERT +Pre-training +Corpora +Student data +Science education +ozone +literature +oxygen +atom +molecule +molenle +oaTitle Suppressed Due to Excessive Length +3 +specific tasks. Therefore, it is reasonable to speculate that ML-based scoring of +students’ scientific writing can be improved if we have a domain-specific lan- +guage model for scientific education. In this case, we need to find the proper +domain-specific data that are directly relevant to student writing. It is impor- +tant to note that student responses are preliminary expressions of general science +knowledge and lack the rigor of academic journal publications. In addition, their +writing is also influenced by the developmental progress of writing skills and the +length of the required tasks. These characteristics of student writing are chal- +lenges for using NLP tools to score students’ writing [9] [6]. Therefore, to further +improve the application of large pre-trained language models to automatically +score students’ scientific writing, we use different datasets to train BERT and +compare their performance on various downstream tasks. In this work, we make +the following contributions: +1. We provide a method to improve model performance on the downstream +tasks by contextualizing BERT with the downstream context in advance. +2. We prove the effectiveness of domain-specific data in improving BERT- +based model performance. +3. We will release our language models, which can be further tested and used +in other science education tasks. +2 +Methodology +2.1 +Architecture/Background +The BERT (Bidirectional Encoder Representations from Transformers) language +model [4] is based on the transformer architecture [13]. It is trained using the +masked language modeling (MLM) objective, which requires the model to predict +missing words in a sentence given the context. This training process is called pre- +training. The pre-training of BERT is unsupervised and only requires unlabeled +text data. During pre-training, word embedding vectors are multiplied with three +sets of weights (query, key and value) to obtain three matrices Q, K, and V, +respectively. These matrices are then used to calculate attention scores, which +are weights that measure the importance among input words. For example, in the +example ”I love my cats.”, the word ”I” should (ideally) be strongly associated +with the word ”my”, since they refer to the same subject. +Fig. 2. An example of BERT’s attention mechanism + +High attention score +love +my +cats.4 +Zhengliang Liu, Xinyu He , Lei Liu, Tianming Liu, and Xiaoming Zhai +For each word, the attention scores are then used to weigh intermediate +outputs that sum up to the final vector representation of this word. +Attention(Q, K, V ) = softmax( QK +√dk +)V +(1) +where dk refers to the dimension of the K matrix. +BERT takes a sequence of words as the input, and outputs a latent represen- +tation of input tokens in the form of word vectors. This latent representation, +or embedding, captures the semantics, positional information, and contextual +information of the input sentence. It can be further used for downstream NLP +tasks. To use BERT for practical natural language understanding applications, +it is necessary to fine-tune the model on the target task. BERT can be fine-tuned +on a wide variety of tasks, such as topic classification and question answering, by +adding task-specific layers on top of this pre-trained transformer. Fine-tuning is +a supervised learning process. During this process, BERT is trained on a labeled +dataset and the parameters of the model are updated in training to minimize +the task-specific loss function. +2.2 +Domain-specific training +BERT is a fundamental building block for language models. In practice, it +has many variants that are tailored to the purposes and peculiarities of spe- +cific domains [3,8,2,12,11]. For example, BioBERT [8] is a large language model +trained on biomedical publications (PubMed) and delivers superior performance +on biomedical and chemical named entity recognition (NER), since it has a large +and contextualized vocabulary of biomedical and chemical terms. +Substantial evidence indicates that language models perform better when +then target and source domains are aligned [8,5]. In other words, continual pre- +training BERT-based models with in-domain corpora could significantly improve +their performance on downstream tasks [5]. In addition, there is much correlation +between model performance and the extent of in-domain training. Specifically, +training with more relevant in-domain text and training-from-scratch can further +improve pre-trained language models [5]. +In this work, we incorporate prior experience in NLP, specifically that of +domain-specific training, to train our SCiEdBERT models designed specifically +for science education tasks. +2.3 +Training Design +We follow a pyramid-shaped training scheme to maximize our models’ utilization +of domain-relevant data. +In Figure 3, we can see that SciBERT [3] is a science-oriented version of +BERT developed through in-domain pre-training. As shown in Table 2, some of +the models we developed for this experiment in this study are further extensions +of SciBERT through continual pre-training on various science education data. + +Title Suppressed Due to Excessive Length +5 +Fig. 3. The pyramid training scheme +The primary benefit of following the pyramid training scheme is to avoid +diluting the relatively scarce in-domain data with the vastly more abundant +general text data. If instead a language model is trained on a combined corpus +of general text and domain-specific data, the effects of in-domain training will +be insignificant. +3 +Experiment +3.1 +Dataset +We employ several datasets to train the language models, including the Academic +Journal Dataset for Science Education (SciEdJ), a large dataset of students’ +written Responses (SR1), and a small dataset of students’ responses to four +argumentation tasks (SR2). Then, we use seven tasks from the large dataset +(7T) and the four argumentation tasks (4T) as two datasets to fine-tune the +trained language model. Below we briefly introduce these datasets. +Training Dataset We use three datasets to train the language model. The Sci- +EdJ is a collection of 2,000 journal articles from journals in science education. We +select ten journals in science education with the highest impact factors according +to Web of Science, including Journal of Research in Science Teaching, Interna- +tional Journal of Science Education, Science Education, etc. For Each journal, +we collect the most recent 200 articles. The SR1 dataset is a collection of over +50,000 student short responses to 49 constructed response assessment tasks in + +Our work: 50k+ +student responses +SciEdBERT +More Specific +Semantic Scholar +SciBERT +3.17B words +More General +Wikipedia +Bookcorpus +BERT +2.5B words +0.8B words +WIKIPEDIA6 +Zhengliang Liu, Xinyu He , Lei Liu, Tianming Liu, and Xiaoming Zhai +science for middle school students. Students are anonymous to researchers and +not traceable. The SR2 dataset is a collection of 2,940 student responses from a +set of argumentation assessment tasks [7]. +Fine-tuning Dataset . We employ two datasets to evaluate the model per- +formance. The 7T dataset includes seven tasks selected from the SR1 dataset, +including short-constructed student responses and human expert-assigned la- +bels. Overall, the 7T dataset includes 5,874 labeled student responses (915 for +task H4-2, 915 for task H4-3, 834 for task H5-2, 883 for task J2-2, 743 for task +J6-2, 739 for tasks J6-3, and 845 for task R1-2). The 4T dataset includes 2940 +student responses and their labels from SR2 dataset (e.g., 770 for item G4, 642 +for item G6, 765 for item S2, and 763 for item S3). All the samples in the two +datasets are written responses from middle school students to explain science +phenomena. Trained experts are employed to assign scores to student responses +according to scoring rubrics developed by science education researchers, and the +inter-rater reliability is at or above satisfactory level (details see [16][15]). +3.2 +Baselines +Our study aims to examine how the context of training data matters to pre- +trained models’ (e.g., BERT) performance and explore strategies to further im- +prove model performance. To achieve this goal, we employ various datasets to +train and fine-tune the models. First, we use the original BERT as the pre- +trained model and 7T as the downstream task. This is the baseline model. We +then train a BERT model from SR1 and use 7T as the downstream task. Given +that the 7T is grounded in the context of SR1, a comparison between the two +fine-tuned models (based on BERT vs. SR1-BERT) can address our goals. +Second, we repeat this training and fine-tuning process using BERT with SR2 +and 4T datasets. To examine the generalization of the findings, we also employ +4T as the downstream task in other pre-trained models, including SciBERT [3], a +BERT model trained on SciEdJ (i.e., SciEJ-BERT), a SciBERT model trained on +SciEdJ (i.e., SciEdJ-SciBERT), a BERT model trained on SR2 (i.e., SR2-BERT), +and a SciBERT model trained on SR2 (i.e., SR2-SciBERT), with increasingly +closer contextualization between the pre-trained models and the downstream +tasks. +3.3 +Results +As Table 1 presents, the average accuracy of SR1-BERT (0.912) is slightly higher +than the accuracy of BERT (0.904). Among the seven tasks, SR1-BERT achieves +higher accuracy than BERT on four tasks and are on par with BERT on the +remaining three tasks. This indicates that the accuracy of automatic scoring can +be improved to a certain extent by training the model with in-domain training +data. + +Title Suppressed Due to Excessive Length +7 +Table 1. Comparing different model performance on 7T task +Item +Accuracy +BERT SR1-BERT +H4-2 +0.913 +0.929 +H4-3 +0.831 +0.831 +H5-2 +0.958 +0.970 +J2-2 +0.920 +0.926 +J6-2 +0.959 +0.973 +J6-3 +0.845 +0.845 +R1-2 +0.864 +0.864 +Average 0.904 +0.912 +This indication is clearer in our second experiment with the 4T dataset. As +Table 2 presents, overall, SR2-SciBERT has the highest average accuracy (0.866), +which indicates training the model with the contexts of the downstream tasks +can improve the accuracy of automatic scoring. +The model with the second highest accuracy (0.852) is SR2-BERT. SR2- +BERT has the same performance as SR2-SciBERT on S3 and even higher accu- +racy (0.821) than SR2-SciBERT (0.815) on G4. On S2, SR2-BERT’s performance +(0.915) is only second to SR2-BERT. Only on G6, SR2-BERT has a lower accu- +racy (0.719) than comparison models. Therefore, although the two models share +the same average accuracy, based on the accuracy results on each individual task, +SR2-BERT performs better than BERT. This is also in line with our previous +findings that context matters in improving model performance. +SciEdJ-SciBERT and SciEdJ-BERT have the lowest average accuracy scores +(0.842) among the models. Only on G4 do these two models perform better +than BERT. This indicates that the context of science education publications +cannot help BERT learn the language of student responses better. In fact, on the +contrary, such context may introduce confusion to the machine learning process. +In summary, SR2-SciBERT and SR2-BERT achieve the best results among +the models, which indicates that contextualizing the language models with the +same language of the downstream tasks can improve the model’s performance. +Table 2. Comparing model performance on the 4T tasks +Item +Accuracy +BERT SciEdJ-BERT SciEdJ-SciBERT SR2-BERT SR2-SciBERT +G4 +0.792 +0.804 +0.815 +0.821 +0.815 +G6 +0.766 +0.727 +0.742 +0.719 +0.766 +S2 +0.895 +0.882 +0.889 +0.915 +0.928 +S3 +0.934 +0.954 +0.921 +0.954 +0.954 +Average 0.847 +0.842 +0.842 +0.852 +0.866 + +8 +Zhengliang Liu, Xinyu He , Lei Liu, Tianming Liu, and Xiaoming Zhai +4 +Conclusions +This study investigates training language models with different contextual data +and compares their performance on eleven constructed response tasks. The re- +sults indicate that using the in-domain data directly related to downstream tasks +to contextualize the language model can improve a pre-trained language model’s +performance. In automatic scoring of students’ constructed responses, this means +continual pre-training the language model on student responses and then fine- +tuning the model with the scoring tasks. In science education, using SciEdBERT +can further improve model performance as SciEdBERT is well-versed in scien- +tific vocabulary. Our study confirms the effectiveness of using domain-specific +data to pre-train models to improve their performance on downstream tasks and +validate a strategy to adapt language models to science education. + +Title Suppressed Due to Excessive Length +9 +References +1. Amerman, H., Zhai, X., Latif, E., He, P., Krajcik, J.: Does transformer deep learn- +ing yield more accurate sores on student written explanations than traditional +machine learning? In: Paper submitted to the Annual Meeting of the American +Educational Research Association. Chicago (2023) +2. Araci, D.: Finbert: Financial sentiment analysis with pre-trained language models. +arXiv preprint arXiv:1908.10063 (2019) +3. Beltagy, I., Lo, K., Cohan, A.: Scibert: A pretrained language model for scientific +text. arXiv preprint arXiv:1903.10676 (2019) +4. Devlin, J., Chang, M.W., Lee, K., Toutanova, K.: Bert: Pre-training of deep bidirec- +tional transformers for language understanding. arXiv preprint arXiv:1810.04805 +(2018) +5. Gu, Y., Tinn, R., Cheng, H., Lucas, M., Usuyama, N., Liu, X., Naumann, T., Gao, +J., Poon, H.: Domain-specific language model pretraining for biomedical natural +language processing. ACM Transactions on Computing for Healthcare (HEALTH) +3(1), 1–23 (2021) +6. Ha, M., Nehm, R.H.: The impact of misspelled words on automated computer +scoring: A case study of scientific explanations. Journal of Science Education and +Technology 25(3), 358–374 (2016) +7. Haudek, K.C., Zhai, X.: Exploring the effect of assessment construct complexity +on machine learning scoring of argumentation (2021) +8. Lee, J., Yoon, W., Kim, S., Kim, D., Kim, S., So, C.H., Kang, J.: Biobert: a +pre-trained biomedical language representation model for biomedical text mining. +Bioinformatics 36(4), 1234–1240 (2020) +9. Litman, D.: Natural language processing for enhancing teaching and learning. In: +Thirtieth AAAI conference on artificial intelligence (2016) +10. Novak, A.M., McNeill, K.L., Krajcik, J.S.: Helping students write scientific expla- +nations. Science Scope 33(1), 54 (2009) +11. Rezayi, S., Dai, H., Liu, Z., Wu, Z., Hebbar, A., Burns, A.H., Zhao, L., Zhu, D., +Li, Q., Liu, W., et al.: Clinicalradiobert: Knowledge-infused few shot learning for +clinical notes named entity recognition. In: International Workshop on Machine +Learning in Medical Imaging. pp. 269–278. Springer (2022) +12. Rezayi, S., Liu, Z., Wu, Z., Dhakal, C., Ge, B., Zhen, C., Liu, T., Li, S.: Agribert: +knowledge-infused agricultural language models for matching food and nutrition. +In: IJCAI. IJCAI (2022) +13. Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A.N., Kaiser, +�L., Polosukhin, I.: Attention is all you need. Advances in neural information pro- +cessing systems 30 (2017) +14. Zhai, X., C Haudek, K., Shi, L., H Nehm, R., Urban-Lurain, M.: From substitution +to redefinition: A framework of machine learning-based science assessment. Journal +of Research in Science Teaching 57(9), 1430–1459 (2020) +15. Zhai, X., Haudek, K.C., Ma, W.: Assessing argumentation using machine learning +and cognitive diagnostic modeling. Research in Science Education pp. 1–20 (2022) +16. Zhai, X., He, P., Krajcik, J.: Applying machine learning to automatically assess +scientific models. Journal of Research in Science Teaching 59(10), 1765–1794 (2022) +17. Zhai, X., Yin, Y., Pellegrino, J.W., Haudek, K.C., Shi, L.: Applying machine learn- +ing in science assessment: a systematic review. Studies in Science Education 56(1), +111–151 (2020) + diff --git a/TdFLT4oBgHgl3EQfQS8Y/content/tmp_files/load_file.txt b/TdFLT4oBgHgl3EQfQS8Y/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..612bed7d168ebc3cd94ec41460885bbc69dbe5da --- /dev/null +++ b/TdFLT4oBgHgl3EQfQS8Y/content/tmp_files/load_file.txt @@ -0,0 +1,343 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf,len=342 +page_content='Context Matters: A Strategy to Pre-train Language Model for Science Education Zhengliang Liu1∗, Xinyu He1∗, Lei Liu2, Tianming Liu1∗∗, and Xiaoming Zhai1∗∗ 1 University of Georgia, Athens, GA 30666, USA 2 Educational Testing Service, Princeton, NJ, USA Co-First Author ** Corresponding Authors: tliu@uga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='edu xiaoming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='zhai@uga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='edu Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' This study aims at improving the performance of scoring student responses in science education automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' BERT-based lan- guage models have shown significant superiority over traditional NLP models in various language-related tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' However, science writing of students, including argumentation and explanation, is domain-specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' In addition, the language used by students is different from the language in journals and Wikipedia, which are training sources of BERT and its existing variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' All these suggest that a domain-specific model pre- trained using science education data may improve model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' However, the ideal type of data to contextualize pre-trained language model and improve the performance in automatically scoring student written responses remains unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' Therefore, we employ different data in this study to contextualize both BERT and SciBERT models and compare their performance on automatic scoring of assessment tasks for scientific argumentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' We use three datasets to pre-train the model: 1) journal articles in science education, 2) a large dataset of students’ written responses (sample size over 50,000), and 3) a small dataset of students’ written responses of scientific argumentation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' Our exper- imental results show that in-domain training corpora constructed from science questions and responses improve language model performance on a wide variety of downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' Our study confirms the effectiveness of continual pre-training on domain-specific data in the education do- main and demonstrates a generalizable strategy for automating science education tasks with high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' We plan to release our data and SciEdBERT models for public use and community engagement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' 1 Introduction Writing is critical in science learning because it is the medium for students to express their thought processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' In classroom settings, educators have engaged students in writing explanations of phenomena, design solutions, arguments, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' [10][15], with which students develop scientific knowledge and competence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' How- ever, it is time-consuming for teachers to review and evaluate natural language arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='12031v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='AI] 27 Jan 2023 2 Zhengliang Liu, Xinyu He , Lei Liu, Tianming Liu, and Xiaoming Zhai writing, thus preventing the timely understanding of students’ thought processes and academic progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' Recent development in machine learning (ML), especially natural language processing (NLP), has proved to be a promising approach to promoting the use of writing in science teaching and learning [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' For example, various NLP methods have been employed in science assessment practices that involve constructed responses, essays, simulation, or educational games [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' In this rapidly developing domain, the state-of-the-art Bidirectional Encoder Rep- resentations from Transformers (BERT) model [4], a transformer-based machine learning architecture developed by Google, demonstrates superiority over other machine learning methods in scoring student responses to science assessment tasks [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' The SciEdBERT framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' A student response instance is classified based on the latent representation of word vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' Studies have shown that the performance on NLP tasks can be improved by using domain-specific data to contextualize language models [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' Several BERT- based language models, such as SciBERT [3], AgriBERT [12], BioBERT [8], and ClinicalRadioBERT [11], have demonstrated significant success on domain- + Score = 0 + Score = 1 + Score = 2 + Score = 3 Classification Encoding .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' ！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' Embedding ！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' ！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' ！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' ！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' 1>" ↑ t T SciEdBERT Pre-training Corpora Student data Science education ozone literature oxygen atom molecule molenle oaTitle Suppressed Due to Excessive Length 3 specific tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' Therefore, it is reasonable to speculate that ML-based scoring of students’ scientific writing can be improved if we have a domain-specific lan- guage model for scientific education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' In this case, we need to find the proper domain-specific data that are directly relevant to student writing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' It is impor- tant to note that student responses are preliminary expressions of general science knowledge and lack the rigor of academic journal publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' In addition, their writing is also influenced by the developmental progress of writing skills and the length of the required tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' These characteristics of student writing are chal- lenges for using NLP tools to score students’ writing [9] [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' Therefore, to further improve the application of large pre-trained language models to automatically score students’ scientific writing, we use different datasets to train BERT and compare their performance on various downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' In this work, we make the following contributions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' We provide a method to improve model performance on the downstream tasks by contextualizing BERT with the downstream context in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' We prove the effectiveness of domain-specific data in improving BERT- based model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' We will release our language models, which can be further tested and used in other science education tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' 2 Methodology 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='1 Architecture/Background The BERT (Bidirectional Encoder Representations from Transformers) language model [4] is based on the transformer architecture [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' It is trained using the masked language modeling (MLM) objective, which requires the model to predict missing words in a sentence given the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' This training process is called pre- training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' The pre-training of BERT is unsupervised and only requires unlabeled text data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' During pre-training, word embedding vectors are multiplied with three sets of weights (query, key and value) to obtain three matrices Q, K, and V, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' These matrices are then used to calculate attention scores, which are weights that measure the importance among input words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' For example, in the example ”I love my cats.”, the word ”I” should (ideally) be strongly associated with the word ”my”, since they refer to the same subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' An example of BERT’s attention mechanism High attention score love my cats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='4 Zhengliang Liu, Xinyu He , Lei Liu, Tianming Liu, and Xiaoming Zhai For each word, the attention scores are then used to weigh intermediate outputs that sum up to the final vector representation of this word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' Attention(Q, K, V ) = softmax( QK √dk )V (1) where dk refers to the dimension of the K matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' BERT takes a sequence of words as the input, and outputs a latent represen- tation of input tokens in the form of word vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' This latent representation, or embedding, captures the semantics, positional information, and contextual information of the input sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' It can be further used for downstream NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' To use BERT for practical natural language understanding applications, it is necessary to fine-tune the model on the target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' BERT can be fine-tuned on a wide variety of tasks, such as topic classification and question answering, by adding task-specific layers on top of this pre-trained transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' Fine-tuning is a supervised learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' During this process, BERT is trained on a labeled dataset and the parameters of the model are updated in training to minimize the task-specific loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='2 Domain-specific training BERT is a fundamental building block for language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' In practice, it has many variants that are tailored to the purposes and peculiarities of spe- cific domains [3,8,2,12,11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' For example, BioBERT [8] is a large language model trained on biomedical publications (PubMed) and delivers superior performance on biomedical and chemical named entity recognition (NER), since it has a large and contextualized vocabulary of biomedical and chemical terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' Substantial evidence indicates that language models perform better when then target and source domains are aligned [8,5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' In other words, continual pre- training BERT-based models with in-domain corpora could significantly improve their performance on downstream tasks [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' In addition, there is much correlation between model performance and the extent of in-domain training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' Specifically, training with more relevant in-domain text and training-from-scratch can further improve pre-trained language models [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' In this work, we incorporate prior experience in NLP, specifically that of domain-specific training, to train our SCiEdBERT models designed specifically for science education tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='3 Training Design We follow a pyramid-shaped training scheme to maximize our models’ utilization of domain-relevant data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' In Figure 3, we can see that SciBERT [3] is a science-oriented version of BERT developed through in-domain pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' As shown in Table 2, some of the models we developed for this experiment in this study are further extensions of SciBERT through continual pre-training on various science education data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' Title Suppressed Due to Excessive Length 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' The pyramid training scheme The primary benefit of following the pyramid training scheme is to avoid diluting the relatively scarce in-domain data with the vastly more abundant general text data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' If instead a language model is trained on a combined corpus of general text and domain-specific data, the effects of in-domain training will be insignificant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' 3 Experiment 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='1 Dataset We employ several datasets to train the language models, including the Academic Journal Dataset for Science Education (SciEdJ), a large dataset of students’ written Responses (SR1), and a small dataset of students’ responses to four argumentation tasks (SR2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' Then, we use seven tasks from the large dataset (7T) and the four argumentation tasks (4T) as two datasets to fine-tune the trained language model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' Below we briefly introduce these datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' Training Dataset We use three datasets to train the language model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' The Sci- EdJ is a collection of 2,000 journal articles from journals in science education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' We select ten journals in science education with the highest impact factors according to Web of Science, including Journal of Research in Science Teaching, Interna- tional Journal of Science Education, Science Education, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' For Each journal, we collect the most recent 200 articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' The SR1 dataset is a collection of over 50,000 student short responses to 49 constructed response assessment tasks in Our work: 50k+ student responses SciEdBERT More Specific Semantic Scholar SciBERT 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='17B words More General Wikipedia Bookcorpus BERT 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='5B words 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='8B words WIKIPEDIA6 Zhengliang Liu, Xinyu He , Lei Liu, Tianming Liu, and Xiaoming Zhai science for middle school students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' Students are anonymous to researchers and not traceable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' The SR2 dataset is a collection of 2,940 student responses from a set of argumentation assessment tasks [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' Fine-tuning Dataset .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' We employ two datasets to evaluate the model per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' The 7T dataset includes seven tasks selected from the SR1 dataset, including short-constructed student responses and human expert-assigned la- bels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' Overall, the 7T dataset includes 5,874 labeled student responses (915 for task H4-2, 915 for task H4-3, 834 for task H5-2, 883 for task J2-2, 743 for task J6-2, 739 for tasks J6-3, and 845 for task R1-2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' The 4T dataset includes 2940 student responses and their labels from SR2 dataset (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=', 770 for item G4, 642 for item G6, 765 for item S2, and 763 for item S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' All the samples in the two datasets are written responses from middle school students to explain science phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' Trained experts are employed to assign scores to student responses according to scoring rubrics developed by science education researchers, and the inter-rater reliability is at or above satisfactory level (details see [16][15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='2 Baselines Our study aims to examine how the context of training data matters to pre- trained models’ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=', BERT) performance and explore strategies to further im- prove model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' To achieve this goal, we employ various datasets to train and fine-tune the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' First, we use the original BERT as the pre- trained model and 7T as the downstream task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' This is the baseline model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' We then train a BERT model from SR1 and use 7T as the downstream task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' Given that the 7T is grounded in the context of SR1, a comparison between the two fine-tuned models (based on BERT vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' SR1-BERT) can address our goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' Second, we repeat this training and fine-tuning process using BERT with SR2 and 4T datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' To examine the generalization of the findings, we also employ 4T as the downstream task in other pre-trained models, including SciBERT [3], a BERT model trained on SciEdJ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=', SciEJ-BERT), a SciBERT model trained on SciEdJ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=', SciEdJ-SciBERT), a BERT model trained on SR2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=', SR2-BERT), and a SciBERT model trained on SR2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=', SR2-SciBERT), with increasingly closer contextualization between the pre-trained models and the downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='3 Results As Table 1 presents, the average accuracy of SR1-BERT (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='912) is slightly higher than the accuracy of BERT (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='904).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' Among the seven tasks, SR1-BERT achieves higher accuracy than BERT on four tasks and are on par with BERT on the remaining three tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' This indicates that the accuracy of automatic scoring can be improved to a certain extent by training the model with in-domain training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' Title Suppressed Due to Excessive Length 7 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' Comparing different model performance on 7T task Item Accuracy BERT SR1-BERT H4-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='913 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='929 H4-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='831 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='831 H5-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='958 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='970 J2-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='920 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='926 J6-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='959 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='973 J6-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='845 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='845 R1-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='864 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='864 Average 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='904 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='912 This indication is clearer in our second experiment with the 4T dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' As Table 2 presents, overall, SR2-SciBERT has the highest average accuracy (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='866), which indicates training the model with the contexts of the downstream tasks can improve the accuracy of automatic scoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' The model with the second highest accuracy (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='852) is SR2-BERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' SR2- BERT has the same performance as SR2-SciBERT on S3 and even higher accu- racy (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='821) than SR2-SciBERT (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='815) on G4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' On S2, SR2-BERT’s performance (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='915) is only second to SR2-BERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' Only on G6, SR2-BERT has a lower accu- racy (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='719) than comparison models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' Therefore, although the two models share the same average accuracy, based on the accuracy results on each individual task, SR2-BERT performs better than BERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' This is also in line with our previous findings that context matters in improving model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' SciEdJ-SciBERT and SciEdJ-BERT have the lowest average accuracy scores (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='842) among the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' Only on G4 do these two models perform better than BERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' This indicates that the context of science education publications cannot help BERT learn the language of student responses better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' In fact, on the contrary, such context may introduce confusion to the machine learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' In summary, SR2-SciBERT and SR2-BERT achieve the best results among the models, which indicates that contextualizing the language models with the same language of the downstream tasks can improve the model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' Comparing model performance on the 4T tasks Item Accuracy BERT SciEdJ-BERT SciEdJ-SciBERT SR2-BERT SR2-SciBERT G4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='792 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='804 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='815 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='821 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='815 G6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='766 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='727 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='742 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='719 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='766 S2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='895 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='882 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='889 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='915 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='928 S3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='934 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='954 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='921 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='954 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='954 Average 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='847 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='842 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='842 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='852 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content='866 8 Zhengliang Liu, Xinyu He , Lei Liu, Tianming Liu, and Xiaoming Zhai 4 Conclusions This study investigates training language models with different contextual data and compares their performance on eleven constructed response tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' The re- sults indicate that using the in-domain data directly related to downstream tasks to contextualize the language model can improve a pre-trained language model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' In automatic scoring of students’ constructed responses, this means continual pre-training the language model on student responses and then fine- tuning the model with the scoring tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' In science education, using SciEdBERT can further improve model performance as SciEdBERT is well-versed in scien- tific vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' Our study confirms the effectiveness of using domain-specific data to pre-train models to improve their performance on downstream tasks and validate a strategy to adapt language models to science education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' Title Suppressed Due to Excessive Length 9 References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=' Amerman, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=', Zhai, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdFLT4oBgHgl3EQfQS8Y/content/2301.12031v1.pdf'} +page_content=', Latif, E.' metadata={'source': 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Diffusion Model for Fluid Field Prediction +Gefan Yang * 1 Stefan Sommer * 2 +Abstract +We propose a novel denoising diffusion generative +model for predicting nonlinear fluid fields named +FluidDiff. By performing a diffusion process, the +model is able to learn a complex representation of +the high-dimensional dynamic system, and then +Langevin sampling is used to generate predictions +for the flow state under specified initial conditions. +The model is trained with finite, discrete fluid sim- +ulation data. We demonstrate that our model has +the capacity to model the distribution of simulated +training data and that it gives accurate predictions +on the test data. Without encoded prior knowl- +edge of the underlying physical system, it shares +competitive performance with other deep learning +models for fluid prediction, which is promising +for investigation on new computational fluid dy- +namics methods. +1. Introduction +Computational fluid dynamics (CFD) is the field that in- +volves the use of numerical techniques to solve the gov- +erning equations for fluids, which exhibit chaotic, time- +dependent behaviors known as turbulence due to the nonlin- +ear nature of these equations. Despite the nonlinearity, the +fine spatial and temporal discretization under high-fidelity +conditions is computationally expensive and can consume a +large amount of computational resources. All these inconve- +niences lead to the significant demand for improvements on +CFD methodologies. +Data-driven algorithms, represented by deep learning (DL), +have received extensive attention in the past decade and +have performed well in solving high-dimensional nonlinear +problems. Inspired by previous work, we propose an easily +trainable generative model for solving nonlinear fluid pre- +diction tasks, which is able to give reasonable predictions +without requiring knowledge of the physical laws. At the +same time, our model has good flexibility, which makes it +1Niels Bohr Institute, University of Copenhagen, Copenhagen, +Denmark 2DIKU, University of Copenhagen, Copenhagen, Den- +mark. Correspondence to: Gefan Yang . +only need to be retrained to adapt to different fluid prediction +tasks without changing the model structure. +1.1. Contributions and outline +The paper contributes by 1) developing the denoising diffu- +sion based fluid flow prediction model FluidDiff; 2) demon- +strating the capacity of the model to predict the fluid state +at different time points conditional on input data; 3) being +competitive with other DL based fluid simulation models +for predicting smoke evolution. The paper is organized as +follows. Section 2 introduces the related work, where the +general DL methods and specifically, generative models +used in CFD are reviewed, together with diffusion-based +generative models; Section 3 describes a general fluid field +prediction problem and the principle of denoising diffu- +sion probabilistic model. Section 4 shows FluidDiff, where +the detailed network architecture and algorithms are pre- +sented. Section 5 shows the experimental results on a 2D +floating-smoke case, together with qualified and quantified +discussions; Section 6 concludes the work. +Figure 1. We propose a denoising diffusion generative model for +fluid prediction named FluidDiff. The model takes the initial +condition of the system and makes predictions of fluid states at a +certain time point. The model does not rely on specific physics +governing equations, but only on the data gained through numerical +simulations, and it can adapt to different tasks without changing +the architecture. +arXiv:2301.11661v1 [cs.LG] 27 Jan 2023 + +t=0 +FluidDiff +Velocity Field +Prediction +Density +Source +Density Field2. Related Work +Various DL-based methods have been applied to enhance the +CFD (Bar-Sinai et al., 2019; Kochkov et al., 2021; Li et al., +2020; Ajuria Illarramendi et al., 2020; Weymouth, 2022; +Ling et al., 2016; Jiang et al., 2021; Beck et al., 2019; Murata +et al., 2020; Rojas et al., 2021; Eivazi et al., 2022). Among +them, there are three main areas where the DL can be incor- +porated with physics-principle-based numerical solvers: +• In the direct numerical solver, neural networks (NNs) +are widely used to improve the computational effi- +ciency in approximating spatial derivatives (Bar-Sinai +et al., 2019), finding the correlations between fine and +coarse girds (Kochkov et al., 2021), improving partial +differential equations (PDEs) solvers in coarse grid +(Li et al., 2020), accelerating solving Possion equa- +tions (Ajuria Illarramendi et al., 2020) and pressure +corrections (Weymouth, 2022). +• NNs are used to model the turbulence using Reynolds- +averaged Navier-Stokes models (RANS) (Ling et al., +2016; Jiang et al., 2021) and coarsely-resolved large- +eddy simulations (LES) (Beck et al., 2019) +• DL are also used to develop reduced order models +(ROM). More specifically, proper orthogonal decom- +position (POD) can be achieved through NNs (Murata +et al., 2020; Rojas et al., 2021; Eivazi et al., 2022), +which is used to map the high-dimensional flow space +into a low-dimensional latent space, the latter contains +most of the information of system but with much less +dimensions. +Generative models such as auto-encoders (AEs)(Kingma & +Welling, 2014), energy-based models (EBMs)(LeCun et al., +2006), normalized flows (Dinh et al., 2015) and generative +adversarial networks (GANs)(Goodfellow et al., 2014) can +be regarded as a kind of ROM since they are all able to +convert original flow spaces to certain latent spaces, the +latter usually are of lower dimension. Sampling from the +latent spaces leads to predictions. Among them, GANs +are the most representative ones due to their outstanding +sampling quality and general type. Another merit of GANs +is the loss function can be modified to be more physically +plausible, which is usually achieved by introducing physical +constraints into the loss of generator. Therefore, GANs have +attracted the interest of many researchers and been applied +to fluid modeling and prediction (Farimani et al., 2017; Xie +et al., 2018; Cheng et al., 2020; Akkari et al., 2020; Chu +et al., 2021; Yousif et al., 2021; Drygala et al., 2022; Wu +et al., 2022; Ferreira et al., 2022; Xie et al., 2022). Fari- +mani et al.(Farimani et al., 2017) train a conditional GAN +(cGAN) to generate the steady solution of heat conduction +and impressible fluid flow without any physical knowledge. +Xie et al.(Xie et al., 2018) use a cGAN to address the super- +solution problem for spatio-temporal fluid flows, the model +can enable highly-detailed velocities or vorticities from low- +resolution inputs. Cheng et al. develop a deep convolutional +GAN (DCGAN) to predict spatio-temporal flow distribu- +tions, then apply it to a real-world case and gained good +consistency with rapid computational speed. Akkari et al. +extended Farimani et al.(Farimani et al., 2017)’s work, they +study the use of DCGAN on impressible unsteady fluid flow +in a channel with a moving obstacle inside, the model can +not only memorize the training data, but also given new +reasonable predictions. Chu et al.(Chu et al., 2021) focus +on using cGANs to solve the ill-posed problems of fluids, +which requires model to derive plausible predictions from +sparse inputs, e.g. a single frame of a density field. Also, +the predictions are able to be controlled by modalities like +obstacles and physical parameters. Yousif et al.(Yousif et al., +2021) train a GAN that can reconstruct high-resolution tur- +bulent flows with coarse ones. They also test the possibility +of using transfer learning in flow prediction tasks, which can +be meaningful for reducing the computational cost of CFD. +Meanwhile, Drygala et al.(Drygala et al., 2022) give the +mathematical proof that GANs are able to learn represen- +tations of the chaotic systems from limited state snapshots, +which provides the mathematical fundamentals of the imple- +mentation of GANs. Wu et al.(Wu et al., 2022) incorporate +Naiver-Stokes (NS) equations into GAN’s loss function, +which endows model with more physical meanings, and it +outperforms the similar ones. Ferreira et al.(Ferreira et al., +2022) apply cGANs on simulations of fluid flow in fractured +porous media. They propose using cGAN for upscaling the +permeability of single fractures, which leads to a substan- +tial reduction of computational time. Xie et al.(Xie et al., +2022) develop a two-stage model based on cGANs that can +generate realistic smoke visualizations from hand-written +sketches, together with a user-friendly interface that can +serve for various design scenarios. Apart from GANs, there +are examples of implementations of other generative mod- +els on flow prediction tasks. Kim et al.(Kim et al., 2019) +present a generative model based on AE to synthesize fluid +simulations from a set of reduced parameters, which is used +for 2D and 3D smoke data. Morton et al.(Morton et al., +2021) also use a varientional AE (VAE) for parameterized +fluid prediction tasks. +Diffusion models (Hyv¨arinen & Dayan, 2005; Vincent, +2011; Sohl-Dickstein et al., 2015) have emerged as a new +catalog of deep generative models, which show comparable +or even better performance than GANs in various computer +vision (CV) tasks (Song & Ermon, 2019; Ho et al., 2020; +Dhariwal & Nichol, 2021; Ho et al., 2022; Rombach et al., +2022; Saharia et al., 2022; Gu et al., 2022; Daniels et al., +2021). In general, diffusion models can be divided into +three main categories: (i) Denoising diffusion probabilistic + +models (DDPMs)(Ho et al., 2020); (ii) Score-based diffu- +sion models(Song & Ermon, 2019); (iii) Stochastic differ- +ential equation (SDE) based models (Song et al., 2021b), +where (iii) can be treated as the generalization of (i) and +(ii). Among them, DDPMs and their variants are the most +widely used and well studied. However, to the best of our +knowledge, despite of the popularity of DDPMs in CV, +there is currently no precedent for applying it to fluid flows +prediction tasks, the only application is (Shu et al., 2022), +who used DDPM for super-resolution reconstructions of +fluid simulation. Compared with its competitor GAN, the +biggest advantage of DDPM is that it can avoid training +instability and model collapse. The former comes from the +game competition between the generator network and the +discriminator network in GAN, while the latter is because +the generator in GAN only learns a subset of the entire +data space. All these two drawbacks of GANs requires +researchers to carefully design the network structure and +loss function. Nevertheless, the trade-off of stability is the +slower sampling speed of DDPM, since the sampling of +DDPM is realized by iterative Langevin dynamics, which +requires hundreds of inference steps. So far, some research +have been done to accelerate the sampling speed(Song et al., +2021a; Jolicoeur-Martineau et al., 2021). +3. Background +3.1. A general fluid flow prediction problem +Predicting complex fluid flows is usually addressed through +solving a nonlinear PDE, such as NS equation, Euler equa- +tion, or other equations suit for different circumstances and +approximations under some constraints, also with some cer- +tain initial and boundary conditions. Generally speaking, +they can be ascribed to the following equation set: +∂u +∂τ = f(u, x, ∇xx, ∇2 +xx) +(1) +g(u, x, τ) = 0 +(2) +ϕ1(u, x, τ)|x=x0 = 0 +(3) +ϕ2(u, x, τ)|τ=0 = 0 +(4) +where u is the physical variable we are interested in, which +can be either scalar (e.g. density, pressure) or vector (e.g. +velocity). x is the spatial coordinates, ∇x and ∇2 +x represents +the first and second order spatial derivative respectively. τ +is the time, g is the constraints, which can be physical (e.g. +impressibility, heat conduction) or artificial (e.g. geometric +obstacles, sources). ϕ1, ϕ2 stands for boundary condition +and initial condition respectively. +By solving equation (1) under constraints (2)(3)(4), usually +numerically, one can receive the solver as a mapping: u = +ˆu(x, τ), Further, if g, ϕ1 and ϕ2 can be parameterized to a +parameter vector c, then u = ˆu(x, τ, c). In the conventional +way, a variety of numerical methods have been developed +to obtain ˆu. Under some assumptions, the numerical solver +can be treated as a mapping M : x, τ, c → ˆu(x, τ, c). A +novel approach is to use a data-driven model like neural +network to substitute traditional numerical solver, but also +reaches M. We demonstrate that, after using dedicated- +designed architecture and sufficient training, a DDPM can +approximate M well, so that with a set of given parameters +(x, τ, c), the model can predict u without solving Equation +(1), which demonstrates the principle of fluid flow prediction +problem. +3.2. Denoising diffusion probabilistic model +DDPM consists of two processes: forward (diffusion) pro- +cess and reverse (denoising) process. In the forward process, +the original training data are perturbed by a series of Gaus- +sian noises with different means and variances in a discrete +period t ∈ {1, 2, . . . , T}. In each step the corrupted data +x1, x2, . . . , xT is Markovian: +p(xt|xt−1) =N +� +xt; +� +1 − βt · xt−1, βt · I +� +p(xt|x0) =N +� +xt; +� +ˆαt · x0, (1 − ˆαt) · I +� +(5) +where T is the number of diffusion steps, β1, . . . , βT ∈ +[0, 1) are hyperparameters that control the variance of noise. +I is the identity matrix with the same dimension as the data. +N(x; µ, σ) stands for normal distribution with mean µ and +standard deviation σ. αt = 1 − βt, ˆαt = �t +i=1 αi and U +is the uniform distribution. The variance schedule (βt)T +t=1 +is chosen that ˆβT → 0 and therefore p(xT ) ≃ N(0, I). +Moreover, if the diffusion is small enough, i.e. (βt)T +t=1 ≪ 1, +the reverse transition probability p(xt−1|xt) should have the +same function form as the forward process, which is also +Gaussian +q(xt−1|xt) = N (xt−1; µt(xt, t), σt(xt, t)) +(6) +which means if one starts from a Gaussian noise N(0, I) +and applies Equation (6) progressively, it will finally reach +the original distribution p(x0). In (Ho et al., 2020), σt in +Equation (6) is fixed to be a constant and µ is the function +of the clean data x0 and the corrupted data xt at time t: +µt(xt, x0) = +� +ˆαt−1βt +1 − ˆαt +x0 + +√αt(1 − ˆαt−1) +1 − ˆαt +xt +σt = 1 − ˆαt−1 +1 − ˆαt +βt + +Maximizing likelihood is used in the training with optimiz- +ing the variational lower-bound: +LDDP M = Ep +� +− log qθ(x0:T ) +p(x1:T |x0) +� += Ep[DKL(p(xT |x0)||p(xT )) ++ +� +t>1 +DKL(p(xt−1|xt, x0)||qθ(xt−1|xt)) +− log qθ(x0|x1)] += Ep +� 1 +2σt +∥µt(xt, x0) − µθ(xt, t)∥2 +� ++ C (7) +where DKL represents KL divergence. One shall find that +maximizing likelihood here is essentially to approximate +the mean in the reverse process. With the reparameterizing +trick that xt(x0, z) = √ ˆαtx0 + √1 − ˆαtz for z ∼ N(0, I), +Equation (7) can be adapted to a simpler version: +Lsimple +DDP M = Ex0 +� +||ϵ − ϵθ( +� +ˆαtx0 + +� +1 − ˆαtϵ, t)||2� +x0 ∼ pdata(x0), +ϵ ∼ N(0, I) +(8) +A NN is trained to approximate the noise added on xt. With +the noise network, the reverse process can be implemented +via sampling process: +xt−1 = +1 +√αt +� +xt − 1 − αt +√1 − ˆαt +ϵθ(xt, t) +� ++ √σtz +∀t ∈ {T, . . . , 1}, +z ∼ N(0, I) +4. Methodology +Suppose the fluid field x satisfies a certain posterior distri- +bution p(x|τ, c), where c is the parameterized constraints +and τ denotes the actual time for the physical flow. Note +that τ is different from t in the diffusion process. A condi- +tional denoising diffusion model can be trained to approxi- +mate p(x|τ, c) and demanding states can be sampled from +it. Unlike the unconditional case (8), the posterior is highly +dependent on τ and c. For simplicity we ascribe them all to +the condition y. Now the task is to approximate and sample +from p(x|y), which is usually referred as the conditional +generation task. +In (Rombach et al., 2022), the author argues that a condi- +tional denoising decoder ϵ(xt, t, y) can be used to control +the conditional diffusion process through inputs y, where +y can be texts, semantic maps or other task. They also use +cross attention mechanism to deal with multi-modalities. +In our case, y are spatial and temporal information of the +system that contains two parts, which are modified to fit the +u, as shown in Figure 4. The training object of our model is +still minimizing the L2 loss between ϵ and ϵθ: +L = Ex0,y +� +||ϵ − ϵθ( +� +ˆαtx0 + +� +1 − ˆαtϵ, t, y)||2� +x0, y ∼ pdata(x0, y), +ϵ ∼ N(0, I) +where diffusion time step t serves as the indicator of diffu- +sion sequence. We used the same method as the position +embedding in(Vaswani et al., 2017), which is widely used in +the Transformer architecture. Through this method, a given +scalar s is encoded into a vector v with the formulation: +vi(s) = +� +sin(ωks), i = 2k +cos(ωks), i = 2k + 1 +ωk = +1 +100002k/d +(9) +where vi is the i-th entry of v and d is the size of embedding. +We use it to encode diffusion step t as a vector t. For the +spatial part of y, it is depicted as a matrix with the same reso- +lution as the training snapshots x. The temporal information +τ is also extended to be a matrix with the same size as x. It +is full of identical entries that have been normalized with the +total simulation time. The noise is added only on x to get +xt, finally, we concatenate xt and y to form a multi-channel +input, and feed it together with diffusion time embedding t +into the network. +For the network, we use a U-Net architecture to predict +ϵ(xt, t, y), please see the right part of Figure 4 for details. +Both down and up sampling part consist of 4 main sections, +In each section, there are two Resnet-like blocks, which is +made up of a 3 × 3 convolutional layer, a group normaliza- +tion and a SiLU activation, the residual connections also +exists although they are not explicitly shown in the figure. +Besides the Resnet-like blocks, an embedding MLP is also +used to map the fixed-size diffusion step vector into the +size that suits for element-wise addition with the output +feature maps from Resnet-like blocks. After the addition, +the output will pass through a Transformer layer, which will +do self-attention to extract important representations, and +finally be halved the size by a down-sampling convolutional +layer. The bottleneck block is a ”sandwich” structure, with +two Resnet-like blocks on either side and one Transformer +block in the middle, all of them share the same structures as +their corresponding ones in the down-sampling path. Along +the up-sampling path, the configuration is symmetric to the +down sampling one, except the Resnet-like blocks in the +up-sampling part not only receive the output from former +block, but also the skip connection from the correspond- +ing down-sampling block. The final layer in each section +is replaced by a 3 × 3 transposed convolutional layer that +doubles the size too. The training and sampling process is +implemented by Algorithm 1 and Algorithm 2. +The form of y can be more diverse, as long as it can be +represented as grid data, like geometric obstacles, special +boundary conditions etc. If the problem is more specific, +other types of conditions like forces, temperature, pressure, +viscosity can also be introduced as additional channels. This +feature enable our model with great flexibility for different +fluid simulation prediction tasks. + +Algorithm 1 Training +Input: x0 ∼ p(x0),y ∼ p(y|x0), T, (ˆαt)T +t=1 +Output: ϵθ +repeat +t ∼ U({1, . . . , T}) +ϵ ∼ N(0, I) +L = ||ϵ − ϵθ(√ˆαtx0 + √1 − ˆαtϵ, t/T, y)||2 +Back propagate L using gradient descent +until converged +Algorithm 2 Sampling +Input: ϵθ, T, (αt)T +t=1, (ˆαt)T +t=1, (σt)T +t=1 +Output: ˆx0 ∼ ˆp(u0) +uT ∼ N(0, I) +for t = T, . . . , 1 do +z ∼ N(0, I) if t > 1 else z = 0 +xt−1 = +1 +√αt +� +xt − +1−αt +√1−ˆαt ϵθ(xt, t/T, y) +� ++ √σtz +end for +5. Experiment +5.1. Problem setup +We tested the model on a 2D random floating-smoke case. +In this case, a domain with size of 64×64 is full of randomly +distributed still smoke. The smoke is free to float upward +by a constant buoyancy force in the vertical direction, the +process is shown in Figure 2. The boundaries are closed in +all directions. The movement of smoke can be described by +it’s velocity field ⃗u, which is governed by the incompressible +Navier-Stokes equations: +∇ · ⃗u = 0 +∂⃗u +∂t + (⃗u · ∇)⃗u = −∇p + ν∇2⃗u + η⃗d +⃗u(x, y, 0) = 0 +⃗u(xboundary, yboundary, t) = 0 +where p is the pressure, ⃗d is the unit direction vector towards +y direction, ν is the viscosity coefficient and η is a constant +to approximate the Boussinesq buoyancy. The task is to +predict ⃗u at a certain time with a given initial density config- +uration which will decide the unique p. These equations are +solved using semi-Lagrangian method under φFlow (TUM) +framework to gather training data. Specifically, we start +with 1000 random density scenes. The property of smoke +is set as ν = 0.03 and η = 0.5. The total simulation time +is chosen as 40.0s with sampling time step of 1.0s, i.e. 40 +snapshots for each experiment. The x and y components +of velocity are recorded as 64 × 64 grid data. Therefore, +there are 40000 snapshots for both ux and uy. During the +training phase, we randomly split 1000 experiments into +training and test groups in a 4:1 ratio, each group contains +complete 40 snapshots for the whole simulation period. i.e. +32000 samples for the training set and 8000 samples for the +test set. The total amount of diffusion steps is 400 and β +is chosen within [0.0001, 0.02] as suggested in (Ho et al., +2020). The optimizer is Adam with learning rate of 0.0001, +a cosines decay is also used to adjust the learning rate. The +training is performed for 40 epochs with the batch size of 8 +on a GTX Titan Xp GPU for 10 hours. +Figure 2. The floating-smoke experiment, the incompressible +smoke is floating freely within a closed domain under a constant +buoyancy force. The initial density is distributed randomly in +each sub experiment. One of the sub experiments is shown here, +the evolution of density is depicted while the velocity fields are +omitted. +5.2. Reproducing the training data +Since our model does not directly output the velocity field, +it is worth testing the modelling capacity of the model on +the training set first. Figure 5 shows the predictions of one +experiment in the training set. The absolute errors are also +exhibited in Figure 6. In order to verify the model’s capacity +of learning the distribution of data, we further investigate the +probability density distribution of the predictions, the results +can be seen in Figure 7. These results shows that the model +indeed captures the distribution of training data. However, +with the increase of time, the velocity field becomes more +and more irregular, and the error of the prediction result also +increases gradually. We suspect that this is due to the fact +that in the physics problem we study, different experiments +with different initial conditions tend to evolve to similar +states at the end of simulation, and the model can easily +confuse them and ignore the nuances. Due to the lack of +regularization based on physical laws and obvious prior +guidance, our model produces large long-term prediction + +60 +60 +60 +50 +50 +50 +40 +40 +40 +30 +30 +30 +20 +20 +20 +10 +10 +10 +0 +0 +0 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +t=Os +t= 5.0s +t=10.0s +60 +60 +60 +50 +50 +50 +40 +40 +40 +30 +30 +30 +20 +20 +20 +10 +10 +10 +0 +0 +0 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +t=20.0s +t=30.0s +t=40.0serrors. Different from traditional CV tasks that require +sample diversity, we want to generate single and accurate +results. We hypothesize that the problem can be mitigated +by introducing guidance based on physical laws during the +training phase. +5.3. Predictions on the test set +Since the initialization of the experiments is random, we +assume the samples in the test set are independent of the +training set and can be used to test the generalization perfor- +mance of our model. Figure 8, 9 and 10 show the prediction +results, absolute error and probability density distribution +respectively. As with the training set, on the test set, the +model makes reasonable predictions in the short term, but +has larger errors in the long term. This point is also reflected +in the probability density distribution. We believe that this +points to our model having less ability to capture fine dif- +ferences between samples. When samples are far apart in +the data distribution, i.e. the early stage of fluid evolution, +where the evolution states of different initial states have +obvious distinguishing features, the prior is able to guide +the denoising process towards the correct target predictions +efficiently. However, when the samples are very close at +the end of evolution, the samples may tend to be homoge- +neous. Due to the high dimensional nature of the problem, +the guidance from prior is weak and the model can easily +confuse the predictions which are actually from the wrong +initial condition, resulting in poor performance on the test +set. At the same time, since the model is not restricted by +any physics laws, it occasionally produces unreasonable pre- +dictions that violate physics, such as the sudden appearance +of large velocity y components at t = 40 s. +5.4. Comparison with other models +We also compare the performance of our model with three +other models commonly used in CFD prediction tasks. We +were not able to find open-source available implementations +of these three models specific to the studied problem. There- +fore, we have implemented the models to the best of our +ability closely following the relevant literature. The three +models for the benchmark are: 1) cGAN, consisting of a +U-Net as generator and a PatchGAN discriminator 2) PINN, +a ten-layer fully-connected network that learns the solution +⃗u(x, y, t) 3) U-Net, for directly learning the mapping from +y to u, and L2 loss is used in the optimization process. It +should be pointed out that in the training process of PINN, +since it is different from the training data type of other mod- +els (the input of PINN is the (x, y, t) coordinate tuple), only +the data from a single experiment are used. We flatten 2D +grid data into 1D vector with the length of 64 × 64 × 40 and +randomly select 10% of them as the training set, and the +remaining 90% as the test set. Thus, comparison between +PINN and the other models is only indicative. Figure 11 +Table 1. Prediction errors from different models, a smaller number +means a more accurate prediction +MODEL +MAE +RMSE +CGAN +0.4030 +0.5749 +PINN +0.1324 +0.1767 +U-NET +0.3894 +0.5603 +FLUIDDIFF +0.1975 +0.3137 +and 12 show the qualitative comparisons among models +and Table 1 gives the MAE and RMSE of the predictions. +In addition, Figure 3 shows more detailed RMSE at differ- +ent time points. The PINN can produce the most accurate +predictions because it not only relies on the governing equa- +tions, but also excludes the initial condition. It benefits from +the encoded knowledge of the underlying physics, and it is +thus not applicable when the governing equations are un- +known. Also when the different complex initial conditions +are considered, a PINN can be hard to train. cGANs suffer +from the instability of training and mode collapse, which +indeed brought challenges to our benchmarks: It was hard to +avoid mode collapse. For the U-Net, it shows limitations on +predicting samples outside the training set. Our model has +the merit to generate relatively accurate predictions without +knowing the governing equations, being easy to train (no +mode collapse), and it shows good generalization ability in +velocity predictions. +6. Conclusion +In this paper, we have proposed a denoising diffusion gen- +erative model for fluid prediction, and applied it to a 2D +smoke-floating scenario. We shows it is possible to imple- +ment diffusion-based generative models on computational +fluid dynamics field. The most significant advantage over +popular-used GANs is that they are easy to train while main- +taining high quality generation performance, which make +it a potential competitor. However, the limitations still ex- +ists. The main drawbacks of denoising diffusion generative +model lies on three: 1) low sampling speed, the sampling +speed highly dependents on the number of diffusion steps, +while a sufficiently large number of steps is usually a guar- +antee of high-quality sampling results. Recently, many im- +provements have been proposed to improve the sampling +speed of diffusion models, which can be used to solve this +problem. 2) spatial inaccuracy, since diffusion-based model +is originally developed for image-like spatial invariant data, +which is contradicted with the accurate fluid predictions. +Additional improvements on spatial information should be +considered to generate more controllable results. 3) absence +of physics constraints, The lack of guidance with practical +physical meaning in the generation process can lead the +model to give predictions that seriously violate the laws + +Figure 3. The RMSE of predictions from different models at dif- +ferent time. Except for PINN, the other three models all have the +problem of RMSE increasing with the prediction time. +of physics, although this phenomenon is quite rare in our +experiments so far. 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An overview of our proposed model, where the fluid field data samples x are corrupted via diffusion process (5) to be xt, while +initial condition and desired predicting time are ascribed as y. Then xt and y are concatenated as the input of the network, together with +the diffusion time step embedding t through (9). The network is trained to predict the noise ϵ added on x given t, y and further used in +sampling process, which is not shown here + +Input +(“r) +Eg(u,t,y): +Eo(u,t,y) +[ut(T), Ut(T), P(O), T] +UpSampling +Conv 3x3 +Ouput +TransConv3x3 +e(α,t,y) +GroupNorm +I t +Self-attention +(t)*n +Ut(T) +SiLU +Conv3x3 +Embedding MLP +(1)n +Self-attention +GroupNorm +2(0 +u(T) +DownSampling +SiLU +Conv 3x3 + ~ N(; Vα- ,(1-t) -I) +Concatenation +Skip connection +GroupNorm +GroupNorm +SiLU +Self- +④ +Element-wise addition +SiLU +attention +Position embedding of time stepB. Experiment +Figure 5. The velocity field predictions of a random experiment in the training set + +F 09 +60 +60 +60 - +0g +40 +40 +40- +40 - +Ref. +20 +20 +20 +20 - +1.5 + 0 ++ 0 +0 : +0 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +F 09 +09 +60 +60 + 1.0 +40. +40 - +40- +20 - +20 - +20 - +20 - +20 - + 0.5 + 0 ++ 0 +0: ++0 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 + 09 +60 +60 + 09 +60 +0.0 +40- +40 - +40 - +40 - +Ref. +20 +20 +20 - +20 +20 - + 0 +0 +20 +40 +20 +40 +20 +40 +60 +20 +40 +20 +40 +60 +-0.5 +0 +60 +60 +0 +0 +60 +0 +F 09 +- 09 +F 09 +F 09 +60 - +40 - +40 - +40 - +40 - +-1.0 +20 - +20 - +20 +20 +20 - ++ 0 +-0 ++ 0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +t= 1.0s +t = 10.0s +t = 20.0s +t= 30.0s +t = 40.0sFigure 6. The absolute velocity field prediction errors of a random experiment in the training set. +Figure 7. The probability density distribution of velocity field predictions of a random experiment in the training set. Our model succeeds +in learning the data distribution of the training set. + +60 +09 +09 +09 +60 +xn +0.30 +40 +40 - +40 +40 +40 +0.25 +Abs. +20 +20 +20 +20 +20 +0.20 ++ 0 +0 +0 ++ 0 +0 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +0.15 +-09 +60 +60 +09 +60 +3 +0.10 +Error +40 +40 +40 +40 +40 +0.05 +Abs. +20 +20 +20 +20 +20 +0 ++ 0 +- 0 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +t= 1.0s +t= 10.0s +t = 20.0s +t = 30.0s +t = 40.0sF L'O +0.7 +2.5 - + Ref. + Ref. + Ref. +0.8 + Ref. +1.2 +15 + Ref. + Pred. + Pred. +0.6 + Pred. + Pred. + Pred. +0.6 +0.7 +1.0 +2.0 +0.5 +0.5 +0.6 +0.8 +1.5 +0.4 +0.5 +0.4 +0.4 +0.6 +0.3 +0.3 +1.0 - +0.3 +0.4 +0.2 +0.2 +0.2 +0.5 - +0.1 +0.1 +0.2 +0.1 +JLH +0.0 1 +0.0 +0.0 +0.0 +0.0 +-0.50 -0.250.000.250.50 +-3 +-2 +-1 +0 +1 +2 +-2 +-1 +0 +1 +2 +-1 +0 +-1 +0 +ux,t= 1.0s +ux, t= 10.0s +ux, t= 20.0s +ux, t= 30.0s +ux, t= 40.0s +0.40 +0.7 - + Ref. +Ref. + Ref. +0.6 + Ref. + Ref. +1.2 + Pred. +0.35 + Pred. + Pred. + Pred. +0.6 + Pred. +2.0 - +0.30 +0.5 +1.0 +0.5 +1.5 +0.25 +0.4 +0.8 +0.4 +0.20 +0.3 +0.6 +1.0 +0.3 +0.15 +0.2 +0.4 +0.2 +0.10 - +0.5 +0.1 +0.1 +0.2 +0.05 +. +L +0.0 +0.00 +0.0 +0.0 +0.0 +-0.5 +0.0 +0.5 +-4 +-2 +0 +-2 +-2 +2 +4 +2 +0 +2 +-1 +0 +1 +uy, t= 1.0s +uy,t = 10.0s +uy,t= 20.0s +uy, t = 30.0s +uy,t= 40.0sFigure 8. The velocity field predictions of a random experiment in the training set +Figure 9. The absolute velocity field prediction errors of a random experiment in the test set + +F 09 +60 +-09 +60 +09 +2 +40 +40- +40- +40 +Ref. +20 - +20 - +20 - +20 - ++ 0 +0 +0 +0 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +1 +F 09 +60 +60 +60 +40 +20 - +20 - +20 +20 +.0 ++ 0 ++0 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +- 09 +60 +60 +60 +09 +40 +40 +40 - +40- +Ref. +-1 +20 - +20 - +20 +20 +20 - +F0 +0 +0 +0 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +09 +60 +60 +09 +60 +40 - +40 - +40 - +40 +20 +20 +20 + +20 - +20 ++ 0 +0 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +0 +40 +60 +t= 1.0s +t = 10.0s +t= 20.0s +t= 30.0s +t= 40.0s-09 + 09 + 09 +09 +60 +xn +1.75 +40 - +40 - +40 +40 +40 +1.50 +Abs. +20 +20 - +20 +20 +20 +1.25 +- 0 +0 +-0 +0 +0 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +1.00 +60 + 09 +60 +09 +60 +0.75 +40 - +40 - +40 +40 +40 +0.50 +Abs. +20 - +20 - +20 +20 +20 +0.25 +- 0 +0. +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +t= 1.0s +t= 10.0s +t = 20.0s +t = 30.0s +t = 40.0sFigure 10. The probability density distribution of velocity field predictions of a random experiment in the test set. In the short term, our +model can still capture a relatively accurate velocity distribution, but there is a deviation in the long-term prediction. + +0.7 - +3.0 + Ref. +0.5 + Ref. +Ref. + Ref. + Ref. + Pred. + Pred. + Pred. +0.6 + Pred. + Pred. +1.0 - +2.5 +0.4 - +0.4 +0.5 +0.8 - +2.0 +0.3 - +0.4 +0.3 +0.6 +0.2 +0.3 +0.2 +1.0 +0.4 +0.2 +0.1 +0.5 +0.1 +0.2 +0.1 +0.0 +0.0 +0.0 +0.0 +0.0 +-0.50 -0.250.000.25 +50.50 +-2 +0 +2 +-3 +-2 +-1 +0 +1 +2 +-2 +-1 +0 +1 +2 +-2 +-1 +0 +1 +2 +ux,t= 1.0s +ux, t= 10.0s +ux, t= 20.0s +ux, t= 30.0s +ux, t= 40.0s +1.4 + Ref. + Ref. + Ref. + Ref. + Ref. +2.00 +0.6 - +0.8 + Pred. +0.30 +Pred. + Pred. + Pred. +1.2 + Pred. +1.75 +0.7 +0.5 +0.25 +1.50 +0.6 +1.0 +1.25 +0.4 - +0.20 +0.5 +0.8 +0.3 - +0.4 +0.15 +0.6 +0.75 +0.3 +0.10 +0.2 +0.4 +0.50 +0.2 +0.05 +0.1 - +0.2 +0.25 +0.1 +0.00 +0.00 +0.0 +0.0 +0.0 +-0.50 -0.25 0.000.250.50 +-4 +-2 +0 +4 +-2 +2 +-2 +-1 +0 +1 +2 +-2 +-1 +0 +1 +uy, t = 1.0s +uy, t = 10.0s +uy, t = 20.0s +uy, t = 30.0s +uy,t= 40.0sFigure 11. Qualitative comparison of velocity x component predictions from different models. Compared with other models, ours gives +reasonable results on x component, and its performance lies second only to PINN. Considering that PINN does not generalize to initial +conditions in our benchmark, it can be said that in the test task, our model performs quite well. + +60 +60 - +60 - +60 +60 +50 +50 +50 - +50 + 50 - +40 +40 +40 +40 +40 +Ref. +2 +30 +30 +30 +30 +30 +20 - +20 - +20 +20 +20 +10 +10 - +10 - +10 - +10 +0→ +01 +0- +0 +0→ +0 +20 +40 +60 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +0 +60 +F09 +09 +F09 +F 09 +50 +50 +50 +50 +50 - +40 - +40 +40 +40 +FluidDiff +30 +30 - +30 . +30 +30 +20 - +20 +20 - +20 +20 +10 +10 - +10 - +10 +10 +0→ +0- +01 +01 +0→ +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +F 09 +F09 + 09 +60 +50 +50 +50 +50 - +50 - +Pred. +0 +40 - +40 - +40 - +40 +40 +AN +30 +30 - +30 +30 +30 +G +20 - +20 - +20 - +20 +20 +10 +10 - +10 - +10 +10 - +0→ +T0 +0→ +01 +01 +20 +40 +60 +0 +20 +40 +60 +20 +40 +60 +20 +60 +20 +0 +0 +0 +40 +0 +40 +60 +60 +09 +60 +60 +60 - +-1 +50 +50 - +Pred. +40 +40 +40 +PINN +30 +30 +30 +30 +20 +20 - +20 - +20 +20 +10 +10 + +10 - +10. +10 +0- ++0 +0- ++ 0 +0 +60 +0 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +F09 +F 09 + 09 +F 09 +F 09 +-2 +50 +50 +50 - +50 +50 - +Pred. +40 +40 +40 +40 +40 - +UNet +30 +30 - +30 +30 +30 +20 +20 - +20 - +20 - +20 +10 +10 +10 +10 +10 +0→ +01 +T0 +01 +0→ +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +t= 1.0s +t= 10.0s +t = 20.0s +t= 30.0s +t = 40.0sFigure 12. Qualitative comparison of velocity y component predictions from different models. Although FluidDiff gives unreasonable +results when predicting the later y component, the overall performance of our model is still good compared with cGAN and U-Net + +60 +60 - +60 - +60 - +09 +50 +50 +50 - +50 - +50, +40 +40 +40 - +40 - +40 +Ref. +1.5 +30 +30 +30 +30 - +30 - +20 - +20 - +20 - +20 - +20 - +10 - +10 +10 +10 - +10 +T0 +01 + 0 +01 +0 +20 +40 +60 +0 +20 +40 +60 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +1.0 +0 +60 +- 09 + 09 +F 09 + 09 +50 - +50 - +50 - +50 +50 - +40 - +40 +40 - +40 - +40- +FluidDiff +30 +30 - +30 +30 - +30 + 0.5 +20 - +20 - +20 +20 - +20 - +10 +10 +10 - +10 - +10 +01 +01 +F0 +01 +0 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +20 +40 +60 +0.0 +F 09 +- 09 +60 + +60 +60 +50 - +50 +50 - +50 - +Pred. +40 +40. +40 +40 +40 +cGAN +30 - +30 - +30 +30 - +30 +-0.5 +20 - +20 - +20 +20 - +20 - +10 - +10 +10 +10 - +10 - ++ 0 ++ 0 ++ 0 +0- ++ 0 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +-1.0 +60 - +60 - +60 - +09 +50 +50 - +50 - +50 . +50. + Pred. +40- +40. +40 - +40 - +40 +PINN +30 +30 - +30 +30 - +30 +20 +20 - +20 - +-1.5 +20 - +20 - +10 - +10 +10 - +10 - +10 ++ 0 +0→ +0- +0→ +40 +60 +20 +40 +20 +40 +60 +20 +40 +60 +20 +0 +20 +0 +0 +0 +0 +40 +60 + 09 +F 09 +F 09 + 09 +-2.0 +50 - +50 +50 +50 +50 - +Pred. +40 - +40 +40 +40 - +40 - +UNet +30 - +30 - +30 +30 - +30 +20 - +20 +20 +20 +20 - +2.5 +10 - +10 +10 - +10 +10 - +0→ +0 +F0 +01 +0 +20 +40 +60 +0 +20 +40 +60 +20 +40 +60 +0 +20 +40 +60 +0 +20 +40 +60 +t= 1.0s +t= 10.0s +t = 20.0s +t= 30.0s +t = 40.0s \ No newline at end of file diff --git a/VNFJT4oBgHgl3EQf3S18/content/tmp_files/load_file.txt b/VNFJT4oBgHgl3EQf3S18/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9374292cc60ab98a5c5e5ef7e75bd1cc7cbebdba --- /dev/null +++ b/VNFJT4oBgHgl3EQf3S18/content/tmp_files/load_file.txt @@ -0,0 +1,1043 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFJT4oBgHgl3EQf3S18/content/2301.11661v1.pdf,len=1042 +page_content='A Denoising Diffusion Model for Fluid Field Prediction Gefan Yang * 1 Stefan Sommer * 2 Abstract We propose a novel denoising diffusion generative model for predicting nonlinear fluid fields named FluidDiff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFJT4oBgHgl3EQf3S18/content/2301.11661v1.pdf'} +page_content=' By performing a diffusion process, the model is able to learn a complex representation of the high-dimensional dynamic system, and then Langevin sampling is used to generate predictions for the flow state under specified initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFJT4oBgHgl3EQf3S18/content/2301.11661v1.pdf'} +page_content=' The model is trained with finite, discrete fluid sim- ulation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFJT4oBgHgl3EQf3S18/content/2301.11661v1.pdf'} +page_content=' We demonstrate that our model has the capacity to model the distribution of simulated training data and that it gives accurate predictions on the test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFJT4oBgHgl3EQf3S18/content/2301.11661v1.pdf'} +page_content=' Without encoded prior knowl- edge of the underlying physical system, it shares competitive performance with other deep learning models for fluid prediction, which is promising for investigation on new computational fluid dy- namics methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFJT4oBgHgl3EQf3S18/content/2301.11661v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFJT4oBgHgl3EQf3S18/content/2301.11661v1.pdf'} +page_content=' Introduction Computational fluid dynamics (CFD) is the field that in- volves the use of numerical techniques to solve the gov- erning equations for fluids, which exhibit chaotic, time- dependent behaviors known as turbulence due to the nonlin- ear nature of these equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFJT4oBgHgl3EQf3S18/content/2301.11661v1.pdf'} +page_content=' Despite the nonlinearity, the fine spatial and temporal discretization under high-fidelity conditions is computationally expensive and can consume a large amount of computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFJT4oBgHgl3EQf3S18/content/2301.11661v1.pdf'} +page_content=' All these inconve- niences lead to the significant demand for improvements on CFD methodologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFJT4oBgHgl3EQf3S18/content/2301.11661v1.pdf'} +page_content=' Data-driven algorithms, represented by deep learning (DL), have received extensive attention in the past decade and have performed well in solving high-dimensional nonlinear problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFJT4oBgHgl3EQf3S18/content/2301.11661v1.pdf'} +page_content=' Inspired by previous work, we propose an easily trainable generative model for solving nonlinear fluid pre- diction tasks, which is able to give reasonable predictions without requiring knowledge of the physical laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFJT4oBgHgl3EQf3S18/content/2301.11661v1.pdf'} +page_content=' At the same time, our model has good flexibility, which makes it 1Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark 2DIKU, University of Copenhagen, Copenhagen, Den- mark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VNFJT4oBgHgl3EQf3S18/content/2301.11661v1.pdf'} +page_content=' Correspondence to: Gefan Yang +C[m,n] = (A[m, k] Xint8 B[k, n] * fp32 (a_s *fp32 b_s) + fp32 (a_z[m,k] +Xfp32 B[k, n] * b_s) ) * fp32c_s + int64 c_z +The const weight preprocessing optimization is to exploit +the optimization opportunity that some of the input +tensors are constant at the execution time. For the static +quantization inference use case, the weight tensors and +quantization parameters are constant, so a portion of the +post-transformation equation contains computation over +constant weight, scale, and zero point can be avoided +completely at runtime. The challenge is that the weight +data buffer might not be available during the compilation, +so the compiled code needs to preprocess the constant +weight at the execution time when it first arrives. As a_s, +b_s, c_s, and c_z are constants passed as dequantize op’s +attribute, these constants can be folded in the compile- +time. The equation above can be further transformed as +Figure 5. Graph IR optimization passes +Low-Precision Conversion +Const Weight Preprocess +Layout Propogation +Fusion +DQ +matmul +f32 +DQ +A[m, k] +a_s +a_z +B[k,n] b_s +C[m, n] +* +matmul +I8 +* ++ +matmul +I8 +* +A[m, k] +a_s +a_z +B[k,n] +b_s +B[k,n] +b_s +bcast +a_z[m,k] +Q +c_s c_z +* ++ +c_s c_z +C[m, n] +* +matmul +I8 +* ++ +matmul +I8 +* +A[m, k] +a_s +a_z +B[k,n] +b_s +B[k,n] +b_s +bcast +a_z[m,k] +* ++ +c_s +c_z +C[m, n] +CW: +Compensated +Weight +Subgraph( +matmul, *, ++, +, *) +A[m, k] B[k,n] +CW[n] +Subgraph( +matmul, *, ++, +, *) +C2[n, n2] +B2[n, n2] CW2[n2] +C[m, n] +Subgraph( +matmul, *, ++, +, *) +A[m, k] +B[k,n] +CW[n] +Subgraph( +matmul, *, ++, +, *) +C2[m, n2] +B2[n, n2] +CW2[n2] +C[m’, n’, +MB, NB] +RO +RO +RO +RO +RO +A[m’, k’, +MB, KB] +B[k’, n’ +NB, KB] +C[m’, n’, +MB2, NB2] +B2[n’, n2’, +NB, N2B] +A[m, k] +CW[n] +C2[m, n2] +CW2[n2] +RO +RO +RO +B[k’, n’ +NB, KB] +C[m’, n’, +MB2, NB2] +B2[n’, n2’, +NB, N2B] + +matmul +18matmu +18as +matmu +18a +bs +matmu +18CZa +matmul +18CZmatmu +18 + +below. Figure. 5 also shows the Graph IR after the +constant weight preprocess optimization. +if (init) const_weight_comp = a_z[m,k] Xfp32 B[k, n] * fp32 b_s; +C[m,n] = (A[m, k] Xint8 B[k, n] * fp32 a_s *fp32 b_s + fp32 +const_weight_comp) * fp32c_s + int64 c_z; +The constant weight preprocess optimization recognizes +the constant tensor and builds a special initial function +that preprocesses the constant weight and reuses the +processed weight at the runtime. It recognizes the weight +matrix B is passed as a constant logical tensor in the input +graph, meaning that the weight buffer holds a constant +value and won’t change since the first execution of the +compiled partition. This constant is named a runtime +constant. If a DNN op’s inputs are runtime constant or +compile-time constant, the output tensor is runtime +constant as well. The optimization propagates and marks +all the runtime constants throughout the graph. Later the +lowering generates special code for runtime constants, to +make sure these runtime constants only be executed once +in the first execution, and all future execution will reuse +the processed result. +The layout propagation optimization exploits extra +performance benefits with a sequence of Tunable OP by +allowing Tunable OP to use the most desired block +layout. As Tunable OP relies on the blocked layout to +achieve the best performance on the CPU, very often the +best-performed block layout might be different between +two Tunable ops. It allows the Tunable ops within a +subgraph to use a blocked layout but keep the graph +input/output tensor as a plain layout. It first inserts reorder +operations at the graph boundary to ensure the entry and +exit points using the plain layout. Then it iterates the +DNN computation graph and inserts reorder operation +between two Tunable OPs if they use different blocked +layouts. Before inserting a reorder OP before the Tunable +OP, it first queries the Tunable OP for its desired blocked +layouts, if none of the desired blocked layouts is +consistent with the current layout, it inserts a reorder +layout. The inserted reorder OPs are fused to the previous +Tunable OP. The inserted layout reordering for the input +weights produces runtime constants in the inference +scenario, and they are handled by constant weight +preprocessing like the compensated weight. +The Fusion optimization supports both fine-grain fusion +and coarse-grain fusion. The fine-grain fusion inserts +Fusible ops to anchor points of Tunable op and forms one +fused op, and coarse-grain fusion tries to merge multiple +fused ops together to generate optimized code for an even +larger group of operations together. +Both pre-op fusion and post-op fusion help improve the +memory locality since the Fusible OPs work on tensor +slices instead of tensors after being fused into anchor +points of Tunable OP, and tensor slices have a much +higher possibility in the memory system closer to the +compute function unit. For a Fusible op between two +Tunable ops, it is more desirable to fuse it to the previous +Tunable operation as post-op since the overall cost would +be lower. The pre-op fusion only supports limited cases +like reorder and transpose operations and only be used at +the entry point of the graph. There are typically multiple +Fusible operations that need to be fused as post-op. For +example, matmul ops are usually followed by bias, +activation, or normalization ops. These OPs are first +broken down to a sequence of Fusible op, like +elementwise, broadcast, reorder, and reduction ops, and +then fused into Tunable OP as groups of post-ops. +The fine-grain fusion optimization grows a sequence of +post-ops using a simple heuristic to decide whether the +fusion is profitable. It first considers the immediate +succeeding operations of the Tunable op as post-op +candidates and keeps growing the sequence. The heuristic +simply sets a limit of operations so stop growing the +sequence when the limit is reached. For example, the +post-op sequence can only have one reorder and one +reduction op. As the post-op may involve accessing +additional memory, like the second operand tensor for a +binary op, the heuristic fusion optimization also monitors +the total additional memory being accessed and limit the +potential negative impact on the Tunable op execution. +Then, the fine-grain fusion optimization sorts the post-ops +in topological order and splits them into two groups if the +post-ops contain one reduction op: the post-ops not +dependent on the reduction op, and the reduction op and +its dependent ops. The first group is inserted into the same +post-op anchor. The reduction op may be split into two +anchor points, and its dependent ops are inserted at the +second anchor point where the final reduction result is +collected. +The coarse-grain fusion optimization further merges fused +ops. The larger scope of the DNN graph opens many +possibilities. Multiple Fused ops could be lowered to one +parallel loop, in order to improve data locality or better +exploit the parallelism. For example, the outermost “mpi” +loop of two fused ops may have the same blocking factor, +so that they can be merged as one loop. When the +heuristic chooses the parameters for each Tunable op, it + + + +tries to choose the outermost loop blocking factor best +aligned with core numbers, so each instantiated fused op +has the same blocking factors as its neighbor. When the +coarse-grain fusion optimization decides to merge two +fused ops, it marks the two nested loops in Tensor IR as +“mergeable” during the lowering process. Then Tensor IR +merges two nested loops mechanically as guided by the +Graph IR optimizations. +Tensor IR optimization +Tensor IR is the lowest intermediate representation in the +oneDNN Graph Compiler. At the Tensor IR level, the +DNN computation graph is lowered to a C-like program, +which includes function, statement, expression, and +intrinsic functions. The Fused OP is lowered as a +function, which contains nested loops. A complex +statement describes a structure like a loop, and a simple +statement does computation. Var and Tensor represent +scalar variables and multi-dimension arrays respectively. +Tensor IR supports Graph IR optimization by merging +loops as instructed by Graph IR. Figure 6 shows the +example of Tensor IR for the pseudo-code in figure 4. In +the Tensor IR, the computation on the tensor slices is +represented by either a nested loop or a function call to +the microkernel. The inserted pre-op and post-op are +lowered to nested loops. The two post-op, ReLU and +reorder ops, are merged as one nested loop using the hint +passed by Graph IR. +The main optimizations on Tensor IR are tensor size +optimization and memory buffer optimization. Tensor +size optimization tries to reduce the tensor size of each +temporary tensor. The temporary tensors are introduced in +the pre-op and post-op fusion process. The temporary +tensor was initially introduced as a full-size tensor in the +lowering process and then reduced by the tensor size +optimization. In the example code of figure 6, the post- +ops are fused into one loop nest in the Tensor IR. Since +the accesses of the temporary tensors, C’’ and C’’’, are +local to the innermost loop body, the temporary tensor +could be replaced by a scalar variable. Compared to +accessing global and larger original tensors, the result +code has a much smaller memory footprint and better +cache locality. The temporary tensor introduced by pre-op +fusion can be reduced similarly by analyzing the scope of +the tensor usage. For example, A’[MSN, BS, MB, KB] +could be reduced to A’ [BS, NB, KB], since the producer +of A’ and consume are within the “msi” loop, so there is +no need to save the result along the 2nd dimension of A’. +After tensor size optimization, the multiple-dimension +tensor representation is flattened to a one-dimensional +array to represent the memory buffer. The memory buffer +optimization tries to reuse the memory buffer of +temporary tensors to have a minimum overall temporary +buffer size for the compiled code and tries to improve the +locality of the temporary buffer use. +Figure 6. Lowering to Tensor IR +Var Const MPN, NPN, MSN, NSN, BS, KSN, MB, NB; +Tensor FP32[M, K] A; +Tensor FP32[M/MB, K/KB, MB, KB] A'; +Tensor FP32[K/KB, N/NB, NB, KB] B; +Tensor FP32[NSN, MB, NB] C'; +Tensor FP32[M/MB,N/NB, MB, NB] C'', C'''; +Tensor FP32[M/MB2, N/NB2, MB2, NB2] C; +Var int* A_addr[BS], B_addr[BS]; +Var Index mp, np, ms, ks, ns, nps, mps; +Parallel loop mp i= 0, MPN, 1 { + Parallel loop npi = 0, NPN, 1 { + Loop msi = 0, MSN, 1 { + mpsi = mpi * M/MPN + msi; + Loop nsi = 0, NSN, 1 { + Loop mbi = 0, MB, 1 { + Loop nbi = 0, NB, 1 { + C’[0:NSN, 0:MB, 0:NB] = 0; + } + } + } + Loop ksi = 0, KSN, BS { + Loop bsi = 0, BS, 1 { + ksbi = ksi * BS + bsi; + Loop mbi = 0, NB, 1 { + Loop kbi = 0, KB, 1 { + A’[mpsi, ksbi, mbi, kbi] + = A[mpsi*MB + mbi, ksbi*KB+kbi]; + } + } + } + Loop nsi = 0, NSN, 1 { + npsi = npi * N/NPN + nsi; + Loop bsi = 0, BS, 1 { + A’_addr[bsi] = &A’[mpsi, ksi, 0, 0]; + B_addr[bsi] = &B [ksi, npsi, 0, 0]; + } + C'_addr = &C’[nsi,0, 0] ; + Batch_reduce_gemm(A’_addr, B_addr, + C'_addr, MB, NB, KB, Batch = BS); + } + } + Loop nsi = 0, NSN, 1 { + npsi = npi * N/NPN + nsi; + + + + Loop mbi = 0, MB, 1 { + Loop nbi = 0, NB, 1 { + C’’[mpsi, npsi, mbi, nbi] = C’[nsi, mbi, nbi]; + C’’’[mpsi, npsi, mbi, nbi]= max(C’’[mpsi, npsi, mbi, nbi], 0); + C[(mpsi*MB+mbi)/MB2, (npsi*NB+nbi)/NB2, + (mpsi*MB+mbi)%MB2, (npsi* NB+nbi)%NB2] + = C’’’[mpsi, npsi, mbi, nbi]; + } + } + } + } + } +} + + + +The main target of memory buffer optimization is to reuse +the memory buffer created for the temporary tensors +between fused op. In the inference use case, the output +tensor is only consumed by the next fused op, and so the +buffer could be reclaimed once the next fused op +completed execution. Since the input tensor size is known +to the compilation process, the memory buffer usage can +be tracked at the compile time and optimized to improve +efficiency. +Memory buffer optimization uses life span analysis like +traditional compiler analysis for register allocation based +on the def-use chain. Memory buffer optimization has +extra consideration for buffer reuse since the memory +buffer size used is not uniform as registers. It scans the +Tensor IR, tracks all the memory buffers alive, and then +computes the peak size of the total memory buffers +needed for the entire compiled graph. The algorithm +considers both reusing the hot memory and reducing the +overall peak memory. At each point, when an +intermediate buffer is needed, it tries to reuse the free +intermediate buffers, which are already allocated but not +used anymore. Among multiple choices of reusable +memory buffers, it chooses the one that was used most +recently, so likely the data is still in the cache system. +Experimental results +oneDNN Graph Compiler targets performance-critical +DNN computation graph, which is usually a subgraph of +the whole DNN model graph. We selected two DNN +computation subgraphs as target workloads to evaluate +the performance. The Multilayer Perceptron (MLP) +workload contains multiple matmul ops intermixed with +activation ops like ReLU. The MLP subgraph is the basic +building block for many deep learning models, including +recommendation systems and natural language +processing. The Multi-Head Attention (MHA) subgraph +is the key element to Transformer based deep learning +models like Bert for natural language processing. The +MHA workload focuses on the scaled dot-product +attention portion of the MHA graph, which contains two +batch matmul ops and a softmax as well as other binary +ops between them. Depending on the use case, the MLP +and MHA subgraphs tend to account for more than half of +total model execution time, especially for DLRM or Bert +Large models. + + +We choose the inference use case for the evaluation and +measure the performance for both FP32 and Int8 data +types. We choose several representative data shapes for +weights and input tensors and select a wide range of batch +sizes. The weight sizes for MLP are from the MLPerf +Workload Category +data type +input batch size +sequence length +hidden size +head numbers +MLP_1 +Int8, FP32 +32, 64, 128, 256, 512 +N/A +13x512x256x128 +N/A +MLP_2 +Int8, FP32 +32, 64, 128, 256, 512 +N/A + 479x1024x1024x512x256x1 +N/A +MHA_1 +Int8, FP32 +32, 64, 128 +128 +768 +8 +MHA_2 +Int8, FP32 +32, 64, 128 +128 +768 +12 +MHA_3 +Int8, FP32 +32, 64, 128 +384 +1024 +8 +MHA_4 +Int8, FP32 +32, 64, 128 +512 +1024 +16 +Figure 7. Performance comparison for individual Matmul op +Table 1. Workload parameters +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +32, 13, 512 +64, 13, 512 +128, 13, 512 +256, 13, 512 +512, 13, 512 +32, 512, 256 +64, 512, 256 +128, 512, 256 +256, 512, 256 +512, 512, 256 +32, 256, 128 +64, 256, 128 +128, 256, 128 +256, 256, 128 +512, 256, 128 +32, 479, 1024 +64, 479, 1024 +128, 479, 1024 +256, 479, 1024 +512, 479, 1024 +32, 1024, 1024 +64, 1024, 1024 +128, 1024, 1024 +256, 1024, 1024 +512, 1024, 1024 +32, 1024, 512 +64, 1024, 512 +128, 1024, 512 +256, 1024, 512 +512, 1024, 512 +32, 256, 1 +64, 256, 1 +128, 256, 1 +256, 256, 1 +512, 256, 1 +Matmul kernel performace related to expert-tuned +implementation +Baseline (FP32, Int8) +Compiled Kernel (FP32) +Compiled Kernel (Int8) +speedup +m,k,n + + + +DLRM model, and the sequence length and hidden size +choices for MHA are from the Bert models. The +performance data is collected on an Intel® Xeon® +Platinum 8358 processor with 32 cores. +We first study the fine-grain fusion performance for +individual layers since this is the foundation of the +oneDNN Graph Compiler. The tests evaluate all the +problem sizes used in the MLP tests. The baseline is +heavily optimized and uses oneDNN primitives, which is +the industry standard best-performant expert-tuned +implementation. Both the baseline and oneDNN Graph +Compiler assume weight being pre-packed, compensated, +and preprocessed. The input and output matrixes are in +plain layouts. Figure 7. shows that the performance of +oneDNN Graph Compiler’s automated kernel generated +using the template approach is 6% better than the expert- +tuned primitives for the given test cases. oneDNN Graph +Compiler outperforms the expert-tuned primitives in +many smaller problem sizes and falls behind on certain +cases, particularly with k=479. The performance of +individual layers heavily depends on the algorithm and +heuristic, and we expect the performance gap on specific +cases would be narrowed as oneDNN Graph Compiler +continues to develop the algorithm and heuristic, and vice +versa. +Figure 8 shows performance comparisons for MLP and +MHA tests between oneDNN Graph Compiler and +oneDNN Primitives. The left side shows performance +data for FP32 data type and the right side is Int8. Each +test is named with workload category, batch size, and data +type as shown in Table 1. For each test, we measure the +performance of the baseline, oneDNN Graph Compiler, +and the middle setting which disables the coarse-grain +fusion and evaluates the rest optimizations including the +fine-grain fusion for oneDNN Graph Compiler. The +baseline uses expert-tuned oneDNN primitive with fusion +support and has been integrated into multiple DL +frameworks to accelerate deep learning on the CPU. +Specifically, it uses oneDNN primitives post-op fusion to +fuse matmul op with ReLU for the MLP tests and with +division and addition ops for the MHA tests. For Int8 +tests, before calling the oneDNN primitives, the baseline +applies similar low-precision graph transformation and +maps the graph to low-precision matmul and post-ops. +Besides that, the baseline also performs weight pre- +packing, compensated weight preprocessing, and caches +the result to avoid re-computation at runtime. +The performance results show oneDNN Graph Compiler +significantly improves the performance of the target DNN +computation subgraph. For MLP tests, oneDNN Graph +Figure 8. oneDNN Graph Compiler performance evaluation for MLP and MHA subgraph +0.00 +1.00 +2.00 +3.00 +MHA performance evalution +FP32 & int8 inference on 32-core CPU +Primtives + post-ops +Compiler without coarse-grain fusion +Compiler +0.00 +0.50 +1.00 +1.50 +2.00 +2.50 +3.00 +3.50 +MLP performance evalution +FP32 & int8 inference on 32-core CPU +Primtives + post-ops +Compiler without coarse-grain fusion +Compiler +speedup +speedup + + + +Compiler demonstrates an average of 1.73x speed up on +Int8 data type and 1.22x speed up on FP32. The five +MLP_1 tests for Int8 data type show the highest speedup +at an average of 2.72x. Among a total of 2.72x speed up, +coarse-grain fusion is the main contributor and accounts +for 1.95x. It merges 3 parallel loops, lowered from 3 +matmul ops, into one parallel loop. The coarse-grain +fusion greatly reduces the synchronization overhead and +permits the activation data to be in the fastest cache for +the next matmul op. For the MLP_1 Int8 tests, the entire +activation and weight tensor fit in the L2 cache, so the +coarse-grain fusion performs very well for these cases. +When disabling the coarse-grain fusion, the remaining +optimization accounts for about 1.4x. There are mainly +three reasons. First, although the baseline implements the +same fusion, oneDNN Graph Compiler has better +performance for each individual matmul op in MLP_1, as +shown in Figure 7. Second, the layout propagation allows +all three matmul ops to run with the same blocked layout +without extra reordering. Last, due to the MLP_1 tests +being relatively short, the total API call overhead takes up +to 10% of the execution time for the baselines, which is +reduced by about 3 times since the compiled code needs +only to be called one time. Compared to MLP_1 Int8 +tests, the MLP_1 FP32 tests show an overall 1.47x +performance gain, with 1.15x from coarse-grain fusion +and 1.28x from rest optimizations. +For MLP_2 test cases, the oneDNN Graph Compiler +shows an overall 10% better performance Int8 data type +and 1% on FP32. When the coarse-grain fusion is +disabled, oneDNN Graph Compiler is 1% slower +compared to the baseline with FP32 and Int8 test cases +combined. As we learned from the individual matmul op +analysis, the initial layer of MLP_2 (k=479) has lower +performance for oneDNN Graph Compiler and negatively +impacts the overall performance. MLP_2 tests also get +benefit from the coarse-grain fusion but to a lesser extent. +The coarse-grain fusion is not able to merge all the loop +nests due to the current heuristic limitation. We believe +that the performance for MLP_2 test cases can be further +improved with more heuristics tuning. +For the MHA subgraph, oneDNN Graph Compiler +demonstrates an overall 1.91x performance gain over 24 +MHA tests with 1.99x on Int8 data type and 1.84x on +FP32. In contrast to the MLP performance, the +performance benefit is more significant for the tests with +larger problem sizes, and fine-grain fusion helps more +than coarse-grain fusion. The baseline doesn’t have the +same fusion capability as the fine-grain fusion, as +oneDNN post-op baseline doesn’t fuse the softmax op +with the preceding batch matmul op. oneDNN Graph +Compiler decomposes softmax op to multiple basic +operations, and its fine-grain fusion optimization fuses +them to the preceding batch matmul ops. The basic +operations are divided into two groups of post-ops: a +group of element-wise ops and a group led by a reduction +op. These two groups are inserted into the batch matmul +op. This gives a significant boost to the performance by +an average of 1.51x. The coarse-grain fusion adds another +27% performance gain on top of fine-grain fusion by +merging the two nested loops translated from two batch +matmul ops. +Conclusion +We proposed a hybrid approach to address the unique +challenges of deep learning compilation. It distilled key +ingredients of expert-tuned primitives for compute- +intensive DNN operations like matrix multiplication and +uses compiler techniques on the DNN computation graph +to fully exploit the performance opportunity at the graph +level. The template uses an expert-developed microkernel, +algorithm, and heuristic, to ensure compiler-generated +code achieves comparable performance to expert-tuned +primitives. 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Available: +http://arxiv.org/abs/1903.06498 + diff --git a/WNAzT4oBgHgl3EQfYPzf/content/tmp_files/load_file.txt b/WNAzT4oBgHgl3EQfYPzf/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..057d668fc8f5acb1ae071c5c8b565ee8cd1393b8 --- /dev/null +++ b/WNAzT4oBgHgl3EQfYPzf/content/tmp_files/load_file.txt @@ -0,0 +1,811 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf,len=810 +page_content='oneDNN Graph Compiler: A Hybrid Approach for High-Performance Deep Learning Compilation Jianhui Li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Zhennan Qin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Yijie Mei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Jingze Cui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Yunfei Song,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Ciyong Chen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Yifei Zhang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Longsheng Du,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Xianhang Cheng,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Baihui Jin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Jason Ye,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Eric Lin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Dan Lavery Software and Advanced Technology Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Intel Abstraction With the rapid development of deep learning models and hardware support for dense computing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' the deep learning (DL) workload characteristics changed significantly from a few hot spots on compute-intensive operations to a broad range of operations scattered across the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Accelerating a few compute-intensive operations using the expert-tuned implementation of primitives doesn’t fully exploit the performance potential of AI hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Various efforts are made to compile a full deep neural network (DNN) graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' One of the biggest challenges is to achieve end-to-end compilation by generating expert- level performance code for the dense compute-intensive operations and applying compilation optimization at the scope of DNN computation graph across multiple compute-intensive operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' We present oneDNN Graph Compiler, a tensor compiler that employs a hybrid approach of using techniques from both compiler optimization and expert-tuned kernels for high-performance code generation of the deep neural network graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' oneDNN Graph Compiler addresses unique optimization challenges in the deep learning domain, such as low-precision computation, aggressive fusion, optimization for static tensor shapes and memory layout, constant weight optimization, and memory buffer reuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Experimental results demonstrate up to 2x performance gains over primitives-based optimization for performance-critical DNN computation graph patterns on Intel® Xeon® Scalable Processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Introduction With the vigorous development of AI applications, deep learning software stacks and hardware are rapidly evolving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Data Scientist is continuously exploring new deep neural network (DNN) models to improve the accuracy of the models by increasing the model parameters, using larger training datasets, and exploring innovative DNN structures and operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Deep learning frameworks, like TensorFlow[3] and PyTorch[4], are developed to support the development and deployment of deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' While supporting Data Scientists to develop new models, DL frameworks also need to efficiently use hardware resources to meet the huge computing needs of deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Meanwhile, various hardware supports for deep learning have been introduced, including adding matrix operation units on GPU and CPU, and hardware accelerators dedicated to deep learning computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Deep learning (DL) frameworks provide a rich set of deep neural network (DNN) operations for developers to describe a DNN model and use primitives libraries by default to offload the most performance-critical operations to CPU and GPU [1][2][20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Most of the execution time of DL applications is spent on the DNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' DL frameworks represent the DNN models internally as a computation graph of DNN operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' After performing high-level graph optimizations, the graph is executed operation by operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' On top of their own implementation of DNN ops, DL frameworks use third-party primitives libraries to offload the most performance-critical DNN operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Primitives library offers a simple and effective way to offload deep learning computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' However, its performance benefit doesn’t scale to new AI workloads and hardware due to its limited optimization scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' With the fast evolution of AI software and hardware, the performance characteristics of deep learning workload have been shifted from a few hot spots of concentrated compute-intensive operations to many scattered DNN operations including memory-bound operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' There are multiple sources that contribute to the increasing time percentage on memory-bound operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The deep neural network used for natural language processing [19] and recommendation systems [18] has smaller input data and overall lower compute intensity compared to computer vision models [22] [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Instead of supporting each innovative activation function with a complex DNN operation, DL Frameworks tend to compose multiple existing fine-grain operations, to maintain a balance of scalability and usability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Lastly, hardware acceleration usually focuses on accelerating the dense computation of low-precision data types and relies on the software to optimize memory-bound operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The performance characteristic shift drives the development of low-level graph compilers in the deep learning domain also known as tensor compilers [5][6] [7] [10] [15] [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The tensor compilers focus on optimization techniques specific to the deep learning workload, aiming at generating highly efficient code for a DNN computation graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Tensor compilers view DNN operations as tensor computation and internally represented as nested multi-level loops with the innermost loop body processing each tensor element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' It uses compiler loop transformation techniques to parallelize, vectorize, reorder, and merge the nested loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' It has been a hot research area on how to automatically generate code for compute-intensive primitives and achieve high performance comparable to expert-tuned implementation [9] [14] [16][21] [25] [26] [27] [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Due to the inherent complexity of nested loop transformation, some research uses the autotune method to search for a good solution in a large search space, and some use analytical models to determine optimal tuning parameters for generating high-performance primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The tensor compilation generated code is mainly used in specific use cases where the expert-tuned primitives library can’t offer the required performance and the users are willing to spend extra resources to find a better solution very often via autotuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' oneDNN Graph Compiler is an open-source tensor compiler that automates the code generation for compute- intensive DNN operations like matrix multiplication and achieves the same level of computing efficiency as primitives library implementation [1][11][12][13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Based on that, it further explores more advanced optimization for a subgraph with multiple compute-intensive DNN operations, such as optimizing the whole Multilayer Perception (MLP) network construct containing multiple matrix multiplication operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' oneDNN Graph Compiler applied domain-specific expert knowledge that was distilled from the expert-tuned kernel development process to an automated compilation process and achieved performance in parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' It combines compiler and kernel library techniques and focuses on domain- specific optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' With expert-tuned microkernels and two levels of compiler IR, oneDNN Graph Compiler addresses domain-specific optimization challenges, such as generating efficient code for the compute-intensive kernels of static tensor shapes with blocked memory layout, constant weight optimization, aggressive fusion, and memory buffer reuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Experimental results show that oneDNN Graph Compiler delivers significant performance gains over primitive- based optimization for performance-critical DNN computation graph patterns on CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' High-level Design The principal of oneDNN Graph Compiler high-level design is performance, simplicity, and modularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' oneDNN Graph Compiler has two levels of intermediate representations: Graph IR, and Tensor IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The input DNN computation graph is internally represented as Graph IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The Graph IR optimization module performs a number of transformations that optimize and group the computation graph as a sequence of fused operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Graph IR is further lowered to Tensor IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The Tensor IR doesn’t preserve DNN operation semantics and is close to the C program semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The data structure it operates on is multidimensional arrays, representing tensor buffers in physical memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Tensor IR is then further lowered to LLVM IR and intrinsic calls to Microkernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The Graph IR keeps the DNN OP semantics, so most domain-specific optimizations are done at this level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Instead of lowering the DNN OP to Tensor IR and performing sophisticated loop analysis to achieve the best loop schedule and fusion, oneDNN Graph Compiler uses expert-developed templates, microkernels, and heuristics to guide the code generation of compute-intensive operations and the fusion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The decisions of parallel task decomposition, loop scheduling and tiling, tensor memory layout, and whether to fuse with neighbor operations are based on the Graph IR with DNN OP semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' This simplifies Tensor IR design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The Tensor IR supports mechanically lowering Fused OP to nested loop, but there is no need to support sophisticated loop analysis and complex nested loop transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Tensor IR optimization mainly focuses on tensor buffer optimization and supports low-level code generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The use of compilation techniques, including Graph IR and Tensor IR, helps to increase the optimization scope from individual primitives to a larger subgraph with multiple compute-intensive operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' oneDNN Graph Compiler IR and Optimization GraphIR GraphROptimization(decomposition,transformation,fusion) GraphIRLowering TensorIR TensorIROptimization(Loop,tensor,variable,constant) TensorIR Lowering LLVMIR Microkernel Graph IR uses graph, logical tensor, and OP to describe a computation graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' A graph contains a set of OPs and logical tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Each OP represents an operation in a computation graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' A logical tensor represents the tensor’s metadata, like the element’s data type, shape, and memory layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' OP has kind, category, attributes, and logical tensors for inputs and outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Graph IR optimization module first decomposes complex OPs into basic DNN OPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The complex DNN OPs are OPs with complex semantics which could be composed of simple fundamental operations like addition and multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' They are introduced by DL frameworks to support high-level DNN OP semantics for ease of programming, such as batchnorm, quantize, gelu, and many activation operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The basic DNN OPs are categorized to be either Tunable OP or Fusible OP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Tunable OPs describe DNN operations that use tunable parameters to instantiate a pre-defined template to generate the best-performing code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The example includes compute-intensive operations like matmul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Fusible OP refers to operations that can be fused to Tunable OPs, such as element-wise operations, broadcast, reduction, and data movement operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The decomposition of complex DNN operations simplifies the Graph IR optimization module so it only needs to handle basic DNN operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Besides the general compiler optimizations like common subexpression elimination (CSE), dead code elimination, and constant folding, it includes domain-specific optimizations like low-precision conversion, tensor memory layout propagation, constant weight preprocessing, and fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The fusion optimization pass decides whether it is profitable to fuse two operations and keeps fusing OPs to form a subgraph, which is represented as a Fused OP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The Graph IR is transformed into a graph of Fused OPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The Graph IR is then lowered into Tensor IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Just like the C program, Tensor IR supports function, statement, expression, and intrinsic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The Tensor IR module, lowered from a Graph IR graph, contains multiple functions, each of which represents a lowered Fused OP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The Tensor IR module has an entry function that contains a sequence of calls to other functions lowered from Fused OPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' A Tensor IR function contains multiple statements build on expressions, which operate on constants, variables, and tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Constants and variables represent individual data elements, used to represent scalar data like loop index, tensor shape, address and offset to tensor buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Tensors represent multi- dimension arrays backed by a data buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The intrinsic function is used to represent a microkernel, which is carefully hand-tuned and fulfills a subtask of a DNN OP with data in the fastest cache on a single CPU core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' This paper describes the most effective techniques used in the oneDNN Graph Compiler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' To build a high-quality tensor compiler, there are many details to consider at the implementation level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' We first explain the foundation, using templates to guide the lowering of Tunable OP and Fused OP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Then we describe the Graph IR optimization which maximizes the fusion opportunity to form a large and profitable Fused OP and low-level Tensor IR optimization for efficient code generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Microkernel-Based Template for Tunable OP Lowering Automating the high-performance code generation for Tunable OPs is the foundation of a tensor compiler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' oneDNN Graph Compiler took an approach inherited Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Microkernel based template for Tunable OP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='Index of single-core kernel within multi-core kernel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='mpi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='npi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='kpi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='Number of single-core kernels within multi-core kernel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='MPN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='NPN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='KPN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='Index of microkernel within single-core kernel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='msi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='nsi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='ksi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='Number of microkernel within single-core kernel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='MSN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='NSN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='KSN/BS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='Index of microkernel within multi-core kernel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='mpsi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='npsi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='kpsi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='Number of microkernel within multi-core kernel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='MPSN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='NPSN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='KPSN/BS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='Tensor size ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='Tensor block size ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='MB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='NB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='KB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='Tensor slice size accessed by Microkernel size (batch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='size = BS) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='MB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='NB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='KB * BS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='Tensor slice size accessed by single-core kernel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='MSBN = MB * MSN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='NSBN = NB * NSN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='KSBN = KB * KSN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='Tensor slice size accessed by multi-core kernel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='M = MB * MSN * MPN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='MB * MPSN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='N = NB * NSN * NPN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='= NB * NPSN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='K = KB *KSN * KPN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='= KB * KPSN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='Parallel loop mpi = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' MPN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' 1 { Parallel loop npi = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' NPN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' 1 { Loop msi = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' MSN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' 1 { mpsi = mpi * M/MPN + msi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' C’[0:NSN, 0:MB, 0:NB] = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Loop ksi = 0, KSN, BS { Loop nsi = 0, NSN, 1 { npsi = npi*N/NPN + nsi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' A_addr[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='.BS] = &A[mpsi:1, ksi:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='.BS, 0:MB, 0:KB];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' B_addr[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='.BS] = &B[ksi:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='.BS, npsi:1, 0:NB, 0:KB];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' C’[npsi:1,0:MB, 0:NB] += Batch_reduce_gemm (A_addr[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='.BS], B_addr[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='.BS],Batch = BS);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' } } C[mpsi:1, npi:NSN, 0:MB, 0:NB] = C’[0:NSN, 0:MB, 0:NB];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' } } } single-core kernel micro kernel multi-core kernel Tensor is described with a Tensor name followed by index and size for each dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Tensor A[0:M, 0:K] refers to 2 dimensions tensor starting from the position [0,0] with size [M, K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' A[0:MB, 0:KB] refers to a tensor slice containing a subset of A tensor elements, starting from position 0 to MB-1 along the m dimension, and 0 to NB-1 along the n dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The pseudo-code uses a blocked layout for A, B, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' C[0:MPSN, 0:NPSN, 0:MB, 0:NB] denotes the full C tensor C[0:M, 0:N] reordered with a blocked layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' C[mps:1, np:NSN, 0:MB, 0:NB] denotes a tensor slice which “slice” the C tensor in the first 2 dimensions starting from position “mps” and “np” with size “1” and “NSN”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' A_addr[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='.BS] denotes an array with BS elements from A_addr[0] to A_addr[BS-1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' A[mps:1, ks:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='.BS, 0:MB, 0:KB] denotes an array of BS tensor slices from A[mps:1, ks:0, 0:MB, 0:KB] to A[mps:1, ks:BS-1, 0:MB, 0:KB].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' from the performance library development, which first creates the code templates for a given Tunable OP and then instantiates it with parameters decided by a heuristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The parameters are decided based on the input data tensor shape and hardware sizes of the microarchitecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The template shown above is for a matmul op that does matrix multiplication over A[M, K] and B[K, N] and produces C[M, N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The template is applied to a common deep learning use case where the computation uses multiple cores, and the size of input and output tensor fits within the cache system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The outer parallel loops divide the kernel into multiple subtasks for multi-cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Each subtask is assigned to one single core, named single-core kernel, which is represented by the inner loops which call a microkernel in the innermost loop body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The microkernel and the single-core kernel operate on a tensor slice that represents a subset of tensor elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' For example, the original tensor is represented as A[0:M, 0:N], where the subscription represents starting offset and size for each dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The tensor slice is represented as A[0:MB, 0:NB], where MB and NB refer to the tile size of the tensor slice along m and n dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' A submatrix is a special case of a 2-dimension tensor slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' In the template above, the microkernel produces a small submatrix C[0:MB, 0:NB], and the single-core kernel outputs a larger submatrix C[0:MSN, 0:NSN].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The microkernel is an important element for the oneDNN Graph Compiler to achieve comparable performance to expert-tuned primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' oneDNN Graph Compiler uses the microkernel named batch-reduce GEMM [8][24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The microkernel has two inputs, both representing a batch of 2D matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' It first applies matrix multiplication with each batch element to produce a batch of immediate 2D matrices and then sums them to a final 2D matrix output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' This interface can be used for many variants of matmul op in both inference and training use cases and was adopted by both oneDNN primitives and oneDNN Graph Compiler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The microkernel is fine-tuned to maximize the compute efficiency by fully utilizing the compute function unit and the high bandwidth provided by registers and the L1 cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=" It abstracts the ISA difference so oneDNN Graph Compiler doesn't need to deal with different vector or matrix instructions provided by different CPUs." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' However, the oneDNN Graph Compiler needs to choose the input submatrix sizes for the microkernel so that they are usually multiples of register sizes used by the vector and matrix function units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Also, it needs to choose the batch size for the microkernel so that the whole input and output submatrices fit within the L1 cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' To further streamline the cache access, the input and output tensors are blocked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' To simplify the implementation, the input and output tensors are blocked using the submatrix sizes [MB, NB, KB].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' So, each microkernel accesses a contiguous memory buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The parameters for lowering a matmul op refer to the variable values in the template above: MPN, NPN, MB, NB, KB, BS, and ordering of loops indexed by ms, ks, and ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The other parameters can be derived from the parameters above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' oneDNN Graph Compiler uses an expert-tuned heuristic to decide these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' For a given output matrix size, it first proposes single-core kernel size options, a set of [MPN, NPN], which can use all cores with good load balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' It further proposes microkernel size options, a set of [MB, NB, KB, BS], which ensure good microkernel performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Then the heuristic picks a pair of these sizes, which has the best overall kernel performance for the entire system with all cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' It iteratively searches for the best parameters, based on a cost model which considers multi-core load balancing and single-core kernel efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Heuristic also reports the loop ordering of the inner loops which it assumes when computing the single-core kernel efficiency during the search process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' oneDNN Graph Compiler developed multiple templates for different uses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' One Tunable OP can have multiple templates depending on the use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' For example, for training use cases, the input activation can be significantly larger than the L2 cache size so requires an additional loop level to block the data for the L2 cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' For inference cases, sometimes the use case only processes one data sample with multiple cores so that the template may have to apply “k-slicing” to extract additional parallelism from the reduction axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Very often some dimensions of the input tensors are not multiple of microkernel sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' oneDNN Graph Compiler pads the input tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The padding and un-padding happen at the entry and exit points of the computation graph and are fused into the Tunable OP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The examples used in this paper assume the tensor sizes are aligned with hardware vector and matrix operand sizes to simplify the description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Template with Anchors for Fused OP Lowering Fusion is a very important technique to achieve high performance in the deep learning domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' oneDNN Graph Compiler supports two types of fusion: fine-grain fusion fuses one Tunable OP with multiple adjacent Fusible OPs to a Fused OP, and coarse-grain fusion fuses multiple Fused OPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The fine-grain fusion is supported by the template of Tunable OP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The template contains placeholders, as known as anchors, at the beginning and the end of each loop level for the input and output tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The Graph IR fusion optimization decides whether it is profitable to fuse a Fusible OP to a Tunable OP and which anchor point is assigned to the Fusible OP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The Fused OP lowering pass retrieves anchors for Fusible OPs and directly inserted its corresponding Tensor IR at the anchor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The main benefit of fusion is that the operation being fused only needs to access tensor slices associated with the anchor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The anchors before the microkernel are called pre-op anchors, which work on the input tensors’ tensor slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The anchors after the microkernel are called post- op anchors, which work on the output tensor’s tensor slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The Fusible op is called pre-op if it is inserted into pre-op anchors, and the Fusible op for post-op anchors is called post-op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The fusion optimization evaluates total memory access and computation cost for each anchor point and selects one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' There are multiple choices of commit anchors to insert a pre-op or post-op fusion, depending on the different levels of the loop nest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The commit anchors inside the innermost loop work on the smallest tensor slice, which provides a low per-access cost as the data is in the fastest cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' However, the inner loop bodies also have more computations since these computations are performed redundantly along the orthogonal dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Meanwhile, the fusion process may require allocating additional temporary tensors, which the fusion optimization also needs to evaluate carefully and factor in the potential interference with the cache behavior of Tunable OP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' For example, the pre-op introduces temporary tensors since they can’t work on the original input tensors directly, which causes data conflicts with the read accesses from other parallel cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Figure 3 illustrates the pre-op anchors and post-op anchors within a template and the associated tensor slices for each anchor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The right table in Figure 3 shows the tensor slice working set size for each anchor point which describes the memory size accessed by the fused operation at the anchor point on a single core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' It also shows the formula to compute how many times the fused op is invoked within a single-core kernel and how many total tensor element memory accesses are needed for each anchor point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The concrete number can be deduced when the template is instantiated with the parameters for a Tunable OP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The fusion optimization uses a heuristic to decide which anchor to choose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The heuristic evaluates the cost of a single-cost kernel between all possible anchors and the option of not fusion, and then it chooses the one with the lowest estimated cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Many choices are straightforward so there is no need for over-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' For the template illustrated in Figure 3, anchor #4 is a good option for pre- Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Fused OP template with anchors and cost table Anchor Tensor slice’s working set size per core Access times per core Total memory access per core pre_op_anchor#1 A’ [MSN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' KSN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' MB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' KB] B’ [KSN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='NPSN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' NB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' KB] 1 MSN* MB * KSN * KB NPSN * NB * KSN * KB pre_op_anchor#2 A’ [MSN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' KSN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' MB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' KB] B’ [KSN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='NSN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' NB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' KB] 1 MSN* MB * KSN * KB NSN * NB * KSN * KB pre_op_anchor#3 A’ [KSN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' MB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' KB] B’ [KSN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' NSN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' NB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' KB] MSN MSN * MB * KSN * KB MSN * NSN * NB * KSN * KB pre_op_anchor#4 A’ [BS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' MB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' KB] B’ [BS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' NSN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' NB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' KB] MSN * KSN/BS MSN * MB * KSN * KB MSN * NSN * NB * KSN * KB pre_op_anchor#5 A’ [BS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' MB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' KB] B’ [BS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' KB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' NB] MSN * NSN KSN/ BS MSN * MB * KSN * KB *NSN MSN * NSN * NB * KSN * KB post_op_anchor#1 C[MB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' NSBN] MSN MSBN*NSBN post_op_anchor#2 C[MSBN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' NSBN] 1 MSBN*NSBN post_op_anchor#3 C[MSBN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' N] 1 MSBN * N Parallel loop mpi = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' MPN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' 1 { pre_op_anchor#1 : A[mpi*MSN:MSN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' 0:KSN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' 0:MB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' 0:KB];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' pre_op_anchor#1 : B[0:KSN, 0:NPSN, 0:NB, 0:KB];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Parallel loop npi = 0, NPN, 1 { pre_op_anchor#2 : A[mpi*MSN:MSN, 0:KSN, 0:MB, 0:KB];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' pre_op_anchor#2 : B[0:KSN,npi*NSN:NSN, 0:NB, 0:KB];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Loop msi = 0, MSN, 1 { mpsi = mpi * M/MPN + msi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' pre_op_anchor#3 : A[mpsi:1, 0:KSN, 0:MB, 0:KB];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' pre_op_anchor#3 : B[0:KSN,npi*NSN:NSN, 0:NB, 0:KB];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' C’[0:NSN, 0:MB, 0:NB] = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Loop ksi = 0, KSN, BS { pre_op_anchor#4 : A[mpsi:1, ksi:BS, 0:MB, 0:KB];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' pre_op_anchor#4 : B[ksi:BS, npi*NSN:NSN, 0:NB, 0:KB];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Loop nsi = 0, NSN, 1 { npsi = npi*N/NPN + nsi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' pre_op_anchor#5 : A[mpsi:1, ksi:BS, 0:MB, 0:KB];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' pre_op_anchor#5 : B[ksi:BS, npsi:1, 0:NB, 0:KB];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' A_addr[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='.BS] = &A[mpsi:1, ksi: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='.BS, 0:MB, 0:KB];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' B_addr[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='.BS] = &B[ksi:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='.BS, npsi:1, 0:NB, 0:KB];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' C’[nsi:1,0:MB, 0:NB] += Batch_reduce_gemm (A_addr[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='.BS], B_addr[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='.BS],Batch = BS);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' } } C[mpsi:1, npi:NSN, 0:MB, 0:NB] = C’[0:NSN, 0:MB, 0:NB];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' post_op_anchor#1 : C[mpsi:1, npi:NSN, 0:MB, 0:NB];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' } post_op_anchor#2 : C[mpi*MSN:MSN, npi*NSN:NSN, 0:MB, 0:NB];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' } post_op_anchor#3 : C[mpi*MSN:MSN, 0:NPSN, 0:MB, 0:NB];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' } The template has predefined anchors as placeholders to fuse pre-ops and post-ops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Each anchor point is associated with a tensor slice for the pre-ops and post-ops to work on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Once the blocking parameters are decided, the tensor slice size and access times can be deduced to support the fusion decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The fusion optimization pass chooses anchor points for groups of pre-ops and post-ops according to the estimated computation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' op processing input tensor A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Compared to anchor #5, the associated tensor slice is the same, but there is an additional “nsi” loop which causes redundant and same computation for the tensor slices under each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' However, anchor #5 is a good option for B, since the “nsi” loop reduces the associated tensor slice size to B[BS, NB, KB], compared to B[BS, NSN, NB, KB] from anchor #4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Although the total memory access for B is the same between anchor #4 and #5, the memory access cost for anchor #4 is smaller since the tensor slice is more likely located in the cache closer to the CPU core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The post-op fusion must be done after k-dimension reduction is done, or else the post-op computation interferes with the reduction and produces incorrect results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' So, the first post-op anchor in Figure 3 is not in the innermost loop until the “ksi” loop is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Among the three post-op anchors, anchor #1 is the most profitable choice, as it has the smallest tensor slice so likely the data is still “hot” in the cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' For reduction post-op, it may split into two anchor points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' It first inserts the first half of reduction in anchor #1 , which reduces to a partial result into a temporary tensor, and then inserts the second half at either anchor #2 or #3 which reduces to the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' If the reduction is along “n” dimension, it usually chooses anchor #3 since at this point there is no need to perform synchronization across multiple cores for the final reduction as the value for the “n” dimension is all computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Figure 4 shows a pseudo-code for fusing reorder and ReLU (rectified linear unit) ops to an instantiated GEMM OP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The first reorder op is inserted as pre-op fusion at anchor #4, which converts from a plain layout tensor A to a blocked layout A’ with blocking factors MB and KB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The fused reorder op works on the tensor slice of A’, denoted as A’[mpsi:1, ksi:BS, 0:MB, 0:KB], which starts from the position A’[mpsi, ksi, 0, 0] and has a slice with the size of [BS, MB, KB].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' By fusing the reorder at anchor #4, the single-core kernel handles the small amount of data in the cache and then immediately feeds the data to the micro-kernel as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The example code in Figure 4 fuses two post-ops, a ReLU op followed by a reorder op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Both operations are inserted at the post-op anchor #1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Instead of saving to output tensors, it saves to a temporary tensor, C’’, and applies the ReLU computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Similarly, the ReLU op outputs a temporary tensor C’’’, which is reordered to the result tensor C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The reorder op changes the memory layout of the C tensor from the blocking factor of MB and NB to MB2 and NB2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' These temporary tensors introduced by the post-op fusion are non-essential and will be reduced to minimum size at the Tensor IR optimization stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Graph IR Optimization The Tunable OP template provides the foundation for automating the process of building high-performance compute-intensive primitives and fusing neighbor memory-bound operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' It serves as the core functionality of oneDNN Graph Compiler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' On top of that, oneDNN Graph Compiler also exploits optimization opportunities only available when the computation is offloaded as a computation graph instead of individual operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The Graph IR is first decomposed into a graph of basic DNN operations, applied a number of optimizations, fused to a number of Fused OPs, and then lowered to Tensor IR using the templates for Tunable OPs and Fused OPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' This section introduces a few graph-level optimizations specific to the deep learning domain: low- precision conversion, constant weight preprocess, layout propagation, and fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' 5 illustrates these important optimizations with a quantized multilayer perceptron (MLP) example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The low-precision computation brings significant speedup as it reduces both the computation and memory bandwidth required to compute a deep learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The low-level precision computation graph preserves the compute-intensive operations in the FP32 data type with surrounding type conversion operations inserted by quantization tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Low-precision conversion transforms the input DNN computation graph and converts the compute-intensive operation to low-precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Pseudo code for Fused OP Parallel loop mp i= 0, MPN, 1 { Parallel loop npi = 0, NPN, 1 { Loop msi = 0, MSN, 1 { mpsi = mpi * M/MPN + ms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' C’[0:NSN, 0:MB, 0:NB] = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Loop ksi = 0, KSN, BS { Reorder(A, [1, 1], A’[mpsi:1, ksi:BS, 0:MB, 0:KB], [MB, KB], from=[mpsi, ksi]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Loop nsi = 0, NSN, 1 { npsi = npi * N/NPN + nsi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' A’_addr[0:BS] = &A’[mpsi, ksi:BS, 0, 0];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' B_addr[0:BS] = &B[ksi:BS, npsi, 0, 0];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' C’[nsi:1,0:MB, 0:NB] += Batch_reduce_gemm (A’_addr[0:BS], B_addr[0:BS],Batch = BS);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' } } C’’[mpsi:1, npsi:NSN, 0:MB, 0:NB] = C’[0:NSN, 0:MB, 0:NB];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' C’’’[mpsi:1, npi:NSN, 0:MB, 0:NB]) = Relu(C’’[mpsi:1, npi:NSN, 0:MB, 0:NB]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Reorder( C’’’[mpsi:1, npi:NSN, 0:MB, 0:NB], [MB, NB], C, [MB2, NB2], to=[mpsi, npi]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' } } } Figure 5 starts with the input quantized DNN graph, which contains an FP32 matmul op surrounded by two dequantize ops and a quantize op, denoted by DQ and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The dequantize converts an Int8 data type tensor to FP32 and the quantize op does the reverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The green box indicates a constant tensor and the orange box regular tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The fine line indicates scalar, the middle line indicates Int8 tensor, and the thick line indicates FP32 tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The example shows an asymmetric dynamic quantization case, so the first dequantize op scales A input tensor by a_s and then offset by a_z to adjust the zero point, and the other dequantize op just scales the B matrix with b_s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' B matrix is the weight matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Low-precision conversion optimization first breakdown the quantize and dequantize op to be simple addition and multiply ops and transform the graph to be a mathematically equivalent form that uses Int8 matmul op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The transformation can be illustrated by the following mathematic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Matrix multiplication is denoted by “X”, and elementwise multiply and addition are denoted by “*” and “+”, the broadcast and type conversion needed for Uint8 or Int8 data type processing are omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The transformed equation looks more complex, but it lowers the matmul precision from FP32 to Int8, which is the main goal of the optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' C[m,n] = Quantize (Dequantize (A[m, k], a_s, a_z) Xfp32 Dequantize (B[k, n], b_s), c_s, c_z) => C[m,n] = (A[m, k] Xint8 B[k, n] * fp32 (a_s *fp32 b_s) + fp32 (a_z[m,k] Xfp32 B[k, n] * b_s) ) * fp32c_s + int64 c_z The const weight preprocessing optimization is to exploit the optimization opportunity that some of the input tensors are constant at the execution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' For the static quantization inference use case, the weight tensors and quantization parameters are constant, so a portion of the post-transformation equation contains computation over constant weight, scale, and zero point can be avoided completely at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The challenge is that the weight data buffer might not be available during the compilation, so the compiled code needs to preprocess the constant weight at the execution time when it first arrives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' As a_s, b_s, c_s, and c_z are constants passed as dequantize op’s attribute, these constants can be folded in the compile- time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The equation above can be further transformed as Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Graph IR optimization passes Low-Precision Conversion Const Weight Preprocess Layout Propogation Fusion DQ matmul f32 DQ A[m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' k] a_s a_z B[k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='n] b_s C[m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' n] matmul I8 + matmul I8 A[m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' k] a_s a_z B[k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='n] b_s B[k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='n] b_s bcast a_z[m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='k] Q c_s c_z + c_s c_z C[m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' n] matmul I8 + matmul I8 A[m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' k] a_s a_z B[k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='n] b_s B[k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='n] b_s bcast a_z[m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='k] + c_s c_z C[m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' n] CW: Compensated Weight Subgraph( matmul,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' *,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' *) A[m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' k] B[k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='n] CW[n] Subgraph( matmul,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' *,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' *) C2[n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' n2] B2[n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' n2] CW2[n2] C[m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' n] Subgraph( matmul,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' *,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' +,' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' KB] B[k’,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' n’ NB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' KB] C[m’,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' n’,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' MB2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' NB2] B2[n’,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' n2’,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' NB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' N2B] A[m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' k] CW[n] C2[m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' n2] CW2[n2] RO RO RO B[k’,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' n’ NB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' KB] C[m’,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' n’,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' MB2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' NB2] B2[n’,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' n2’,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' NB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' N2B] matmul 18matmu 18as matmu 18a bs matmu 18CZa matmul 18CZmatmu 18 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' 5 also shows the Graph IR after the constant weight preprocess optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' if (init) const_weight_comp = a_z[m,k] Xfp32 B[k, n] * fp32 b_s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' C[m,n] = (A[m, k] Xint8 B[k, n] * fp32 a_s *fp32 b_s + fp32 const_weight_comp) * fp32c_s + int64 c_z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The constant weight preprocess optimization recognizes the constant tensor and builds a special initial function that preprocesses the constant weight and reuses the processed weight at the runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' It recognizes the weight matrix B is passed as a constant logical tensor in the input graph, meaning that the weight buffer holds a constant value and won’t change since the first execution of the compiled partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' This constant is named a runtime constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' If a DNN op’s inputs are runtime constant or compile-time constant, the output tensor is runtime constant as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The optimization propagates and marks all the runtime constants throughout the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Later the lowering generates special code for runtime constants, to make sure these runtime constants only be executed once in the first execution, and all future execution will reuse the processed result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The layout propagation optimization exploits extra performance benefits with a sequence of Tunable OP by allowing Tunable OP to use the most desired block layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' As Tunable OP relies on the blocked layout to achieve the best performance on the CPU, very often the best-performed block layout might be different between two Tunable ops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' It allows the Tunable ops within a subgraph to use a blocked layout but keep the graph input/output tensor as a plain layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' It first inserts reorder operations at the graph boundary to ensure the entry and exit points using the plain layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Then it iterates the DNN computation graph and inserts reorder operation between two Tunable OPs if they use different blocked layouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Before inserting a reorder OP before the Tunable OP, it first queries the Tunable OP for its desired blocked layouts, if none of the desired blocked layouts is consistent with the current layout, it inserts a reorder layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The inserted reorder OPs are fused to the previous Tunable OP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The inserted layout reordering for the input weights produces runtime constants in the inference scenario, and they are handled by constant weight preprocessing like the compensated weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The Fusion optimization supports both fine-grain fusion and coarse-grain fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The fine-grain fusion inserts Fusible ops to anchor points of Tunable op and forms one fused op, and coarse-grain fusion tries to merge multiple fused ops together to generate optimized code for an even larger group of operations together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Both pre-op fusion and post-op fusion help improve the memory locality since the Fusible OPs work on tensor slices instead of tensors after being fused into anchor points of Tunable OP, and tensor slices have a much higher possibility in the memory system closer to the compute function unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' For a Fusible op between two Tunable ops, it is more desirable to fuse it to the previous Tunable operation as post-op since the overall cost would be lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The pre-op fusion only supports limited cases like reorder and transpose operations and only be used at the entry point of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' There are typically multiple Fusible operations that need to be fused as post-op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' For example, matmul ops are usually followed by bias, activation, or normalization ops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' These OPs are first broken down to a sequence of Fusible op, like elementwise, broadcast, reorder, and reduction ops, and then fused into Tunable OP as groups of post-ops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The fine-grain fusion optimization grows a sequence of post-ops using a simple heuristic to decide whether the fusion is profitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' It first considers the immediate succeeding operations of the Tunable op as post-op candidates and keeps growing the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The heuristic simply sets a limit of operations so stop growing the sequence when the limit is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' For example, the post-op sequence can only have one reorder and one reduction op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' As the post-op may involve accessing additional memory, like the second operand tensor for a binary op, the heuristic fusion optimization also monitors the total additional memory being accessed and limit the potential negative impact on the Tunable op execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Then, the fine-grain fusion optimization sorts the post-ops in topological order and splits them into two groups if the post-ops contain one reduction op: the post-ops not dependent on the reduction op, and the reduction op and its dependent ops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The first group is inserted into the same post-op anchor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The reduction op may be split into two anchor points, and its dependent ops are inserted at the second anchor point where the final reduction result is collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The coarse-grain fusion optimization further merges fused ops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The larger scope of the DNN graph opens many possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Multiple Fused ops could be lowered to one parallel loop, in order to improve data locality or better exploit the parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' For example, the outermost “mpi” loop of two fused ops may have the same blocking factor, so that they can be merged as one loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' When the heuristic chooses the parameters for each Tunable op, it tries to choose the outermost loop blocking factor best aligned with core numbers, so each instantiated fused op has the same blocking factors as its neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' When the coarse-grain fusion optimization decides to merge two fused ops, it marks the two nested loops in Tensor IR as “mergeable” during the lowering process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Then Tensor IR merges two nested loops mechanically as guided by the Graph IR optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Tensor IR optimization Tensor IR is the lowest intermediate representation in the oneDNN Graph Compiler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' At the Tensor IR level, the DNN computation graph is lowered to a C-like program, which includes function, statement, expression, and intrinsic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The Fused OP is lowered as a function, which contains nested loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' A complex statement describes a structure like a loop, and a simple statement does computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Var and Tensor represent scalar variables and multi-dimension arrays respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Tensor IR supports Graph IR optimization by merging loops as instructed by Graph IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Figure 6 shows the example of Tensor IR for the pseudo-code in figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' In the Tensor IR, the computation on the tensor slices is represented by either a nested loop or a function call to the microkernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The inserted pre-op and post-op are lowered to nested loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The two post-op, ReLU and reorder ops, are merged as one nested loop using the hint passed by Graph IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The main optimizations on Tensor IR are tensor size optimization and memory buffer optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Tensor size optimization tries to reduce the tensor size of each temporary tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The temporary tensors are introduced in the pre-op and post-op fusion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The temporary tensor was initially introduced as a full-size tensor in the lowering process and then reduced by the tensor size optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' In the example code of figure 6, the post- ops are fused into one loop nest in the Tensor IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Since the accesses of the temporary tensors, C’’ and C’’’, are local to the innermost loop body, the temporary tensor could be replaced by a scalar variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Compared to accessing global and larger original tensors, the result code has a much smaller memory footprint and better cache locality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The temporary tensor introduced by pre-op fusion can be reduced similarly by analyzing the scope of the tensor usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' For example, A’[MSN, BS, MB, KB] could be reduced to A’ [BS, NB, KB], since the producer of A’ and consume are within the “msi” loop, so there is no need to save the result along the 2nd dimension of A’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' After tensor size optimization, the multiple-dimension tensor representation is flattened to a one-dimensional array to represent the memory buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The memory buffer optimization tries to reuse the memory buffer of temporary tensors to have a minimum overall temporary buffer size for the compiled code and tries to improve the locality of the temporary buffer use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Lowering to Tensor IR Var Const MPN, NPN, MSN, NSN, BS, KSN, MB, NB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Tensor FP32[M, K] A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=" Tensor FP32[M/MB, K/KB, MB, KB] A';" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Tensor FP32[K/KB, N/NB, NB, KB] B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=" Tensor FP32[NSN, MB, NB] C';" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=" Tensor FP32[M/MB,N/NB, MB, NB] C'', C''';" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Tensor FP32[M/MB2, N/NB2, MB2, NB2] C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Var int* A_addr[BS], B_addr[BS];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Var Index mp, np, ms, ks, ns, nps, mps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Parallel loop mp i= 0, MPN, 1 { Parallel loop npi = 0, NPN, 1 { Loop msi = 0, MSN, 1 { mpsi = mpi * M/MPN + msi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Loop nsi = 0, NSN, 1 { Loop mbi = 0, MB, 1 { Loop nbi = 0, NB, 1 { C’[0:NSN, 0:MB, 0:NB] = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' } } } Loop ksi = 0, KSN, BS { Loop bsi = 0, BS, 1 { ksbi = ksi * BS + bsi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Loop mbi = 0, NB, 1 { Loop kbi = 0, KB, 1 { A’[mpsi, ksbi, mbi, kbi] = A[mpsi*MB + mbi, ksbi*KB+kbi];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' } } } Loop nsi = 0, NSN, 1 { npsi = npi * N/NPN + nsi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Loop bsi = 0, BS, 1 { A’_addr[bsi] = &A’[mpsi, ksi, 0, 0];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' B_addr[bsi] = &B [ksi, npsi, 0, 0];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=" } C'_addr = &C’[nsi,0, 0] ;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=" Batch_reduce_gemm(A’_addr, B_addr, C'_addr, MB, NB, KB, Batch = BS);" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' } } Loop nsi = 0, NSN, 1 { npsi = npi * N/NPN + nsi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Loop mbi = 0, MB, 1 { Loop nbi = 0, NB, 1 { C’’[mpsi, npsi, mbi, nbi] = C’[nsi, mbi, nbi];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' C’’’[mpsi, npsi, mbi, nbi]= max(C’’[mpsi, npsi, mbi, nbi], 0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' C[(mpsi*MB+mbi)/MB2, (npsi*NB+nbi)/NB2, (mpsi*MB+mbi)%MB2, (npsi* NB+nbi)%NB2] = C’’’[mpsi, npsi, mbi, nbi];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' } } } } } } The main target of memory buffer optimization is to reuse the memory buffer created for the temporary tensors between fused op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' In the inference use case, the output tensor is only consumed by the next fused op, and so the buffer could be reclaimed once the next fused op completed execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Since the input tensor size is known to the compilation process, the memory buffer usage can be tracked at the compile time and optimized to improve efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Memory buffer optimization uses life span analysis like traditional compiler analysis for register allocation based on the def-use chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Memory buffer optimization has extra consideration for buffer reuse since the memory buffer size used is not uniform as registers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' It scans the Tensor IR, tracks all the memory buffers alive, and then computes the peak size of the total memory buffers needed for the entire compiled graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The algorithm considers both reusing the hot memory and reducing the overall peak memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' At each point, when an intermediate buffer is needed, it tries to reuse the free intermediate buffers, which are already allocated but not used anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Among multiple choices of reusable memory buffers, it chooses the one that was used most recently, so likely the data is still in the cache system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Experimental results oneDNN Graph Compiler targets performance-critical DNN computation graph, which is usually a subgraph of the whole DNN model graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' We selected two DNN computation subgraphs as target workloads to evaluate the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The Multilayer Perceptron (MLP) workload contains multiple matmul ops intermixed with activation ops like ReLU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The MLP subgraph is the basic building block for many deep learning models, including recommendation systems and natural language processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The Multi-Head Attention (MHA) subgraph is the key element to Transformer based deep learning models like Bert for natural language processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The MHA workload focuses on the scaled dot-product attention portion of the MHA graph, which contains two batch matmul ops and a softmax as well as other binary ops between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Depending on the use case, the MLP and MHA subgraphs tend to account for more than half of total model execution time, especially for DLRM or Bert Large models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' We choose the inference use case for the evaluation and measure the performance for both FP32 and Int8 data types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' We choose several representative data shapes for weights and input tensors and select a wide range of batch sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The weight sizes for MLP are from the MLPerf Workload Category data type input batch size sequence length hidden size head numbers MLP_1 Int8, FP32 32, 64, 128, 256, 512 N/A 13x512x256x128 N/A MLP_2 Int8, FP32 32, 64, 128, 256, 512 N/A 479x1024x1024x512x256x1 N/A MHA_1 Int8, FP32 32, 64, 128 128 768 8 MHA_2 Int8, FP32 32, 64, 128 128 768 12 MHA_3 Int8, FP32 32, 64, 128 384 1024 8 MHA_4 Int8, FP32 32, 64, 128 512 1024 16 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Performance comparison for individual Matmul op Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Workload parameters 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='4 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' 256,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' 1 Matmul kernel performace related to expert-tuned implementation Baseline (FP32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Int8) Compiled Kernel (FP32) Compiled Kernel (Int8) speedup m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='n DLRM model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' and the sequence length and hidden size choices for MHA are from the Bert models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The performance data is collected on an Intel® Xeon® Platinum 8358 processor with 32 cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' We first study the fine-grain fusion performance for individual layers since this is the foundation of the oneDNN Graph Compiler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The tests evaluate all the problem sizes used in the MLP tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The baseline is heavily optimized and uses oneDNN primitives, which is the industry standard best-performant expert-tuned implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Both the baseline and oneDNN Graph Compiler assume weight being pre-packed, compensated, and preprocessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The input and output matrixes are in plain layouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' shows that the performance of oneDNN Graph Compiler’s automated kernel generated using the template approach is 6% better than the expert- tuned primitives for the given test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' oneDNN Graph Compiler outperforms the expert-tuned primitives in many smaller problem sizes and falls behind on certain cases, particularly with k=479.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The performance of individual layers heavily depends on the algorithm and heuristic, and we expect the performance gap on specific cases would be narrowed as oneDNN Graph Compiler continues to develop the algorithm and heuristic, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Figure 8 shows performance comparisons for MLP and MHA tests between oneDNN Graph Compiler and oneDNN Primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The left side shows performance data for FP32 data type and the right side is Int8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Each test is named with workload category, batch size, and data type as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' For each test, we measure the performance of the baseline, oneDNN Graph Compiler, and the middle setting which disables the coarse-grain fusion and evaluates the rest optimizations including the fine-grain fusion for oneDNN Graph Compiler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The baseline uses expert-tuned oneDNN primitive with fusion support and has been integrated into multiple DL frameworks to accelerate deep learning on the CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Specifically, it uses oneDNN primitives post-op fusion to fuse matmul op with ReLU for the MLP tests and with division and addition ops for the MHA tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' For Int8 tests, before calling the oneDNN primitives, the baseline applies similar low-precision graph transformation and maps the graph to low-precision matmul and post-ops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Besides that, the baseline also performs weight pre- packing, compensated weight preprocessing, and caches the result to avoid re-computation at runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The performance results show oneDNN Graph Compiler significantly improves the performance of the target DNN computation subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' For MLP tests, oneDNN Graph Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' oneDNN Graph Compiler performance evaluation for MLP and MHA subgraph 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='00 MHA performance evalution FP32 & int8 inference on 32-core CPU Primtives + post-ops Compiler without coarse-grain fusion Compiler 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='50 MLP performance evalution FP32 & int8 inference on 32-core CPU Primtives + post-ops Compiler without coarse-grain fusion Compiler speedup speedup Compiler demonstrates an average of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='73x speed up on Int8 data type and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='22x speed up on FP32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The five MLP_1 tests for Int8 data type show the highest speedup at an average of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='72x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Among a total of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='72x speed up, coarse-grain fusion is the main contributor and accounts for 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='95x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' It merges 3 parallel loops, lowered from 3 matmul ops, into one parallel loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The coarse-grain fusion greatly reduces the synchronization overhead and permits the activation data to be in the fastest cache for the next matmul op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' For the MLP_1 Int8 tests, the entire activation and weight tensor fit in the L2 cache, so the coarse-grain fusion performs very well for these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' When disabling the coarse-grain fusion, the remaining optimization accounts for about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='4x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' There are mainly three reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' First, although the baseline implements the same fusion, oneDNN Graph Compiler has better performance for each individual matmul op in MLP_1, as shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Second, the layout propagation allows all three matmul ops to run with the same blocked layout without extra reordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Last, due to the MLP_1 tests being relatively short, the total API call overhead takes up to 10% of the execution time for the baselines, which is reduced by about 3 times since the compiled code needs only to be called one time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Compared to MLP_1 Int8 tests, the MLP_1 FP32 tests show an overall 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='47x performance gain, with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='15x from coarse-grain fusion and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='28x from rest optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' For MLP_2 test cases, the oneDNN Graph Compiler shows an overall 10% better performance Int8 data type and 1% on FP32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' When the coarse-grain fusion is disabled, oneDNN Graph Compiler is 1% slower compared to the baseline with FP32 and Int8 test cases combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' As we learned from the individual matmul op analysis, the initial layer of MLP_2 (k=479) has lower performance for oneDNN Graph Compiler and negatively impacts the overall performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' MLP_2 tests also get benefit from the coarse-grain fusion but to a lesser extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The coarse-grain fusion is not able to merge all the loop nests due to the current heuristic limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' We believe that the performance for MLP_2 test cases can be further improved with more heuristics tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' For the MHA subgraph, oneDNN Graph Compiler demonstrates an overall 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='91x performance gain over 24 MHA tests with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='99x on Int8 data type and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='84x on FP32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' In contrast to the MLP performance, the performance benefit is more significant for the tests with larger problem sizes, and fine-grain fusion helps more than coarse-grain fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The baseline doesn’t have the same fusion capability as the fine-grain fusion, as oneDNN post-op baseline doesn’t fuse the softmax op with the preceding batch matmul op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' oneDNN Graph Compiler decomposes softmax op to multiple basic operations, and its fine-grain fusion optimization fuses them to the preceding batch matmul ops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The basic operations are divided into two groups of post-ops: a group of element-wise ops and a group led by a reduction op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' These two groups are inserted into the batch matmul op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' This gives a significant boost to the performance by an average of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='51x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The coarse-grain fusion adds another 27% performance gain on top of fine-grain fusion by merging the two nested loops translated from two batch matmul ops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Conclusion We proposed a hybrid approach to address the unique challenges of deep learning compilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' It distilled key ingredients of expert-tuned primitives for compute- intensive DNN operations like matrix multiplication and uses compiler techniques on the DNN computation graph to fully exploit the performance opportunity at the graph level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The template uses an expert-developed microkernel, algorithm, and heuristic, to ensure compiler-generated code achieves comparable performance to expert-tuned primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' The compiler uses two-level intermediate representations at the level of both DNN op graph and C program to support domain-specific optimizations needed for deep learning computation, including low-precision, constant weight, tensor memory layout, fine-grain fusion, coarse-grain fusion, and tensor memory buffer reuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Performance evaluation shows up to 2x performance gain for performance critical DNN computation graph in CPU inference usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Reference [1] oneDNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='com/oneapi-src/oneDNN [2] cuDNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' https://developer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='nvidia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='com/cudnn [3] Tensorflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='tensorflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='org/ [4] Pytorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' https://pytorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content='org/ [5] XLA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' : 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' pp 1–25 [12] Tze Meng Low, Francisco D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Igual, Tyler M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Smith, Enrique S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Quintana-Orti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Analytical Modeling Is Enough for High-Performance BLIS, ACM Transactions on Mathematical Software Volume 43, Issue 2, June 2017, Article No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' : 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' pp 1–18 [13] Tyler M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Smith, Robert van de Geijn, Mikhail Smelyanskiy, Jeff R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Hammond, and Field G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Van Zee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Anatomy of High-Performance Many-Threaded Matrix Multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' IPDPS , page 1049-1059.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' IEEE Computer Society, (2014) [14] Navdeep Katel, Vivek Khandelwal, Uday Bondhugula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' MLIR-based code generation for GPU tensor cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' CC 2022: 117-128 [15] Philippe Tillet, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Kung, David Cox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Triton: an intermediate language and compiler for tiled neural network computations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' MAPL 2019: Proceedings of the 3rd ACM SIGPLAN International Workshop on Machine Learning and Programming Languages,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' June 2019,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WNAzT4oBgHgl3EQfYPzf/content/2301.01333v1.pdf'} +page_content=' Pages 10–19 [16] Sanket Tavarageri,' metadata={'source': 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Symbolic Representation for Video Event-Relation +Prediction +Andrew Lu*, Xudong Lin*, Yulei Niu, Shih-Fu Chang +Columbia University +{ayl2148,xudong.lin,yn2338,sc250}@columbia.edu +Abstract +Understanding event relationships in videos requires a +model to understand the underlying structures of events (i.e. +the event type, the associated argument roles, and corre- +sponding entities) along with factual knowledge needed for +reasoning. Structural symbolic representation (SSR) based +methods directly take event types and associated argument +roles/entities as inputs to perform reasoning. +However, +the state-of-the-art video event-relation prediction system +shows the necessity of using continuous feature vectors from +input videos; existing methods based solely on SSR inputs +fail completely, event when given oracle event types and ar- +gument roles. In this paper, we conduct an extensive em- +pirical analysis to answer the following questions: 1) why +SSR-based method failed; 2) how to understand the eval- +uation setting of video event relation prediction properly; +3) how to uncover the potential of SSR-based methods. We +first identify the failure of previous SSR-based video event +prediction models to be caused by sub-optimal training set- +tings. Surprisingly, we find that a simple SSR-based model +with tuned hyperparameters can actually yield a 20% abso- +lute improvement in macro-accuracy over the state-of-the- +art model. Then through qualitative and quantitative analy- +sis, we show how evaluation that takes only video as inputs +is currently unfeasible, and the reliance on oracle event in- +formation to obtain an accurate evaluation. Based on these +findings, we propose to further contextualize the SSR-based +model to an Event-Sequence Model and equip it with more +factual knowledge through a simple yet effective way of re- +formulating external visual commonsense knowledge bases +into an event-relation prediction pretraining dataset. The +resultant new state-of-the-art model eventually establishes +a 25% Macro-accuracy performance boost. +*Equal contribution. +1. Introduction +Event understanding has been thoroughly explored in the +past decade [5,20,23,24,27,30,34,37,43,44] due to its great +importance in our daily lives. An event is typically repre- +sented as a verb (indicating the event type) and several argu- +ments, each of which has a role name and an associated en- +tity. Researchers have been devoted to extracting events and +labeling the argument roles [5,20,30,34,38,39,43] in vision, +language, and multimodal domains. These event extraction +and argument role labeling models build the foundation for +higher-level understanding of the relations between events. +Relations between events [1,2,9,28,36,42] have been thor- +oughly studied in the language domain. A specific relation +and event coreference has also been explored in the mul- +timodal setting [5, 20]. However, video event-relation pre- +diction is still a new and challenging task [34] which re- +quires both good representations of video events and com- +monsense knowledge to reason between events. +Structural symbolic representation [12,14,20,22,41,45] +has been widely adopted on various downstream tasks such +as visual question answering [45], image captioning [41], +and action recognition [14] for its high interpretability and +generalization ability [22, 45]. In this work, using struc- +tural symbolic representation, we refer to representations +consisting of discrete tokens organized with certain struc- +tures. When applied to event-relation prediction, it is usu- +ally formulated as an event type (verb) with associated ar- +gument role names and corresponding entities. This event- +argument structural symbolic representation has shown to +be very effective on predicting event relations in the lan- +guage domain [7]. However, the models using structural +symbolic representation for video events are easily biased +by the dominant relations in the VidSitu dataset [34] and +predicts the dominant class for all the classes, which con- +tradicts the success of existing structural symbolic repre- +sentation based methods on various tasks including event- +relation prediction using text. +To answer why structural symbolic representation based +models fails on video event-relation prediction, we first an- +arXiv:2301.03410v1 [cs.CV] 6 Jan 2023 + +alyze if there are any possible patterns that the model could +have leveraged to avoid being misled (Sec. 4.2). Surpris- +ingly, we find that even if the model only memorizes the +dominant relation for pairs of event types, the model is +clearly not supposed to predict only Enables (the most fre- +quent relation in the dataset). Based on this finding, we +bootstrap the failed symbolic representation based model +and the state-of-the-art model with two experiments: uti- +lizing a balanced training data/objective and tuning hyper- +parameters for better training the model. By only changing +the learning rate, we find that the failed symbolic represen- +tation based model has already outperformed the state-of- +the-art by 20%. The state-of-the-art model that heavily re- +lies on video features enjoys a much smaller gain through +the better training setting we identified. +The different behaviors motivate us to carefully analyze +the task and the dataset (Sec 4.3). We evaluate two state- +of-the-art video-language models, HERO [19] and Clip- +BERT [16], on video event-relation prediction. By control- +ling inputs with the help of strong image-text contrastive +models [31], we found that oracle information is needed +to accurately evaluate the models due to the presence of +multiple events co-occurring simultaneously in the same +video. Even with strong pretrained video-language mod- +els, the event-type and argument role descriptions are still +more important than the video feature vectors. +Based on these observations, we propose to contextu- +alize the simple pairwise SSR-based models to an Event- +Sequence model in order to leverage context information +within sequences of events for more accurate event-relation +prediction. Furthermore, we explore leveraging the external +visual knowledge base, VisualCOMET [29], to teach the +model commonsense knowledge about the evolution pro- +cess of events. We propose a simple yet effective approach +to reformulate the annotations of VisualCOMET into event +sequences suitable for even-relation prediction. The contex- +tualized sequence model with pretraining on VisualCOMET +establishes a new state-of-the-art, which has a margin of +25% improvement in terms of Macro-accuracy compared +to the best existing ones. +Our contributions can be summarized as follows: +• We identify why symbolic representation based models +fail on video event-relation prediction and provide a sim- +ple solution which yields a 20% improvement in Macro- +accuracy. +• We identify proper settings needed to evaluate video +event-relation prediction models on VidSitu through +quantitative and qualitative analysis of different model +variations and the dataset. +• We propose a contextualized Event-Sequence model, +coupled with a pretraining technique on VisualCOMET, +to fully utilize the rich contextual information in event se- +quences and commonsense knowledge from the existing +knowledge base, which establishes the new state-of-the- +art 34.2% → 59.2%. +2. Related Work +Visual Event Understanding aims to recognize, extract, +and structure the actions or activities happening in images +or videos. Previous studies simply represent visual events +as verbs or subject-verb-object triplets [4, 6, 10, 15, 21, 34]. +Recent works further study more structural and semantic +representations of visual events. For example, M2E2 [20] +and VideoM2E2 [5] handle extracting events from image- +text and video-text pairs, respectively. Importantly, the vi- +sual situation recognition task aims to identify not only the +activity in an image [30, 43] or video [34], but also the en- +tities including persons and objects (i.e., semantic roles) +associated with the activity. Another benchmark, Visual- +COMET [29], proposes to depict person-centric images as +a graph of commonsense descriptions, including before- +event, intent of people, and next-event. In this work, we +focus on the video event relation prediction task and con- +duct empirical studies to evaluate the roles of event type, +argument roles, and video features in relation prediction. +We also explore using VisualCOMET as an external visual +knowledge base for pretraining. +Structural Symbolic Representation denotes the rep- +resentation consisting of discrete tokens with certain struc- +tures. Structural symbolic representation has been applied +in various tasks in computer vision and natural language +processing. For example, the visual scene in visual ques- +tion answering can be modeled as the structural represen- +tations of objects with their associated attributes and lo- +cations [45]. As for event representation, spatio-temporal +scene graphs [14, 33] decompose each event as a temporal +sequence of spatial scene graphs, where each spatial scene +graph is a set of subject-predicate-object triplets. Further- +more, recent NLP studies show the potential of structural +symbolic representation in event-relation prediction in text, +e.g. part-of-speech (POS) and XML tags [7]. In this work, +we follow VidSitu to represent each video event as event +type/verb and its associated argument roles and entities. +Event-relation Prediction in both Text and Video. +Events in texts are often ordered by temporal relation [3], +causal relation [26], or narrative order [13]. Hong et al. [11] +define event relations in text as 5 main types, along with +21 sub-types, covering inheritance, expansion, contingency, +comparison, and temporality. Based on prior work in cross- +document event relations, VidSitu [34], the only available +video event relation dataset, defines four types of relations: +no relation, causality, enabled, and relation to. In this paper, +we focus on video event-relation prediction and use VidSitu + +Figure 1. The pipeline of the Event-Sequence model for event-relation prediction. Special characters * and ** to denote the target events +the moodel is required to predict relation between. Detailed illustration is in Sec. 3.2 +as a case study. We follow VidSitu for the definition of event +relations. +3. Technical Approach +In this section, we first introduce some preliminaries of +event-relation prediction, then we describe the variants of +the model designs. Later, we describe the training tech- +niques explored in this paper. +3.1. Preliminaries +Structural Symbolic Representation. Structural symbolic +representation generally refers to a representation where el- +ements are discrete tokens and have certain structures, e.g. +scene graphs [14]. In this paper, to represent an event ef- +fectively, the models need to know the event type (usu- +ally a verb), its associated argument roles, and the actual +entities for each argument role. +Therefore, we consider +the sequence of text tokens with the following structure +as the structural symbolic representation of an event x: +x = {v, a1, e1, a2, e2, ..., aM, eM}, where v ∈ W is the +event type/verb, am ∈ W, em ∈ W are the mth argument +role and associated entity, and M is the number of argument +roles for this event. Such a sequence is essentially a traverse +of the graph with v as the root node and am, em as the mth +edge and leaf node. +Event-Relation Prediction. The model is required to pre- +dict the relationship between two events, x1 and x2. Since +VidSitu [34] is the only dataset available for video event- +relation prediction, we adopt its specific setting. +Each +video event sequence consists of five consecutive events +{x1, x2, x3, x4, x5}, each of which is from a two-second +video segment yi ∈ RH×W ×3×F . H, W, F are the height, +width and number of frames of the video segment respec- +tively. In VidSitu, only the relationship between the center +event x3 and other events xi, i ̸= 3 are annotated. +Baseline Model. We follow the RoBERTa variant [34] to +build the baseline model. Based on the structural symbolic +representation, a model F : WL −→ RC takes the se- +quence of text tokens as input and predicts a distribution +over the C possible relationship classes, where L is the +length of the sequence consisting of symbolic text repre- +sentations of xi and x3. The model F is initialized with +pretrained weights from RoBERTa [25]. +Baseline + Video Features. The state-of-the-art [34] shows +that video features are more effective than directly using +symbolic representations. It takes both video features and +the text tokens as inputs G : WL × RD×F −→ RC to pre- +dict the distribution over the C classes. An off-the-shelf +video feature extractor H is used to extract continuous fea- +ture vectors from the video segment yi when the event hap- +pens. The feature vector is concatenated with the output + +dive +* +Argo +AScn +Event 1 +ArgM +man in +lake +wetsuit +downwards +Ev1 +breathe +Causes E +Argo +AScn +RoBERTa +Classifier +ArgM +Event 2 +man in +lake +wetsuit +aggressively +Ev3 +talk +** +Argo +ArgM +Event 3 +Arg1 +brunette +casually +girl +brunette +boy +Event 4 +Event 5 +RoBERTa +Classification +Source Video +Inputs +Output +Model +Head +Segmentsembedding from the text tokens before being fed into the +final classifier G. We denote it as Baseline + Video Features +in the following discussion. When not specified, the video +feature extractor is SlowFast [8,34]. +3.2. Contextualized Event-Sequence Model +When people perceive the visual world, rich contex- +tual information such as neighboring events, location of the +events, manner of the events, etc. are important in under- +standing the relationship between events. We also analyze +the event sequence in the dataset and indeed find patterns, as +presented in Sec. 4.2. Motivated by intuition and these find- +ings, we propose a contextualized Event-Sequence model +for event-relation prediction and explore various ways of +utilizing context. +Event-Sequence Model. Instead of only feeding the model +with two events between which to predict event relations, +we propose to exploit the rich contextual information in a +full sequence of all the five events (shown in Figure 1), +p = F(x1, x2, x3, x4, x5), +(1) +where p ∈ RC is the predicted distribution over the C +classes of relations. Note that to inform the model which +two events between which we want to predict the relation, +we add an extra special token “*” before each of them. +Event-Sequence Model + Video Features. Video features +could be also considered contextual information, as the con- +tinuous feature vectors obtained from pretrained video fea- +ture extractors may convey fine-grained information of the +actual visual scene. We follow the same method of video +feature integration as in [34], +p = G(x1, x2, x3, x4, x5, H(yi), H(y3)), +(2) +where the video features are fused with contextualized em- +beddings before the final classification layer. +Sequence-to-Sequence Model. +Inspired by recent ad- +vances of sequence-to-sequence modeling [18, 22, 32] on +both language and video domains, we also explore a variant +of directly generating the sequence of relationships given +the sequence of events as input, +p1,3, p2,3, p4,3, p5,3 = S(x1, x2, x3, x4, x5), +(3) +where pi,3 is the predicted distribution for the relationship +between the ith event and the middle event x3. Note that we +follow the common strategy of teacher-forcing [40] to han- +dle the conditional generation problem, which means during +training +pk,3 = S(x1, x2, x3, x4, x5, l1, ..., lk−1), +(4) +the ground-truth “historical” event-relations l1, ..., lk−1 are +used for predicting the next relation. During testing, we use +beam search to decode the actual event relation sentence. +This variant doesn’t directly use additional contextual in- +formation as it does not directly take any extra inputs. The +motivation behind this variant is leveraging conditional gen- +eration as a constraint to prevent the model from only pre- +dicting the dominant class as reported in [34]. +Auxiliary Arguments. +Previous state-of-the-art mod- +els [34] only use base arguments (arguments tied to the verb +through direct semantic relations) such the agent and the tar- +get. However, additional contextual information could be +clearly provided by extra argument roles that were not used +in the state-of-the-art model like AMnr, ADir and AScn +(manner, scene and direction). We append them after the +base arguments when using them as additional input to the +model. +3.3. Training +The standard training objective is to use the cross- +entropy loss to train the model for event-relation prediction, +min +θ +− log pl, +(5) +where θ is the parameter to be updated in the model, l is +the ground-truth index of the relation, and p is the predicted +relation type. +Balancing Data/Loss. In [34], the poor performance of the +structural symbolic representation model is attributed to the +possible imbalanced relation distribution, which leads the +model to only predict the dominant relation type. To tackle +this possible issue, we also explore two versions of solu- +tions: re-constructing a balanced dataset or using a balanced +loss. +For the balanced dataset, we aim at keeping the same +number of event pairs in each class by removing videos +containing multiple samples of dominant relations. After +this process, about 70% of the dataset is kept. +For the balanced loss, we adopt the commonly used +weighted cross entropy loss for optimization, +min +θ +−βl log pl, +(6) +where we set βl as the inverse of the proportion of this rela- +tion l in the training set. +VisualCOMET Pretraining. VisualCOMET [29] records +1.4 million commonsense inferences for current visual +events under three types of event relations: Before, Intent +and After, which corresponds to inferring the past, reason, +or future event. +We re-formulate the dataset for event- +relationship pretraining; we use the current event as the x3, +then we randomly sample from the three types of annotated +events to construct an event sequence {x1, x2, x3, x4, x5}. +The relation label could be automatically generated from +the type of annotations. Note that during random sampling, +to simulate a real event sequence, we restrict x1, x2 to be + +either Before or Intent events and restrict x4, x5 to be Intent +or After events. +4. Experiment Results and Discussion +4.1. Dataset and Evaluation Metric +For our main experiments, we use VidSitu [35], a +large-scale dataset containing 29,000 10-second video clips +where each video clip is divided into five 2-second seg- +ments and the most salient verbs and arguments are anno- +tated along with the most dominant event relation. We eval- +uate event-relation prediction by computing Top-1 accuracy +on predicted relations as well as Top-1 accuracy macro- +averaged across the four different relation classes: Causes, +Enables, Reaction To, and No Relation [35]. +We use VLEP [17] to evaluate the generalization of our +model on future event prediction, which is formulated as a +multiple-choice problem. We follow the official setting and +report accuracy on the validation set. +4.2. Why SSR-Based Methods Fail +Preliminary Analysis. +To understand why using struc- +tural symbolic representation fails on the VidSitu dataset, +we analyze possible patterns that the model could have +leveraged to outperform baselines that only predict the dom- +inant class. First, we check whether there are event-pairs +that have a dominant relation other than Enables, which is +the overall dominant relation in the dataset. As the two +examples in Figure 2 show, we found that 66% of event +type pairs satisfy this constraint, which indicates that if the +model simply memorizes the dominant event relation for +each event type pair, it should achieve a macro-accuracy +higher than 25% [34]. This evidence motivates us to fur- +ther explore different hyperparameters to check if the fail- +ure comes from the optimization side. +To understand the possible benefit we can obtain from a +sequence of events as inputs, we study the pattern between +the relation type and the relative distance from x3 to the +event. As shown in Figure 3, the distance distribution of No +Relation and the distribution of Causes are quite different: +No Relation has an almost symmetric distribution w.r.t. the +distance value and is significantly more frequent when the +distance is 2; however, Causes has a clear peak at distance +-1 and decays when the distance value increases, which is +consistent with the temporal order. This study verifies that +feeding the full sequence of events as inputs is beneficial +to the structural symbolic representation models that only +predict Enables. +Tuning Baseline Training Configurations. +This study +focuses on exploiting the potential training configuration +issues of the baseline model that were reported to fail to +predict meaningful event relations [34]. Since we observe +(a) Speak-Respond +(b) Walk-Walk +Figure 2. Histograms of the event relations for two event type +pairs. (Best viewed on a monitor when zoomed in.) +(a) No Relation +(b) Causes +Figure 3. Distribution over the relative distance for No Relation +and Causes. For example, x1 to x3 has a distance value of -2 and +a distance of 2. (Best viewed on a monitor when zoomed in.) +a moderate imbalance in the distribution of event relations +classes, we reevaluate the baseline model after training +on the same dataset class-balanced through undersampling. +This does not improve baseline model performance, which +still always predicts one relation class for all events scoring +25% validation accuracy. +Interestingly, we then discovered the reason behind the +lack of performance on state-of-the-art baseline models to +be a sub-optimally adjusted learning rate rather than lack of +patterns in the data. By simply adjusting the learning rate +from 1e-4 to 1e-5, performance improves dramatically, by +≈ 29% (Table 1). Similar adjustments on state-of-the-art + +speak.01-respond.01 +140 +120 +140 +uanba +40 +0- +SaSTED +Enables +ReBction To +NoRel +Event Relationwalk.01-walk.01 +400 +300 +140 +0 +SaSTED +Enables +ReBction To +NoRel +Event RelationModel +LR +Data +Val Macro Top 1 Acc(%) +Val Top 1 Acc(%) +Baseline +1e-4 +Original +25.00 +39.43 +Baseline +1e-4 +Balanced +25.00 +39.43 +Baseline + Video Features (SOTA) +1e-4 +Original +33.73 +43.71 +Baseline +1e-5 +Original +53.61 +58.92 +Baseline +1e-5 +Original + balanced loss +53.98 +60.47 +Baseline + Video Features +1e-5 +Original +44.08 +49.75 +Event-Sequence +1e-5 +Original + balanced loss +55.38 +61.59 +Table 1. Comparison between baseline model trained with original unbalanced data, baseline model trained with data balanced on event +relation class frequency through undersampling, and baseline model with adjusted learning rate trained on original data. SOTA is short for +state-of-the-art. +multimodal models using text annotations and video fea- +tures also yield a significant performance improvement. But +it is noteworthy that the improvement is much less signifi- +cant compared to our baseline model, which is built upon +structural symbolic representations of events. +4.3. Evaluation Challenges +Simultaneous Events in Videos. +One explanation as to +why adding visual features actually decreases performance +despite being rich in context by nature is the large amount of +noise contained within videos. In visual features of complex +and busy events, multiple salient verbs may emerge to dom- +inate a scene. In such a scenario, the specific verb chosen +will have a drastic influence on relevant argument labels of +entity co-references and subsequent relationships between +events. Figure 4 depicts such a scene where multiple rele- +vant verbs and their corresponding arguments emerge. +The presence of multiple salient events simultaneously +makes effectively evaluating video-only event relation pre- +diction very challenging. Since event relation annotations +are directly tied to annotated event types and argument +roles, many instances occur where predicted event relations +are accurate to the scene, but are evaluated as incorrect since +they differ from the event chosen by the annotator. We ex- +plore this further in Section 4.4 (Performance Analyses Be- +tween Predicted and Human Annotations). +Quantify the Effect of Simultaneous Events. +We evalu- +ate two state-of-the-art video-language models, HERO [19] +and ClipBERT [16], on video event-relation prediction. We +leverage pretrained CLIP-VIT-32 [31] to perform frame se- +lection and region selection given event and argument role +as the textual input, respectively. The details of frames and +region selection are provided in the supplementary mate- +rial. In both models, as shown in Table 2, we observe a +performance increase when frame selection and region se- +lection are utilized which indicates a significant presence of +irrelevant frames in one video segment and also shows the +negative effect cause by simultaneous events. +Model +Frame-level +Region-level +Val Macro Acc(%) +HERO [19] + + +42.15 +HERO + + +48.03 +HERO + + +48.48 +ClipBERT [16] + + +47.62 +ClipBERT + + +49.76 +ClipBERT + + +50.52 +Table 2. Performance of HERO and ClipBERT when with and +without Frame Selection or Region Selection based on CLIP. +4.4. Improving SSR-Based Methods +Adding Additional Contextual Information. +When pre- +dicting the relationship between two events within a five +event sequence from VidSitu, baseline models only take the +two target events as inputs. A simple approach to increasing +the richness of context information contained within inputs +is to increase the temporal duration of the inputs. By giving +the model all five events within a sequence rather than just +the two target events, we are able to leverage patterns seen +in distances between events as well as events preceding and +succeeding the target events, resulting in a minor accuracy +improvement (53.85% → 55.38%) (Table 3). +Previous state-of-the-art baselines also receive inputs +containing only verbs and base arguments (arguments tied +to the verb through direct semantic relations). Providing +specific context in the form of auxiliary arguments describ- +ing the scene of the event, mannerisms and goals of enti- +ties performing actions, etc., yields further performance in- +creases (55.38% → 58.60%) (Table 3). When evaluating +event-sequence input models given only verbs and only base +arguments, performance is still substantially better than ran- +dom guessing, showing that verbs alone without entity co- +references and vice versa provide useful information for +event-relation prediction. +Surprisingly, adding context through video features de- +grades performance rather than showing further improve- +ment. We see a sizable decrease in performance in both +the learning rate adjusted baseline (53.58% → 44.08%) +and event-sequence input models (58.60% → 55.64%) (Ta- + +Figure 4. Example scene containing multiple salient events. A large explosion dominates the background while a man is running away +from the explosion in the foreground. The verb and argument labels are highly dependent on which event is selected. +Model +Verbs +Base Args +Aux Args +Val Macro Acc(%) +Baseline (1e-5 lr) + + + +53.85 +Baseline + Video Features (1e-5 lr) + + + +44.08 +Event-Sequence Only Verb + + + +42.53 +Event-Sequence Only Args + + + +34.73 +Event-Sequence Baseline + + + +55.38 +Event-Sequence All Args + + + +58.60 +Event-Sequence + vid features (SlowFast) + + + +55.64 +Event-Sequence + vid features (CLIP) + + + +45.98 +HERO [19] + + + +42.15 +ClipBERT [16] + + + +47.62 +Table 3. Comparison between model given inputs with various forms of context. +ble 3). We also observe a decrease in performance when +switching to a CLIP-based feature extractor on video inputs +instead of SlowFast. In addition, we provide comparisons +with more powerful vision-language cross-modal baselines +such as HERO [19] and ClipBERT [16] that have recently +shown promise on other video event understanding tasks. +Interestingly, neither HERO nor ClipBERT performs even +close compared to our Event-Sequence model, which veri- +fies its strong effectiveness. +Additional Model Constraints. +We found that adding +further constraints to event prediction models by employ- +ing Sequence-to-Sequence models using BART [18] did +not improve performance beyond our best event-sequence +RoBERTa models (Table 4). However, pretraining on Vi- +sualCOMET [29] yielded a further improvement, showing +that commonsense knowledge of event evolution in Visu- +alCOMET could further help our contextualized sequence +model to better reason the event relation. +Performance Analyses Between Predicted and Human +Annotations. +To demonstrate an end-to-end approach, we +train and evaluate our event-sequence model using pre- +dicted verb and argument roles generated by employing +a joint feature extractor and text encoder on the video +frames [34]. +Results comparing the models with oracle +information, predicted events, and previous state-of-the- +art video only baselines [34] are summarized in Table 5. +Notably, our event-sequence model using predicted events +outperforms previous video encoder baselines showing the +promise of structural symbolic representations even when +they are generated. We observe a significant improvement +when annotated verbs are used to generate argument roles, +which verifies the effectiveness of structural symbolic rep- +resentation and indicates the “noisy” nature of the task and +dataset. +Specifically, many instances occur where predicted verb +and arguments describe a different event within a scene +than human annotations, on which ground truth event re- +lation labels are based. In the specific example shown in +Figure 5, both ground-truth and predicted events accurately +describe the same event, the two men fighting, in Event 3. + +O-2 Seconds +Description1 +Description 2 +Verb +run (walk quickly) +Verb +explode (go boom) +Argo (runner) +man in grey outfit +Arg1 (thing exploding) +castle gate +ArgM (direction) +away from explosion +ArgM (location) +castleModel +Val Macro Top 1 Acc(%) +Val Top 1 Acc(%) +RoBERTa Event-Sequence Baseline +55.38 +61.59 +BART Seq-to-Seq +52.91 +57.43 +RoBERTa Event-Sequence + All Args +58.60 +62.01 +RoBERTa Event-Sequence + VisCom pretraining + All Args +59.21 +62.65 +Table 4. Comparison between RoBERTa model given event-sequence inputs, BART Seq-to-Seq, and RoBERTa Event-Sequence model +pretrained on VisualCOMET data. +Model +Val Macro Acc(%) +Test Macro Acc(%) +Event-Sequence + Annotated verb + Annotated args +55.38 +54.47 +Event-Sequence + Annotated verb + Predicted args +43.30 +42.75 +Event-Sequence + Predicted verb + Predicted args +35.46 +34.94 +Vid features only (SlowFast) +33.78 +30.54 +Table 5. Comparison between event-sequence baseline trained and evaluated on human annotations versus predicted events. +2 Seconds +Event +3 +4s-6s +Event +5 +8s-10s +Ground Truth Annotation +Verb +watch (look at) +Arg0 (observer) +man with long hair +Arg1 (entity observed) +battle +Predicted Verb and Args +Verb +fire (shoot) +Arg0 (shooter) +man in white +ArgM (direction) +towards man in black +Ground Truth Annotation +Verb +shoot (propel projectile) +Arg0 (shooter) +man in beige robe +Arg1 (projectile) +lightning +Predicted Verb and Args +Verb +strike (attack) +Arg0 (attacker) +man in white +Arg1 (target) +man in black +Ev5 is a reaction +to Ev3 +Ev3 enables Ev5 +Figure 5. Example where predicted events differ from ground truth human annotations, but predicted verbs and arguments are salient since +they describe different events within the same scene. The resulting event-relation prediction using predicted events is also ”accurate” but +different from the ground truth relation since they are relating different, equally salient event descriptions. +In Event 5, human annotations describe the man in the fore- +ground watching the battle in the background, which per the +ground truth label is a Reaction To the fighting in Event 3. +However, predicted verb and arguments describe the back- +ground battle in Event 5 and subsequently classifies Event 3 +as Enabling Event 5 since the battle is a continuation of the +previous fighting. In this case, Enables is also an accurate +relationship descriptor but considered incorrect. Such dis- +crepancies between annotations likely explain some of the +performance degradation seen when using predicted events. +4.5. Applications in Downstream Tasks +We adopt the RoBERTa-based model from [17] as the +baseline, which achieves 65.8% accuracy on the VLEP val- +idation. It is a challenging task, as prertaining RoBERTa on +ATOMIC [36] only improved accuracy by 0.5%. However, +when using our event-sequence model as the pretrained +weights for the RoBERTa model, we observe a 0.9% im- +provement of accuracy, which makes video event-relation +prediction a better knowledge source of pretraining than +the comprehensive textual knowledge base, ATOMIC. This +again verifies the effectiveness and the generalization of the +proposed contextualized sequence model. +5. Conclusion +In this paper, we defend the effectiveness of structural +symbolic representations for video event-relation prediction +and we identify sub-optimal training configurations as the +key reason previous models fail. We further provide in- +depth analysis and find that video features are noisy because +of simultaneous events and irrelevant background. It is nec- +essary to use oracle event information as input to ensure +proper evaluation. 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Advances in neural information processing sys- +tems, 31, 2018. 1, 2 + diff --git a/ZNE1T4oBgHgl3EQfwQX2/content/tmp_files/load_file.txt b/ZNE1T4oBgHgl3EQfwQX2/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..86d3b107a7690f828e2ba9ab6da19cab863a5678 --- /dev/null +++ b/ZNE1T4oBgHgl3EQfwQX2/content/tmp_files/load_file.txt @@ -0,0 +1,529 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf,len=528 +page_content='In Defense of Structural Symbolic Representation for Video Event-Relation Prediction Andrew Lu*, Xudong Lin*, Yulei Niu, Shih-Fu Chang Columbia University {ayl2148,xudong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='lin,yn2338,sc250}@columbia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='edu Abstract Understanding event relationships in videos requires a model to understand the underlying structures of events (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' the event type, the associated argument roles, and corre- sponding entities) along with factual knowledge needed for reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Structural symbolic representation (SSR) based methods directly take event types and associated argument roles/entities as inputs to perform reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' However, the state-of-the-art video event-relation prediction system shows the necessity of using continuous feature vectors from input videos;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' existing methods based solely on SSR inputs fail completely, event when given oracle event types and ar- gument roles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' In this paper, we conduct an extensive em- pirical analysis to answer the following questions: 1) why SSR-based method failed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' 2) how to understand the eval- uation setting of video event relation prediction properly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' 3) how to uncover the potential of SSR-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' We first identify the failure of previous SSR-based video event prediction models to be caused by sub-optimal training set- tings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Surprisingly, we find that a simple SSR-based model with tuned hyperparameters can actually yield a 20% abso- lute improvement in macro-accuracy over the state-of-the- art model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Then through qualitative and quantitative analy- sis, we show how evaluation that takes only video as inputs is currently unfeasible, and the reliance on oracle event in- formation to obtain an accurate evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Based on these findings, we propose to further contextualize the SSR-based model to an Event-Sequence Model and equip it with more factual knowledge through a simple yet effective way of re- formulating external visual commonsense knowledge bases into an event-relation prediction pretraining dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' The resultant new state-of-the-art model eventually establishes a 25% Macro-accuracy performance boost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Equal contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Introduction Event understanding has been thoroughly explored in the past decade [5,20,23,24,27,30,34,37,43,44] due to its great importance in our daily lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' An event is typically repre- sented as a verb (indicating the event type) and several argu- ments, each of which has a role name and an associated en- tity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Researchers have been devoted to extracting events and labeling the argument roles [5,20,30,34,38,39,43] in vision, language, and multimodal domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' These event extraction and argument role labeling models build the foundation for higher-level understanding of the relations between events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Relations between events [1,2,9,28,36,42] have been thor- oughly studied in the language domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' A specific relation and event coreference has also been explored in the mul- timodal setting [5, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' However, video event-relation pre- diction is still a new and challenging task [34] which re- quires both good representations of video events and com- monsense knowledge to reason between events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Structural symbolic representation [12,14,20,22,41,45] has been widely adopted on various downstream tasks such as visual question answering [45], image captioning [41], and action recognition [14] for its high interpretability and generalization ability [22, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' In this work, using struc- tural symbolic representation, we refer to representations consisting of discrete tokens organized with certain struc- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' When applied to event-relation prediction, it is usu- ally formulated as an event type (verb) with associated ar- gument role names and corresponding entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' This event- argument structural symbolic representation has shown to be very effective on predicting event relations in the lan- guage domain [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' However, the models using structural symbolic representation for video events are easily biased by the dominant relations in the VidSitu dataset [34] and predicts the dominant class for all the classes, which con- tradicts the success of existing structural symbolic repre- sentation based methods on various tasks including event- relation prediction using text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' To answer why structural symbolic representation based models fails on video event-relation prediction, we first an- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='03410v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='CV] 6 Jan 2023 alyze if there are any possible patterns that the model could have leveraged to avoid being misled (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Surpris- ingly, we find that even if the model only memorizes the dominant relation for pairs of event types, the model is clearly not supposed to predict only Enables (the most fre- quent relation in the dataset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Based on this finding, we bootstrap the failed symbolic representation based model and the state-of-the-art model with two experiments: uti- lizing a balanced training data/objective and tuning hyper- parameters for better training the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' By only changing the learning rate, we find that the failed symbolic represen- tation based model has already outperformed the state-of- the-art by 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' The state-of-the-art model that heavily re- lies on video features enjoys a much smaller gain through the better training setting we identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' The different behaviors motivate us to carefully analyze the task and the dataset (Sec 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' We evaluate two state- of-the-art video-language models, HERO [19] and Clip- BERT [16], on video event-relation prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' By control- ling inputs with the help of strong image-text contrastive models [31], we found that oracle information is needed to accurately evaluate the models due to the presence of multiple events co-occurring simultaneously in the same video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Even with strong pretrained video-language mod- els, the event-type and argument role descriptions are still more important than the video feature vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Based on these observations, we propose to contextu- alize the simple pairwise SSR-based models to an Event- Sequence model in order to leverage context information within sequences of events for more accurate event-relation prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Furthermore, we explore leveraging the external visual knowledge base, VisualCOMET [29], to teach the model commonsense knowledge about the evolution pro- cess of events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' We propose a simple yet effective approach to reformulate the annotations of VisualCOMET into event sequences suitable for even-relation prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' The contex- tualized sequence model with pretraining on VisualCOMET establishes a new state-of-the-art, which has a margin of 25% improvement in terms of Macro-accuracy compared to the best existing ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Our contributions can be summarized as follows: We identify why symbolic representation based models fail on video event-relation prediction and provide a sim- ple solution which yields a 20% improvement in Macro- accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' We identify proper settings needed to evaluate video event-relation prediction models on VidSitu through quantitative and qualitative analysis of different model variations and the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' We propose a contextualized Event-Sequence model, coupled with a pretraining technique on VisualCOMET, to fully utilize the rich contextual information in event se- quences and commonsense knowledge from the existing knowledge base, which establishes the new state-of-the- art 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='2% → 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Related Work Visual Event Understanding aims to recognize, extract, and structure the actions or activities happening in images or videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Previous studies simply represent visual events as verbs or subject-verb-object triplets [4, 6, 10, 15, 21, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Recent works further study more structural and semantic representations of visual events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' For example, M2E2 [20] and VideoM2E2 [5] handle extracting events from image- text and video-text pairs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Importantly, the vi- sual situation recognition task aims to identify not only the activity in an image [30, 43] or video [34], but also the en- tities including persons and objects (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=', semantic roles) associated with the activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Another benchmark, Visual- COMET [29], proposes to depict person-centric images as a graph of commonsense descriptions, including before- event, intent of people, and next-event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' In this work, we focus on the video event relation prediction task and con- duct empirical studies to evaluate the roles of event type, argument roles, and video features in relation prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' We also explore using VisualCOMET as an external visual knowledge base for pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Structural Symbolic Representation denotes the rep- resentation consisting of discrete tokens with certain struc- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Structural symbolic representation has been applied in various tasks in computer vision and natural language processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' For example, the visual scene in visual ques- tion answering can be modeled as the structural represen- tations of objects with their associated attributes and lo- cations [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' As for event representation, spatio-temporal scene graphs [14, 33] decompose each event as a temporal sequence of spatial scene graphs, where each spatial scene graph is a set of subject-predicate-object triplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Further- more, recent NLP studies show the potential of structural symbolic representation in event-relation prediction in text, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' part-of-speech (POS) and XML tags [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' In this work, we follow VidSitu to represent each video event as event type/verb and its associated argument roles and entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Event-relation Prediction in both Text and Video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Events in texts are often ordered by temporal relation [3], causal relation [26], or narrative order [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' [11] define event relations in text as 5 main types, along with 21 sub-types, covering inheritance, expansion, contingency, comparison, and temporality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Based on prior work in cross- document event relations, VidSitu [34], the only available video event relation dataset, defines four types of relations: no relation, causality, enabled, and relation to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' In this paper, we focus on video event-relation prediction and use VidSitu Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' The pipeline of the Event-Sequence model for event-relation prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Special characters * and ** to denote the target events the moodel is required to predict relation between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Detailed illustration is in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='2 as a case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' We follow VidSitu for the definition of event relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Technical Approach In this section, we first introduce some preliminaries of event-relation prediction, then we describe the variants of the model designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Later, we describe the training tech- niques explored in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Preliminaries Structural Symbolic Representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Structural symbolic representation generally refers to a representation where el- ements are discrete tokens and have certain structures, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' scene graphs [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' In this paper, to represent an event ef- fectively, the models need to know the event type (usu- ally a verb), its associated argument roles, and the actual entities for each argument role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Therefore, we consider the sequence of text tokens with the following structure as the structural symbolic representation of an event x: x = {v, a1, e1, a2, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=', aM, eM}, where v ∈ W is the event type/verb, am ∈ W, em ∈ W are the mth argument role and associated entity, and M is the number of argument roles for this event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Such a sequence is essentially a traverse of the graph with v as the root node and am, em as the mth edge and leaf node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Event-Relation Prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' The model is required to pre- dict the relationship between two events, x1 and x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Since VidSitu [34] is the only dataset available for video event- relation prediction, we adopt its specific setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Each video event sequence consists of five consecutive events {x1, x2, x3, x4, x5}, each of which is from a two-second video segment yi ∈ RH×W ×3×F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' H, W, F are the height, width and number of frames of the video segment respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' In VidSitu, only the relationship between the center event x3 and other events xi, i ̸= 3 are annotated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Baseline Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' We follow the RoBERTa variant [34] to build the baseline model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Based on the structural symbolic representation, a model F : WL −→ RC takes the se- quence of text tokens as input and predicts a distribution over the C possible relationship classes, where L is the length of the sequence consisting of symbolic text repre- sentations of xi and x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' The model F is initialized with pretrained weights from RoBERTa [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Baseline + Video Features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' The state-of-the-art [34] shows that video features are more effective than directly using symbolic representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' It takes both video features and the text tokens as inputs G : WL × RD×F −→ RC to pre- dict the distribution over the C classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' An off-the-shelf video feature extractor H is used to extract continuous fea- ture vectors from the video segment yi when the event hap- pens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' The feature vector is concatenated with the output dive Argo AScn Event 1 ArgM man in lake wetsuit downwards Ev1 breathe Causes E Argo AScn RoBERTa Classifier ArgM Event 2 man in lake wetsuit aggressively Ev3 talk ** Argo ArgM Event 3 Arg1 brunette casually girl brunette boy Event 4 Event 5 RoBERTa Classification Source Video Inputs Output Model Head Segmentsembedding from the text tokens before being fed into the final classifier G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' We denote it as Baseline + Video Features in the following discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' When not specified, the video feature extractor is SlowFast [8,34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Contextualized Event-Sequence Model When people perceive the visual world, rich contex- tual information such as neighboring events, location of the events, manner of the events, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' are important in under- standing the relationship between events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' We also analyze the event sequence in the dataset and indeed find patterns, as presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Motivated by intuition and these find- ings, we propose a contextualized Event-Sequence model for event-relation prediction and explore various ways of utilizing context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Event-Sequence Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Instead of only feeding the model with two events between which to predict event relations, we propose to exploit the rich contextual information in a full sequence of all the five events (shown in Figure 1), p = F(x1, x2, x3, x4, x5), (1) where p ∈ RC is the predicted distribution over the C classes of relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Note that to inform the model which two events between which we want to predict the relation, we add an extra special token “*” before each of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Event-Sequence Model + Video Features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Video features could be also considered contextual information, as the con- tinuous feature vectors obtained from pretrained video fea- ture extractors may convey fine-grained information of the actual visual scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' We follow the same method of video feature integration as in [34], p = G(x1, x2, x3, x4, x5, H(yi), H(y3)), (2) where the video features are fused with contextualized em- beddings before the final classification layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Sequence-to-Sequence Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Inspired by recent ad- vances of sequence-to-sequence modeling [18, 22, 32] on both language and video domains, we also explore a variant of directly generating the sequence of relationships given the sequence of events as input, p1,3, p2,3, p4,3, p5,3 = S(x1, x2, x3, x4, x5), (3) where pi,3 is the predicted distribution for the relationship between the ith event and the middle event x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Note that we follow the common strategy of teacher-forcing [40] to han- dle the conditional generation problem, which means during training pk,3 = S(x1, x2, x3, x4, x5, l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=', lk−1), (4) the ground-truth “historical” event-relations l1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=', lk−1 are used for predicting the next relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' During testing, we use beam search to decode the actual event relation sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' This variant doesn’t directly use additional contextual in- formation as it does not directly take any extra inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' The motivation behind this variant is leveraging conditional gen- eration as a constraint to prevent the model from only pre- dicting the dominant class as reported in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Auxiliary Arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Previous state-of-the-art mod- els [34] only use base arguments (arguments tied to the verb through direct semantic relations) such the agent and the tar- get.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' However, additional contextual information could be clearly provided by extra argument roles that were not used in the state-of-the-art model like AMnr, ADir and AScn (manner, scene and direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' We append them after the base arguments when using them as additional input to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Training The standard training objective is to use the cross- entropy loss to train the model for event-relation prediction, min θ − log pl, (5) where θ is the parameter to be updated in the model, l is the ground-truth index of the relation, and p is the predicted relation type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Balancing Data/Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' In [34], the poor performance of the structural symbolic representation model is attributed to the possible imbalanced relation distribution, which leads the model to only predict the dominant relation type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' To tackle this possible issue, we also explore two versions of solu- tions: re-constructing a balanced dataset or using a balanced loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' For the balanced dataset, we aim at keeping the same number of event pairs in each class by removing videos containing multiple samples of dominant relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' After this process, about 70% of the dataset is kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' For the balanced loss, we adopt the commonly used weighted cross entropy loss for optimization, min θ −βl log pl, (6) where we set βl as the inverse of the proportion of this rela- tion l in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' VisualCOMET Pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' VisualCOMET [29] records 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='4 million commonsense inferences for current visual events under three types of event relations: Before, Intent and After, which corresponds to inferring the past, reason, or future event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' We re-formulate the dataset for event- relationship pretraining;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' we use the current event as the x3, then we randomly sample from the three types of annotated events to construct an event sequence {x1, x2, x3, x4, x5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' The relation label could be automatically generated from the type of annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Note that during random sampling, to simulate a real event sequence, we restrict x1, x2 to be either Before or Intent events and restrict x4, x5 to be Intent or After events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Experiment Results and Discussion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Dataset and Evaluation Metric For our main experiments, we use VidSitu [35], a large-scale dataset containing 29,000 10-second video clips where each video clip is divided into five 2-second seg- ments and the most salient verbs and arguments are anno- tated along with the most dominant event relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' We eval- uate event-relation prediction by computing Top-1 accuracy on predicted relations as well as Top-1 accuracy macro- averaged across the four different relation classes: Causes, Enables, Reaction To, and No Relation [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' We use VLEP [17] to evaluate the generalization of our model on future event prediction, which is formulated as a multiple-choice problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' We follow the official setting and report accuracy on the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Why SSR-Based Methods Fail Preliminary Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' To understand why using struc- tural symbolic representation fails on the VidSitu dataset, we analyze possible patterns that the model could have leveraged to outperform baselines that only predict the dom- inant class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' First, we check whether there are event-pairs that have a dominant relation other than Enables, which is the overall dominant relation in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' As the two examples in Figure 2 show, we found that 66% of event type pairs satisfy this constraint, which indicates that if the model simply memorizes the dominant event relation for each event type pair, it should achieve a macro-accuracy higher than 25% [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' This evidence motivates us to fur- ther explore different hyperparameters to check if the fail- ure comes from the optimization side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' To understand the possible benefit we can obtain from a sequence of events as inputs, we study the pattern between the relation type and the relative distance from x3 to the event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' As shown in Figure 3, the distance distribution of No Relation and the distribution of Causes are quite different: No Relation has an almost symmetric distribution w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' the distance value and is significantly more frequent when the distance is 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' however, Causes has a clear peak at distance 1 and decays when the distance value increases, which is consistent with the temporal order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' This study verifies that feeding the full sequence of events as inputs is beneficial to the structural symbolic representation models that only predict Enables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Tuning Baseline Training Configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' This study focuses on exploiting the potential training configuration issues of the baseline model that were reported to fail to predict meaningful event relations [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Since we observe (a) Speak-Respond (b) Walk-Walk Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Histograms of the event relations for two event type pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' (Best viewed on a monitor when zoomed in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=') (a) No Relation (b) Causes Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Distribution over the relative distance for No Relation and Causes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' For example, x1 to x3 has a distance value of -2 and a distance of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' (Best viewed on a monitor when zoomed in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=') a moderate imbalance in the distribution of event relations classes, we reevaluate the baseline model after training on the same dataset class-balanced through undersampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' This does not improve baseline model performance, which still always predicts one relation class for all events scoring 25% validation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Interestingly, we then discovered the reason behind the lack of performance on state-of-the-art baseline models to be a sub-optimally adjusted learning rate rather than lack of patterns in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' By simply adjusting the learning rate from 1e-4 to 1e-5, performance improves dramatically, by ≈ 29% (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Similar adjustments on state-of-the-art speak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='01-respond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='01 140 120 140 uanba 40 0- SaSTED Enables ReBction To NoRel Event Relationwalk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='01-walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='01 400 300 140 0 SaSTED Enables ReBction To NoRel Event RelationModel LR Data Val Macro Top 1 Acc(%) Val Top 1 Acc(%) Baseline 1e-4 Original 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='00 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='43 Baseline 1e-4 Balanced 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='00 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='43 Baseline + Video Features (SOTA) 1e-4 Original 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='73 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='71 Baseline 1e-5 Original 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='61 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='92 Baseline 1e-5 Original + balanced loss 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='98 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='47 Baseline + Video Features 1e-5 Original 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='08 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='75 Event-Sequence 1e-5 Original + balanced loss 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='38 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='59 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Comparison between baseline model trained with original unbalanced data, baseline model trained with data balanced on event relation class frequency through undersampling, and baseline model with adjusted learning rate trained on original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' SOTA is short for state-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' multimodal models using text annotations and video fea- tures also yield a significant performance improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' But it is noteworthy that the improvement is much less signifi- cant compared to our baseline model, which is built upon structural symbolic representations of events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Evaluation Challenges Simultaneous Events in Videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' One explanation as to why adding visual features actually decreases performance despite being rich in context by nature is the large amount of noise contained within videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' In visual features of complex and busy events, multiple salient verbs may emerge to dom- inate a scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' In such a scenario, the specific verb chosen will have a drastic influence on relevant argument labels of entity co-references and subsequent relationships between events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Figure 4 depicts such a scene where multiple rele- vant verbs and their corresponding arguments emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' The presence of multiple salient events simultaneously makes effectively evaluating video-only event relation pre- diction very challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Since event relation annotations are directly tied to annotated event types and argument roles, many instances occur where predicted event relations are accurate to the scene, but are evaluated as incorrect since they differ from the event chosen by the annotator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' We ex- plore this further in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='4 (Performance Analyses Be- tween Predicted and Human Annotations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Quantify the Effect of Simultaneous Events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' We evalu- ate two state-of-the-art video-language models, HERO [19] and ClipBERT [16], on video event-relation prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' We leverage pretrained CLIP-VIT-32 [31] to perform frame se- lection and region selection given event and argument role as the textual input, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' The details of frames and region selection are provided in the supplementary mate- rial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' In both models, as shown in Table 2, we observe a performance increase when frame selection and region se- lection are utilized which indicates a significant presence of irrelevant frames in one video segment and also shows the negative effect cause by simultaneous events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Model Frame-level Region-level Val Macro Acc(%) HERO [19] \x17 \x17 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='15 HERO \x13 \x17 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='03 HERO \x13 \x13 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='48 ClipBERT [16] \x17 \x17 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='62 ClipBERT \x13 \x17 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='76 ClipBERT \x13 \x13 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='52 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Performance of HERO and ClipBERT when with and without Frame Selection or Region Selection based on CLIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Improving SSR-Based Methods Adding Additional Contextual Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' When pre- dicting the relationship between two events within a five event sequence from VidSitu, baseline models only take the two target events as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' A simple approach to increasing the richness of context information contained within inputs is to increase the temporal duration of the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' By giving the model all five events within a sequence rather than just the two target events, we are able to leverage patterns seen in distances between events as well as events preceding and succeeding the target events, resulting in a minor accuracy improvement (53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='85% → 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='38%) (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Previous state-of-the-art baselines also receive inputs containing only verbs and base arguments (arguments tied to the verb through direct semantic relations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Providing specific context in the form of auxiliary arguments describ- ing the scene of the event, mannerisms and goals of enti- ties performing actions, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=', yields further performance in- creases (55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='38% → 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='60%) (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' When evaluating event-sequence input models given only verbs and only base arguments, performance is still substantially better than ran- dom guessing, showing that verbs alone without entity co- references and vice versa provide useful information for event-relation prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Surprisingly, adding context through video features de- grades performance rather than showing further improve- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' We see a sizable decrease in performance in both the learning rate adjusted baseline (53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='58% → 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='08%) and event-sequence input models (58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='60% → 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='64%) (Ta- Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Example scene containing multiple salient events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' A large explosion dominates the background while a man is running away from the explosion in the foreground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' The verb and argument labels are highly dependent on which event is selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Model Verbs Base Args Aux Args Val Macro Acc(%) Baseline (1e-5 lr) \x13 \x13 \x17 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='85 Baseline + Video Features (1e-5 lr) \x13 \x13 \x17 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='08 Event-Sequence Only Verb \x13 \x17 \x17 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='53 Event-Sequence Only Args \x17 \x13 \x13 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='73 Event-Sequence Baseline \x13 \x13 \x17 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='38 Event-Sequence All Args \x13 \x13 \x13 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='60 Event-Sequence + vid features (SlowFast) \x13 \x13 \x13 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='64 Event-Sequence + vid features (CLIP) \x13 \x13 \x13 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='98 HERO [19] \x13 \x13 \x13 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='15 ClipBERT [16] \x13 \x13 \x13 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='62 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Comparison between model given inputs with various forms of context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' ble 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' We also observe a decrease in performance when switching to a CLIP-based feature extractor on video inputs instead of SlowFast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' In addition, we provide comparisons with more powerful vision-language cross-modal baselines such as HERO [19] and ClipBERT [16] that have recently shown promise on other video event understanding tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Interestingly, neither HERO nor ClipBERT performs even close compared to our Event-Sequence model, which veri- fies its strong effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Additional Model Constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' We found that adding further constraints to event prediction models by employ- ing Sequence-to-Sequence models using BART [18] did not improve performance beyond our best event-sequence RoBERTa models (Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' However, pretraining on Vi- sualCOMET [29] yielded a further improvement, showing that commonsense knowledge of event evolution in Visu- alCOMET could further help our contextualized sequence model to better reason the event relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Performance Analyses Between Predicted and Human Annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' To demonstrate an end-to-end approach, we train and evaluate our event-sequence model using pre- dicted verb and argument roles generated by employing a joint feature extractor and text encoder on the video frames [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Results comparing the models with oracle information, predicted events, and previous state-of-the- art video only baselines [34] are summarized in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Notably, our event-sequence model using predicted events outperforms previous video encoder baselines showing the promise of structural symbolic representations even when they are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' We observe a significant improvement when annotated verbs are used to generate argument roles, which verifies the effectiveness of structural symbolic rep- resentation and indicates the “noisy” nature of the task and dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Specifically, many instances occur where predicted verb and arguments describe a different event within a scene than human annotations, on which ground truth event re- lation labels are based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' In the specific example shown in Figure 5, both ground-truth and predicted events accurately describe the same event, the two men fighting, in Event 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' O-2 Seconds Description1 Description 2 Verb run (walk quickly) Verb explode (go boom) Argo (runner) man in grey outfit Arg1 (thing exploding) castle gate ArgM (direction) away from explosion ArgM (location) castleModel Val Macro Top 1 Acc(%) Val Top 1 Acc(%) RoBERTa Event-Sequence Baseline 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='38 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='59 BART Seq-to-Seq 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='91 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='43 RoBERTa Event-Sequence + All Args 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='60 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='01 RoBERTa Event-Sequence + VisCom pretraining + All Args 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='21 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='65 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Comparison between RoBERTa model given event-sequence inputs, BART Seq-to-Seq, and RoBERTa Event-Sequence model pretrained on VisualCOMET data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Model Val Macro Acc(%) Test Macro Acc(%) Event-Sequence + Annotated verb + Annotated args 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='38 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='47 Event-Sequence + Annotated verb + Predicted args 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='30 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='75 Event-Sequence + Predicted verb + Predicted args 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='46 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='94 Vid features only (SlowFast) 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='78 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='54 Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Comparison between event-sequence baseline trained and evaluated on human annotations versus predicted events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='2 Seconds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='Event ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='4s-6s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='Event ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='8s-10s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='Ground Truth Annotation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='Verb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='watch (look at) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='Arg0 (observer) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='man with long hair ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='Arg1 (entity observed) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='battle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='Predicted Verb and Args ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='Verb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='fire (shoot) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='Arg0 (shooter) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='man in white ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='ArgM (direction) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='towards man in black ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='Ground Truth Annotation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='Verb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='shoot (propel projectile) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='Arg0 (shooter) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='man in beige robe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='Arg1 (projectile) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='lightning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='Predicted Verb and Args ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='Verb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='strike (attack) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='Arg0 (attacker) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='man in white ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='Arg1 (target) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='man in black ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='Ev5 is a reaction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='to Ev3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='Ev3 enables Ev5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Example where predicted events differ from ground truth human annotations, but predicted verbs and arguments are salient since they describe different events within the same scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' The resulting event-relation prediction using predicted events is also ”accurate” but different from the ground truth relation since they are relating different, equally salient event descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' In Event 5, human annotations describe the man in the fore- ground watching the battle in the background, which per the ground truth label is a Reaction To the fighting in Event 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' However, predicted verb and arguments describe the back- ground battle in Event 5 and subsequently classifies Event 3 as Enabling Event 5 since the battle is a continuation of the previous fighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' In this case, Enables is also an accurate relationship descriptor but considered incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Such dis- crepancies between annotations likely explain some of the performance degradation seen when using predicted events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Applications in Downstream Tasks We adopt the RoBERTa-based model from [17] as the baseline, which achieves 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='8% accuracy on the VLEP val- idation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' It is a challenging task, as prertaining RoBERTa on ATOMIC [36] only improved accuracy by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' However, when using our event-sequence model as the pretrained weights for the RoBERTa model, we observe a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content='9% im- provement of accuracy, which makes video event-relation prediction a better knowledge source of pretraining than the comprehensive textual knowledge base, ATOMIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' This again verifies the effectiveness and the generalization of the proposed contextualized sequence model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Conclusion In this paper, we defend the effectiveness of structural symbolic representations for video event-relation prediction and we identify sub-optimal training configurations as the key reason previous models fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' We further provide in- depth analysis and find that video features are noisy because of simultaneous events and irrelevant background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' It is nec- essary to use oracle event information as input to ensure proper evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' We propose a contextualized sequence model with a pretraining technique on VisualCOMET to further explore the potential of structural symbolic repre- sentation, which yields a new state-of-the-art with an ab- solute improvement of 25%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Finally, our model trained on video event-relation prediction is also useful in downstream tasks such as future video event prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' References [1] Allison Badgett and Ruihong Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Extracting subevents via an effective two-phase approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' In Proceedings of the 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[41] Ning Xu, An-An Liu, Jing Liu, Weizhi Nie, and Yuting Su.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Scene graph captioner: Image captioning based on structural visual representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Journal of Visual Communication and Image Representation, 58:477–485, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' 1 [42] Wenlin Yao, Zeyu Dai, Maitreyi Ramaswamy, Bonan Min, and Ruihong Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Weakly supervised subevent knowl- edge acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' In Proceedings of the 2020 Confer- ence on Empirical Methods in Natural Language Processing (EMNLP), 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' 1 [43] Mark Yatskar, Luke Zettlemoyer, and Ali Farhadi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Situa- tion recognition: Visual semantic role labeling for image understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 5534–5542, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' 1, 2 [44] Guangnan Ye, Yitong Li, Hongliang Xu, Dong Liu, and Shih-Fu Chang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Eventnet: A large scale structured concept library for complex event detection in video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' In Proceedings of the 23rd ACM international conference on Multimedia, pages 471–480, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' 1 [45] Kexin Yi, Jiajun Wu, Chuang Gan, Antonio Torralba, Push- meet Kohli, and Josh Tenenbaum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Neural-symbolic vqa: Disentangling reasoning from vision and language under- standing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' Advances in neural information processing sys- tems, 31, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} +page_content=' 1, 2' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE1T4oBgHgl3EQfwQX2/content/2301.03410v1.pdf'} diff --git a/atE_T4oBgHgl3EQfyxzf/content/tmp_files/2301.08320v1.pdf.txt b/atE_T4oBgHgl3EQfyxzf/content/tmp_files/2301.08320v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..56dfa169d25bafce6c01d2d353a5ae1efee63dfe --- /dev/null +++ b/atE_T4oBgHgl3EQfyxzf/content/tmp_files/2301.08320v1.pdf.txt @@ -0,0 +1,2759 @@ +arXiv:2301.08320v1 [math.CA] 19 Jan 2023 +FURTHER PROPERTIES OF BALL PROLATES AND APPROXIMATION +OF RELATED ALMOST BAND-LIMITED FUNCTIONS +AHMED SOUABNI +Abstract. In this paper we aim to investigate the approximation of almost band limited +functions using their expansions in the base of ball polynomials and the base of ball prolate +spheroidal wave functions. To do this, we start by giving some required properties on the +ball prolate spheroidal wave functions for our proposed proof. Note that these functions are +both an extension of the classical PSWFs (d = 1) and the ball polynomials (c = 0). +MSC : 42C10, 65L70, 41A10. +Keywords: Ball prolate spheroidal wave functions, Ball polynomials, Finite Fourier trans- +form, Almost band-limited functions. +1. Introduction +Time-limited and band-limited functions are fundamental tools in signal processing. By +Heisenburg’s uncertainty principle, a signal can not be time and band-limited simultaneously. +That is why a natural assumption is that a signal is almost band-limited. This issue has been +initially carried thought Landau, Pollak and Slepian since their pioneer work in the 1960’s, +where prolate spheroidal wave functions have been introduced as the optimal orthogonal +system to represent almost band-limited functions [6] [10] [11] [21]. From the investigation +of the above problem, Slepian was the first to note that PSWFs are the eigenfunctions of the +finite Fourier transform operator corresponding to the eigenvalue λ, i.e +� 1 +−1 +eicxtψ(t)dt = λψ(x) +x ∈ I = (−1, 1). +Slepian et al.[21] proved that this integral operator commutes with some Sturm-Liouville +operator. Hence, PSWFs are also solutions of the second order differential equation +� +(1 − x2)ψ′(x) +�′ ++ (χ − c2x2)ψ = 0 +recovered also by separation of variables for solving the Helmholtz equation in spherical co- +ordinates. +This point is fundamental because that it is a perturbation of the Legendre’s differential +equation and in that way we link up PSWFs with orthogonal polynomials. +We are interested in the theory of prolate spheroidal wave functions because they have a wide +range of applications and remarkable properties. Many extensions of the time-frequency con- +centration problem on the finite interval to other geometries like the disk, 3D ball, sphere, +triangle... have been considered. The reader may consult for example [8] [18] [19] [23]. +This work was supported in part by the DGRST research grant LR21ES10 and the PHC-Utique research +project 20G1503. +1 + +2 +AHMED SOUABNI +We are interested in the extension given by Slepian in [20] where this problem has been +extended to the d-dimensional case . In contrast with the one dimensional case, the prob- +lem of time-frequency concentration over bounded higher dimension domain has not received +enough attention. +In the first part of this work, we will be interested in the prolate spheroidal wave functions in +the multidimensional ball. Note that the first who studied this issue is Slepian in [20] by ex- +tending the finite Fourier transform to the d-dimension. Recently, in [24], authors have given +a very important contribution consisting on writing the Sturm-Liouville operator defining +ball prolate spheroidal wave functions in a suitable form allowing to preserve the key features +of the one-dimensional case. More precisely, they expressed the Sturm-Liouville operator +of interest as a perturbation to the order c2∥x∥2 of the one defining the ball polynomials. +Thus, we have all ingredients to develop spectral methods relative to the study of prolate +spheroidal wave functions. We should mention here that whereas a more general context has +been considered in [24], the aim of this work is to give some refined bounds of the eigenvalues +and eigenfunctions of the integral operator and to establish some other properties of the ball +PSWFs. +The second purpose of this work is to study the quality of approximation of almost band- +limited functions by ball prolate spheroidal wave functions series expansions. In spite of their +important properties, we can’t handle ball PSWFs in a straightforward way because there +is no explicit formula to compute them. That is why one classical scheme is to compute ex- +plicitly their coefficients in terms of ball polynomials basis. Then, it is convenient to develop +almost band-limited functions directly in the base of ball polynomials and see what happens +with the quality of approximation in this framework. +Let us now be a little more specific, ball prolate spheroidal wave functions (ψ(m,c) +k,ℓ +) are defined +as solutions of the following concentration problem +Find f = arg max +f∈Bc +� +Bd |f(x)|2dx +� +Rd |f(x)|2dx. +Here +Bc := {f ∈ L2(Rd) : ˆf(u) = 0 +∀u ̸∈ Bd(0, c)}, +Bd(0, c) := {x ∈ Rd : ∥x∥ ≤ c} +and +Bd := Bd(0, 1). +The solutions of this problem are eigenfunctions of the finite Fourier transform given by +Fc.f(x) = +� +Bd e−icf(y)dy. +and then, by commutativity, eigenfunctions of +Lc,x = −∇.(1 − ∥x∥2)∇ − ∆0 + c2∥x∥2. +Namely, +Fc.ψ(m,c) +k,ℓ += µn(c)ψ(m,c) +k,ℓ +, +Lc,x.ψ(m,c) +k,ℓ += χ(m) +n +(c)ψ(m,c) +k,ℓ +. +Note that by the form under which this last differential operator is given, by Zhang et al. in +[24], the ball PSWFs extend the orthogonal ball polynomials (c=0) +P (m) +k,ℓ (x) = �P +(0,m+ d +2 −1) +k +(2∥x∥2−1)Y m +l (ˆx), +x ∈ Bd, +1 ≤ l ≤ 2m + d − 2 +m +�m + d − 3 +m − 1 +� +, +k, m ∈ N. + +BALL PROLATE SPHEROIDAL WAVE FUNCTIONS +3 +and also provide a Bouwkamp spectral algorithm for the computation of ball PSWFs. Our +first result is an estimation of +���ψ(m,c) +k,ℓ +��� +∞. +Theorem A : Let c > 0. For any integer k such that χ(m) +k +> max{ c2 + 8 +(2m + d), (2/3)6 +� +2π +m + d +2 − 1 +�2 ++ +4(m + d +2)(m + d +2 − 1) − 4 + c2} + m(m + d), we have +max +x∈Bd +���ψ(m,c) +k,ℓ +(x) +��� ≤ +3 +� +3 +� +m + d +2 − 1 +� +2 +� +N(d, m) +Ωd−1 +� +χ(m) +k +(c). +As mentioned before, and as application of this first part, we will give the quality of approx- +imation by ball PSWFs and by ball polynomials. We should mention here that this question +has been solved in the one-dimensional case in [7]. +At first, let us define the concept of +almost-band-limited function. +Definition 1.1. Let c > 0, Ω = Bd(0, c) and ǫΩ > 0. A function f ∈ L2(Rd) is said to be +ǫΩ-band-limited function if +� +∥x∥>c +| ˆf(x)|2dx ≤ ǫ2 +Ω∥f∥2 +L2(Rd). +The approximation of almost band-limited functions by ball prolate spheroidal functions +and by ball polynomials are given by the two following theorems. +Theorem B : Let f ∈ L2(Rd) be an ǫΩ-band-limited function. Then for any positive integer +N ≥ ec +4 , we have +∥f − SN.f∥L2(Bd) ≤ +� +2ǫΩ + Cm,d|µN(c)| +� +χ(m) +N (c) +�1/2� +∥f∥L2(Rd), +where Cm,d = 3 +2 +� +c +(4π)1/4 +�d � +3(m + d +2 − 1) +d +2 + 1 +and SNf is the orthogonal projection of a func- +tion f ∈ L2(Rd) on the space spanned by the N +1 first ball prolate spheroidal wave functions +. +Theorem C : Let f ∈ L2(Rd) be an ǫΩ band-limited function with Ω = B(0, c). Then, for +any positive integer N ≥ +ec−m− d+1 +2 +2 +, we have +∥f − ΠN.f∥L2(Bd) ≤ + +2ǫΩ + CN +� +ec +2(N + 1) + m + d+1 +2 +�2(N+1)+m+ d+1 +2 + + ∥f∥L2(Rd), +where CN,m,d = +1 +22N+m+ d +2 +3� +ec(4N + 3m + d + 4) + +1 + +1 +4 ln +� +ec +2N+m+2+ d+1 +2 +� + + +1/2 +and ΠNf +is the orthogonal projection of a function f ∈ L2(Rd) on the space spanned by the N +1 first + +4 +AHMED SOUABNI +ball polynomials. +The remainder of the paper is organized as follows. Section 2 is devoted to some preliminary +results that will be useful afterwards. In section 3, we give some spectral properties of ball +prolate spheroidal wave functions, namely the behaviour of the eigenvalues of the associated +integral operator and some local estimates giving an upper bound of +���ψ(m,c) +k,ℓ +��� +∞. We conclude, +in section 4, by the quality of approximation in the ball PSWFs basis comparing with the +ball polynomials one. +2. Mathematical preliminaries about some special functions +In this section, we recall some important properties about some special functions, mainly, +the ball polynomials. +For this purpose, we introduce some preliminaries about spherical +harmonics which appear in the definition of ball polynomials. Furthermore, we recall some +properties about Bessel functions which will be frequently used throughout the forthcoming +sections. +2.1. Bessel functions. For α > − 1 +2, the Bessel functions Jα are the bounded solutions of +the ordinary differential equation given by, (see for example [22]), +x2y′′ + xy′ + (x2 − α2)y = 0, +x > 0, +which is equivalent to : +(xy′)′ + +� +x − α2 +x +� +y = 0. +Many recurrence relations for Bessel function of the first kind exist in literature (we refer +reader for example to [22]), among which we cite +(2.1) +d +dx[xαJα(x)] = xαJα−1(x), +(2.2) +d +dx[x−αJα] = −x−αJα−1(x). +Bounds and local estimates of Jα are frequently used in this paper. A first simple and useful +local estimate is given by, see [15] +sup +x≥0 +√x|Jα(x)| ≤ cα, +with +cα = +� � +2/π +if |α| ≤ 1/2 +0.675 +� +α1/3 + +1.9 +α1/3 + 1.1 +α +if α > 1/2. +A second well known estimate of the Bessel function is given in [[16] p.227] by +(2.3) +���Jν(x) +xν +��� ≤ +1 +2νΓ(ν + 1). +An other estimate of the Bessel function, when the argument is less than the order, is given +by +(2.4) +1 ≤ Jν(νx) +xνJν(ν) ≤ eν(1−x) +ν > 0 +and +0 < x ≤ 1. + +BALL PROLATE SPHEROIDAL WAVE FUNCTIONS +5 +In [17], author has given a more precise inequality where it has been shown that +(2.5) +exp +�ν2(1 − x2) +4ν + 4 +� +≤ Jν(νx) +xνJν(ν) ≤ exp +�ν2(1 − x2) +2ν + 4 +� +ν > 0 +and +0 < x ≤ 1. +The following inequality gives us a lower bound of Jν(ν) (we refer reader to [5]) +Jν(ν) ≥ +Γ(1/3) +22/331/6π(ν + α0)1/3 +α0 ∼= 0.0943498 +Thus, by combining the last two inequalities, one gets +(2.6) +Jν(νx) ≥ +Γ(1/3) +22/331/6π(ν + α0)1/3 xν exp +�ν2(1 − x2) +4ν + 4 +� +. +(ν > 0 +and +0 < x ≤ 1). +The spherical Bessel functions are defined as +(2.7) +j(α) +n,c (x) = +� +2(2n + α + 1)J2n+α+1(cx) +√cx +, +x ∈ (0, ∞). +This later set of functions satisfy the orthogonality relation, +� +∞ +0 +j(α) +n,c (x)j(α) +m,c(x)x. = δn,m. +Recall that the Hankel transform of a function f ∈ L2(0, ∞) is given by +Hα.f(x) := +� ∞ +0 +√xyJα(xy)f(y)dy; +α > −1/2. +The Hankel transform of the spherical Bessel functions are given by, see for example [20] +Hα.j(α) +n,c (x) += +� +2(2n + α + 1) +c +�x +c +�α+ 1 +2 P (α,0) +n +� +1 − 2 +�x +c +�2� +χ[0,c](x). +(2.8) +By noticing that the Hankel transform is an involution, one can write (2.8) in a more suitable +form +(2.9) +� 1 +0 +yα+1Jα(cxy)P (0,α) +n +(2y2 − 1)dy = (−1)n J2n+α+1(cx) +cx +. +2.2. Spherical harmonics. Let Rd be the d-dimensional Euclidean space, x will denote the +column vector (x1, · · · xd)T . We will denote the inner product over Rd, for x, y ∈ Rd, by +< x, y >:= +d +� +i=1 +xiyi and ∥x∥ will denote the Euclidean associated norm ∥x∥ := √< x, x > = +� +x2 +1 + · · · + x2 +d. We also introduce its polar spherical coordinates (r := ∥x∥, ˆx := x +r ). +The unit sphere Sd−1 and the unit ball Bd of Rd are denoted respectively by +Sd−1 := {ˆx ∈ Rd : ∥ˆx∥ = 1}, +Bd := {x ∈ Rd : ∥x∥ ≤ 1}. +The inner product of L2(Sd−1) is defined as +< f, g >Sd−1:= +� +Sd−1 f(ˆx)g(ˆx)dσ(ˆx), + +6 +AHMED SOUABNI +where dσ is the surface measure. +Let Hd +n be the space of harmonic homogeneous polynomials of degree n and N(d, n) := +dim Hd +n. It is well known that N(d, n) = 2n + d − 2 +n +�n + d − 3 +n − 1 +� +. Note that the radial and the +angular dependence of a function Hn ∈ Hd +n can be separated : Hn(x) = Hn(rˆx) = rnHn(ˆx). +Definition 2.1. A spherical harmonic of degree n denoted Yn(ˆx) is a harmonic homogeneous +polynomial of degree n in d variables restricted to the unit (d − 1)-sphere. +It is well known that the spherical harmonics satisfy +∆0.Yn = −n(n + d − 2)Yn. +In other words, Yn are eigenfunctions of the angular part of the Laplace operator given by +∆0 = +� +1≤j) +C +( d−2 +2 ) +n +(1) +; +Ωd−1 := σ(Sd−1) = 2π +d+1 +2 +Γ(d+1 +2 ), +which shows that the ultra-spherical polynomial C +( d−2 +2 ) +n +is the basic spherical harmonic in d +dimensions analogous to cos in the case d = 2. +Recall that the ultra-spherical polynomials are given by +C(λ) +n (x) := Γ(λ + 1/2) +Γ(2λ) +Γ(n + 2λ) +Γ(n + λ + 1/2)P +(λ− 1 +2,λ− 1 +2) +n +(x); +C(λ) +n (1) = +Γ(n + 2λ) +Γ(2λ)Γ(n + 1) +(n ≥ 1). +Here P (α,β) +n +are the Jacobi polynomials defined by +P (α,β) +n +(x) = (−1)n +2nn! (1 − x)−α(1 + x)−β dn +dxn +� +(1 − x)n+α(1 + x)n+β� +. +We will re-write the normalization of the ultra-spherical polynomials under the following form +Lemma 2.2. For any positive integer n and under the above notations, we have +(2.11) +� +Sd−1 |C +( d−2 +2 ) +n +(< ˆx, ˆy >)|2dσ(ˆy) = +Ωd−1 +N(d, n) +� +C +( d−2 +2 ) +n +(1) +�2 +. + +BALL PROLATE SPHEROIDAL WAVE FUNCTIONS +7 +Proof. This proof is simply based on (2.10) and the normalization of {Yn}n. In fact, +� +Sd−1 |C +( d−2 +2 ) +n +(< ˆx, ˆy >)|2dσ(ˆy) += +� Ωd−1 +N(d, n) +�2 � +C +( d−2 +2 ) +n +(1) +�2 � +Sd−1 + + +N(d,n) +� +j=1 +Y (n) +j +(ˆx) Y (n) +j +(ˆy) + + +2 +dσ(ˆy) += +� Ωd−1 +N(d, n) +�2 � +C +( d−2 +2 ) +n +(1) +�2 N(d,n) +� +j=1 +Y (n) +j +(ˆx) Y (n) +j +(ˆx) += +� Ωd−1 +N(d, n) +�2 � +C +( d−2 +2 ) +n +(1) +�2 N(d, n) +Ωd−1 +□ +Let Yn be any spherical harmonic of degree n. Using the orthogonality of Yn, together +with (2.10), one gets +Yn(ˆx) = +N(d, n) +C(λ) +n (1)Ωd−1 +� +Sd−1 Yn(ˆy)C +( d−2 +2 ) +n +(< ˆx, ˆy >) dσ(ˆy). +By taking into account the normalization of ultra-spherical polynomials (2.11) and by +Cauchy-Schwarz inequality, one gets +(2.12) +|Yn(ˆx)| ≤ +� +N(d, n) +Ωd−1 +. +To finish with this part, it is useful to note that the finite Fourier transform (over the unit +sphere) of spherical harmonics is given by the following explicit expression given in [14], [1], +(2.13) +� +Sd−1 e−iw<�x.�y>Y m +ℓ (�x)dσ(�x) = (2π)d/2(−i)m +w +d−2 +2 +Jm+ d−2 +2 (w)Y m +ℓ (�y) +�x, �y ∈ Sd−1, +w > 0 +2.3. Ball polynomials: Orthogonal polynomials on Bd. The ball polynomials are de- +fined as +P (m) +k,ℓ (x) = �P +(0,m+ d +2 −1) +k +(2∥x∥2 − 1)Y m +l (x), +x ∈ Bd +1 ≤ ℓ ≤ N(d, m) +k, m ∈ N. +Here +(2.14) +�P (α,β) +k +(x) = +1 +√hk +P (α,β) +k +(x), +hk = +2α+β+1Γ(k + α + 1)Γ(k + β + 1) +k!(2k + α + β + 1)Γ(k + α + β + 1). +It has been shown in [14] that the ball polynomials are orthogonal with respect to the usual +inner product. Recall also that the total degree of P (m) +k,ℓ +is m + 2k. In addition, the ball +polynomials are eigenfunctions of the following differential operator : +(2.15) +Lx.P (m) +k,ℓ (x) = +� +− ∇.(I − xxt)∇ +� +P (m) +k,ℓ (x) = (m + 2k)(m + 2k + d)P (m) +k,ℓ (x). + +8 +AHMED SOUABNI +Note that in [24], authors have proven that this last operator can be written in different more +suitable forms given by +Lx += +−∇.(1 − ∥x∥2)∇ − ∆0 += +−(1 − r2)∂2 +r − d − 1 +r +∂r + (d + 1)r∂r − 1 +r2 ∆0, +(2.16) +Next, we will compute the finite Fourier transform of the ball polynomials P (m) +k,ℓ +in what +follows +Lemma 2.3. For all y = τˆy ∈ Bd, we have +(2.17) +� +Bd e−icP (m) +j,ℓ (x)dx = (2π)d/2(−i)m(−1)j +2 +J2j+m+d/2(cτ) +(cτ) +d +2 +Y (m) +ℓ +(ˆy). +Proof. Let x = ρˆx and y = τ ˆy, then using (2.13) together with (2.9) one gets +� +Bd e−icP (m) +j,ℓ (x)dx += +� 1 +0 +ρm+d−1P (0,m+d/2−1) +k +(2ρ2 − 1) +� +Sd−1 e−icρτ<ˆx,ˆy>Y (m) +ℓ +(ˆx)dσ(ˆx)dρ += +(2π)d/2(−i)m +(cτ)d/2−1 +� 1 +0 +ρm+d/2Jm+ d +2 −1(cρτ)P +(0,m+ d +2 −1) +j +(2ρ2 − 1)dρ.Y (m) +ℓ +(ˆy) += +(2π)d/2(−i)m(−1)j +2(cτ) +d−1 +2 +J2j+m+d/2(cτ) +√cτ +Y (m) +ℓ +(ˆy). +(2.18) +□ +3. Ball prolate spheroidal wave functions : Definitions and spectral +properties +3.1. Definition and normalization. We should mention here that the equivalent defini- +tions given in this paper of ball prolate spheroidal wave functions are an association of those +given by Slepian in [20] and recently by Zhang and co-authors in [24]. We introduce the ball +PSWFs in the classical way, namely as solutions of an energy maximization problem there- +fore as eigenfunctions of an integral operator and finally, by a ”commutativity property”, as +eigenfunctions of a suitable differential operator. +In this work, we adopt the following definition of the Fourier transform over Rd, +ˆf(x) = +1 +(2π)d/2 +� +Rd e−if(y)dy. +Recall that, with this normalization, one has +��� ˆf +��� +L2(Rd) = ∥f∥L2(Rd) and the inversion formula +is then +f(x) = +1 +(2π)d/2 +� +Rd ei ˆf(y)dy. +We are dealing with the issue of most concentrated band-limited functions on the unit ball, +that is +(3.19) +Find f = arg max +f∈Bc +� +Bd |f(x)|2dx +� +Rd |f(x)|2dx. + +BALL PROLATE SPHEROIDAL WAVE FUNCTIONS +9 +From (2.13), and by writing the integral in the spherical-polar coordinates x = rˆx, +ˆx ∈ +Sd−1, one gets +� +Bd eicdx = +(2π)d/2 +� +c∥y − z∥ +� d +2 −1 +� 1 +0 +r +d +2 J d +2 −1(cr∥y − z∥)dr. +On the other hand, +� 1 +0 +r +d +2 J d +2 −1(cr∥y − z∥)dr += +1 +� +c∥y − z∥ +� 1 +2 +� 1 +0 +(cr∥y − z∥) +1 +2J d +2 −1(cr∥y − z∥)dr += +1 +� +c∥y − z∥ +� 1 +2 +� +R +(r∥y − z∥) +1 +2 J d +2 −1(r∥y − z∥) +�r +c +� d−1 +2 χ[0,c](r)dr += +Jd/2 +� +c∥y − z∥) +� +c∥y − z∥ +. +(3.20) +Note that the last equality is from n=0 in (2.9). Finally, combining the last two equations +gives us +� +Bd eicdx = (2π)d/2 Jd/2 +� +c∥y − z∥) +� +� +c∥y − z∥ +� d +2 +. +We retain this result, that will be useful in the sequel, in the following lemma +Lemma 3.1. For any positive real number c, and for all y, z ∈ Bd, we have +(3.21) +� +Bd eicdx = (2π)d/2 Jd/2 +� +c∥y − z∥) +� +� +c∥y − z∥ +� d +2 +. +Let f ∈ Bc, +� +Bd |f(x)|2dx +� +Rd |f(x)|2dx += +1 +(2π)d +� +Bd +� � +Rd ei ˆf(y)dy +�� � +Rd e−i ˆf(z)dz +� +dx +� +Rd +ˆf(x) ˆf(x)dx += +� 1 +2π +�d +� +Bd(0,c) +� � +Bd(0,c) +� � +Bd eidx +� +ˆf(y)dy +� +ˆf(z)dz +� +Rd +ˆf(x) ˆf(x)dx += +� 1 +2π +�d/2 +� +Bd(0,c) +� � +Bd(0,c) +Jd/2(∥y − z∥) +� +∥y − z∥ +� d +2 +ˆf(y)dy +� +ˆf(z)dz +� +Bd(0,c) +ˆf(x) ˆf(x)dx +. +(3.22) + +10 +AHMED SOUABNI +Hence, by a straightforward change of variable and function, the solutions of (2π) +d +2 +� +Bd |f(x)|2dx +� +Rd |f(x)|2dx +are the eigenfunctions of the integral operator Qc given by +(3.23) +Qc.f(x) = +� c +2π +�d � +Bd(2π)d/2 Jd/2(c∥y − z∥) +� +c∥y − z∥ +� d +2 +f(y)dy. +One can easily check that Qc = +� c +2π +�d F∗ +c ◦Fc where Fc is the finite Fourier integral operator +defined on L2(Bd) by +(3.24) +Fc.f(x) = +� +Bd e−icf(y)dy. +We denote by λ(m) +n +(c) the eigenvalues corresponding to ψ(m,c) +k,ℓ +as eigenfunction of Qc, i.e +λ(m) +n +(c)ψ(m,c) +k,ℓ +(z) = +� c +2π +�d � +Bd(2π)d/2 Jd/2(c∥y − z∥) +� +c∥y − z∥ +� d +2 +ψ(m,c) +k,ℓ +(y)dy. +We define the ball spheroidal wave functions as solutions of the energy maximization problem +(3.19) which are at the same time eigenfunctions of both operators Qc and Fc. +On the other hand, in [24], it has been shown that the finite Fourier integral operator com- +mutes with the following positive self-adjoint differential operator +Lc,x += +−∇.(1 − ∥x∥2)∇ − ∆0 + c2∥x∥2 += +−(1 − r2)∂2 +r − d − 1 +r +∂r + (d + 1)r∂r − 1 +r2 ∆0 + c2r2. +(3.25) +Thus, one can also say that the ball PSWFs are also the eigenfunctions of (3.25). That is +Lc,xψ(m,c) +k,ℓ +(x) = χ(m) +k +(c)ψ(m,c) +k,ℓ +(x), +where χm +k (c) are the corresponding eigenvalues and c is the bandwidth parameter. Note also +that ψ(m,c) +k,ℓ +has a separated form given by +(3.26) +ψ(m,c) +k,ℓ +(x) = rmφ(m,c) +k +(2r2 − 1)Y m +ℓ (ˆx), +where φ(m,c) +k +satisfies +(3.27) +−1 +ω0,βm,d(η)∂η +� +ω1,βm,d+1(η)∂ηφ(m,c) +k +� ++c2(η + 1) +8 +φ(m,c) +k += 1 +4 +� +χ(m) +k +(c)−γm,d +� +φ(m,c), +k +with +γm,d = m(m+d). +Remark 3.2. We use in this remark the separated form of the ball prolate spheroidal wave +functions to show that the radial part of these functions are also eigenfunctions of the finite +Hankel transform. We should mention that this remark has been given differently in [20]. + +BALL PROLATE SPHEROIDAL WAVE FUNCTIONS +11 +Thanks to (3.26) and (2.13), one can write +λ(m) +k +rmφ(m,c) +k +(2r2 − 1) = (2π)d/2(−i)m +� 1 +0 +Jm+ d +2 −1(crτ) +(crτ) +d +2 −1 +τ m+d−1φ(m,c) +k +(2r2 − 1)dτ. +It is convenient to make the substitution ϕ(m,c) +k +(r) := rm+ d−1 +2 φ(m,c) +k +(2r2 − 1) to obtain +λ(m) +k +ϕ(m,c) +k +(r)√c +� c +2π +�d/2(i)m = +� 1 +0 +Jm+ d +2 −1(crτ)√crτϕ(m,c) +k +(τ)dτ. +Then +(3.28) +α(m) +k +ϕ(m) +k +(r) = +� 1 +0 +Jν(crτ)√crτϕ(m) +k +(τ)dτ = H(α) +c +.ϕ(m) +k +(r); +ν = m+d +2−1; +α(m) +k += √c +� c +2π +�d/2λ(m) +k +(c). +Here H(α) +c +is the finite Hankel transform given by +H(α) +c +.f(x) = +� 1 +0 +√cxyJα(cxy)f(y)dy. +It may be useful to note that the problem of the finite Hankel transform has been extensively +studied, see for example [4], [13] and [3]. +We note finally that the ball PSWFs are normalized through the following rule : +� +Bd +� +ψ(m,c) +k,ℓ +(x) +�2dx = 1, +or equivalently, in terms of the radial part: +� 1 +0 +r2m+d−1� +φ(m,c) +k +(2r2 − 1) +�2 +dr = 1 +or +� 1 +−1 +(1 + t)m+ d +2 −1|φ(m,c) +k +|2(t)dt = 2m+ d +2 −1. +Finally, we will write the ball prolate in term of limiting operators. Let U be a set of finite +measure in Rd and D(U) be the subspace of L2(Rd) formed by the functions supported in U, +D(U) = {f ∈ L2(Rd) : f(x) = 0 +∀x ̸∈ U}, +and recall that Bc is Paley-Wiener space formed by functions whose Fourier transform are +supported in Bd(0, c), +Bc = {f ∈ L2(Rd) : F.f(u) = 0 +∀u ̸∈ Bd(0, c)}. +Let DU.f(x) = χU(x)f(x) be the orthogonal projection of L2(Rd) onto D(U) and Bc.f(x) = +F−1χBd(0,c)F.f(x) be the orthogonal projection of L2(Rd) onto Bc. +Using (3.21), +Bc.f(x) = +� c +2π +�d/2 � +Rd(2π)d/2 Jd/2(c∥x − y∥) +(c∥x − y∥)d/2 f(y)dy +Then, one can write the integral operator Qc in terms of the limiting operators as +Qc = DBdBcDBd + +12 +AHMED SOUABNI +3.2. Behaviour of λ(m) +n +(c). Recently in [3], the author gives a well precise behaviour of the +eigenvalues α(m) +k +which are related, by remark 3.2, to the eigenfunctions corresponding to ball +prolate functions. Mainly, one can derive an asymptotic super-exponential decay of λm +n (c) +directly by writing theorem 3.2 of [3] under our notations : +Proposition 3.3. For given real numbers m, c > 0, there exists a constant A depending only +on c, such that for every n > ec +4 , we have +(3.29) +λ(m) +n +(c) ≤ A +� +ec +4n + 2m + d +�4n+2m+d +. +In the following proposition, we provide the reader with a lower bound of the eigenvalues +λm +n (c). +Proposition 3.4. For any positive real number c > 0 and for any n ≥ ec +4 , we have +λm +n (c) ≥ C +� +ec +4n + 2m + d +�4n+2m+d +, +where C = +� +Γ(1/3) +22/331/6π(2k + α + α0 + 1)1/3 +�2 (2π)d/2 +c +d+1 +2 +Proof. We will prove this lower bound for the eigenfunctions of the finite Hankel transform +and use again the remark giving the relation between the eigenfunctions of the ball PSWFs +and those of the circular ones. For this purpose, we recall that the eigenfunctions of Q(α) +c +:= +cH(α) +c +◦ H(α) +c +are characterized, using Courant-Fischer max-min theorem, by +α(m) +n +(c) = max +V ∈Gn +min +y∈V ;∥y∥=1 < Q(α) +c +y; y > +Since +� +T (α) +n +:= +� +2(2n + α + 1)xα+1/2P (0,α) +n +(2x2 − 1) +� +n is an orthonormal basis of L2(0, 1), +then +α(m) +n +(c) ≥< Q(α) +c +T (α) +n +, T (α) +n +>L2([0,1]) . +On the other hand, using (2.8) together with (2.7), one gets +���H(α) +c +.T (α) +k +��� +2 +2 = +���j(α) +k +��� +2 +2 = 2(2k + α + 1) +� 1 +0 +����� +J2k+α+1(cx) +√cx +����� +2 +dx. +Thanks to (2.6), one can write, +J2k+α+1(cx) ≥ +Γ(1/3) +22/331/6π(2k + α + α0 + 1)1/3 e +−(2k+α+1) +� +ln(2k+α+1)−1/4 +� +(cx)2k+α+1e +−(cx)2 +4(2k+α+1) + +BALL PROLATE SPHEROIDAL WAVE FUNCTIONS +13 +Therefore, +���H(α) +c +.T (α) +k +��� +2 +≥ +C2e +−2(2k+α+1) +� +ln(2k+α+1)−1/4 +� � c +0 +x2(2k+α+1/2)e− +x2 +2(2k+α+2) dx +≥ +C2e +−2(2k+α+1) +� +ln(2k+α+1) +� � c +0 +x2(2k+α+1/2)e− +x2 +2(2k+α+2) dx +≥ +C2� +ec +2(2k + α + 1) +�2k+α+1 +. +To conclude for the proof, it suffices to use remark 3.2. +□ +We will give in the next proposition a brief description, to the first order, of the counting +number of the eigenvalues λ(m) +n +(c).We use here the well known Landau’s technique [9] based +on computing the trace and the Hilbert-Schmidt norm of Qc. +Proposition 3.5. Let 0 < δ < 1 and let Mc(δ) denote the number of eigenvalues λk(c) not +smaller than δ. Then +Mc(δ) = cd +2d +1 +Γ2(d +2 + 1) + o(cd). +Proof. We start by computing the trace of Qc by using Mercer’s theorem together with the +fact that K(0) = +1 +2d/2Γ(d +2 + 1) and µ(Bd) = +πd/2 +Γ(d +2 + 1), +(3.30) +Tr(Qc) = +� +n +λ(m) +n +(c) = +� +c +√ +2π +�d +µ(Bd)K(0) = +cd +2dΓ2(d +2 + 1). +Here K(x) := +Jd/2(∥x∥) +(∥x∥)d/2 . On the other hand, to compute an estimates of ∥Qc∥HS, we proceed +as follows +∥Qc∥HS = +� +n +� +λ(m) +n +(c) +�2 += +� +c +√ +2π +�2d � +Bd +� +Bd |K(c∥x − y∥)|2dxdy. +Applying the change of variable y = σ, x = σ + τ +c gives us +∥Qc∥HS = +� c +2π +�d � +Bd +� � +c(Bd−σ) +|K(τ)|2dτ +� +dσ. +Note that +� +c(Bd−σ) +|K(τ)|2dτ +≤ +� +Rd |K(τ)|2dτ = +� +Rd +J2 +d/2(∥τ∥) +∥τ∥d +dτ += +2πd/2 +Γ(d/2) +� ∞ +0 +J2 +d/2(t) +t +dt = +πd/2 +Γ(d +2 + 1). +(3.31) +(The last equality is due to [[16] eq 22.58 page 244]) Moreover, Bd − σ contains some B(0, α) +then, by Lebesgue’s dominated convergence theorem one has lim +c→∞ +1 +cd ∥Qc∥HS = +1 +2dΓ(d +2 + 1), + +14 +AHMED SOUABNI +that is +(3.32) +∥Qc∥HS = +cd +2dΓ(d +2 + 1) + o(cd). +Next, we notice that +(3.33) +Tr(Qc) ≥ +Mc(δ) +� +k=0 +λk ≥ δMc(δ), +and using Marzo’s formula (see [12]), one gets +(3.34) +Mc(δ) ≥ Tr(Qc) − +1 +1 − δ +� +Tr(Qc) − ∥Qc∥HS +� +. +Let M+ := lim sup +c→∞ +Mc(δ) +cd +and M− := lim inf +c→∞ +Mc(δ) +cd +. The next step is to prove that both M+ +and M− are independent of δ for all 0 < δ < 1. For this claim, we start by noticing that the +number of eigenvalues not close to 0 or 1 is o(cd) by considering +Jc := +∞ +� +k=0 +λk(c)(1 − λk(c)) = Tr(Qc) − ∥Qc∥HS = o(cd). +Now let δ and γ be two fixed real such that 0 < δ < γ < 1. +Taking into account that +each eigenvalue δ < λk(c) < γ has contribution to Jc at least by δ(1 − γ), one has δ(1 − +γ) [Mc(δ) − Mc(γ)] ≤ Jc = o(cd) and consequently M+ and M− are both independent of δ. +Collecting together (3.33), (3.34), (3.32), (3.32) and the fact that M+ and M− are independent +of δ gives us +(3.35) +1 +2d +1 +Γ2(d +2 + 1) ≤ M− ≤ M+ ≤ 1 +2d +1 +δΓ2(d +2 + 1). +To conclude for the proof it suffices to choose δ near 1 on the right. +□ +3.3. Further basic properties. We study here some other basic properties of the ball +prolate spheroidal wave functions. +First, we give the bounds of the eigenvalues corresponding to the Sturm-Liouville operator. +Lemma 3.6. For any positive real number c, we have +(m + 2k)(m + 2k + d) ≤ χ(m) +k +(c) ≤ (m + 2k)(m + 2k + d) + c2 +n ≥ 0. +Proof. Using the Min-Max theorem, one gets +χ(m) +n +(c) = +min +dim H=n +max +u∈H;∥u∥=1 < Lc,x.u, u > . +Then +min +dim H=n +max +u∈H;∥u∥=1 < L0,x.u, u >≤ χ(m) +n +(c) ≤ +min +dim H=n +max +u∈H;∥u∥=1 < L0,x.u, u > +c2∥u∥2. +Once again, the Min-Max theorem applied to χ(m) +n +(0) gives +χ(m) +n +(0) ≤ χ(m) +n +(c) ≤ χ(m) +n +(0) + c2. +To conclude for the proof, it suffices to use (2.15). +□ + +BALL PROLATE SPHEROIDAL WAVE FUNCTIONS +15 +The next proposition proves that we keep the fundamental property of double orthogonality +already seen in the classical case, and this, by computing the Fourier transform of the ball +prolates spheroidal wave functions. +Lemma 3.7. The Fourier transform of the ball PSWFs are given by +F.ψ(m,c) +k,ℓ +(x) = +(2π)d +cd +� +λ(m) +k +(c) +ψ(m,c) +k,ℓ +�−x +c +� +χ(Bd) +�x +c +� +. +Proof. By the inverse Fourier transform, one has for f ∈ L2(Rd), +f(x) = +1 +(2π)d +� +Rd +� +Rd eif(y)dydx. +On the other hand, from +ψ(m,c) +k,ℓ +(x) = +1 +µ(m) +k +(c) +� +Bd e−icψ(m,c) +k,ℓ +(y)dy, +one gets +F.ψ(m,c) +k,ℓ +(x) += +1 +µ(m) +k +(c) +� +Rd +� +Bd e−icψ(m,c) +k,ℓ +(y)dye−idx += +1 +µ(m) +k +(c) +� +Rd +� � +Rd e−icψ(m,c) +k,ℓ +(-y/c)χ(Bd)(y/c)dy +� +dx. +(3.36) +□ +3.4. Some explicit estimates and bounds of eigenfunctions. The purpose of this para- +graph is to give an explicit upper bound of the ball prolate spheroidal wave functions. To do +this, we start by showing that under some conditions, these functions reached their maximum +at the unit sphere Sd−1. Let c > 0, recall that +ψ(m,c) +k,ℓ +(x) = rmφ(m,c) +k +(η)Y (m) +ℓ +(ˆx), +x = rˆx +η := 2r2 − 1. +Lemma 3.8. Let α(m) +k +:= 1 +4 +� +χ(m) +k +(c) − m(m + d) +� +and c be a positive real number. If α(m) +k +> +c2 +4 then we have +(3.37) +sup +η∈[am,d,1] +|φ(m) +k +(η)| = |φ(m) +k +(1)| +am,d = 2m + d − 2 +2m + d +. +Proof. For sake of clarity, we will write, throughout the proofs of this section, φk and αk +instead of φ(m,c) +k +and α(m) +k +respectively. Hence (3.27) can be written under the form +(3.38) +� +p(η)φ′ +k(η) +�′+qk(η)φk(η) = 0; +p(η) = (1−η)(1+η)m+ d +2 , qk(η) = αk +� +1 − c2(1 + η) +8αk +� +(1 + η)m+ d +2 −1 . + +16 +AHMED SOUABNI +Straightforward computations show that the auxiliary function Zk(η) := φ2 +k(η) + p(η) +qk(η)φ′2 +k (η) +admits a first order derivative only on φ′2 +k given by +Z′ +k(η) = − +1 +q2 +k(η) (p(η)qk(η))′ φ′2 +k (η). +Then, in our case +Z′ +k(η) = − +1 +q2 +k(η)(1+η)2m+d−2 +�� +αk − c2(1 + η) +8 +�� +2m+d−2−t(2m+d) +� +− c2 +8 (1−η2) +� +φ′2 +k (η). +To conclude for the proof, it suffices to note that under the condition α(m) +k +> +c2 +4 , Zk is +increasing over [am,d, 1] and +���φ2 +k(η) +��� ≤ Zk(1) ≤ +���φ2 +k(1) +���. +□ +Lemma 3.9. Under conditions of the previous lemma, we have +(3.39) +sup +η∈[am,d,1] +� +(1 − η)(1 + η)m+ d +2 ��φk(η) +�� ≤ +� +2m+ d +2 +1(m + d +2 − 1). +Proof. The proof of this lemma is based on a fairly well known technique for the Sturm- +Liouville theory consisting on the use of the following auxiliary function Kk(η) = p(η)Zk(η). +Straightforward computations shows that +K′ +k(η) = p′(η)φ2 +k(η) − p2(η)q′(η) +q2(η) +(φ′ +k)2(η). +Taking into account that αk > c2 +4 and the fact that η ≥ 2m+d−2 +2m+d , one has +K′ +k(η) ≥ (1 + η)m+ d +2 −1�� +m + d +2 − 1 +� +− +� +m + d +2 + 1 +� +η +� +φ2 +k(η). +Hence, +Kk(η) = Kk(η) − Kk(1) ≤ 2 +� 1 +−1 +(1 + t)m+ d +2 −1φ2 +k(t)dt = 2m+ d +2 +1(m + d +2 − 1). +□ +Proposition 3.10. Let c be a real positive number. If α(m) +k +> c2 +4 , then +(3.40) +sup +η∈[am,d,1] +��φk(η) +�� ≤ 3 +√ +3 +2 +� +2m+ d +2 +1(m + d +2 − 1) +� +α(c) +k,m +�1/2 +Proof. Without loss of generality, one may assume that ψ(m,c) +k,ℓ +(1) > 0. By (3.27), one has +� +(1 − η2)(1 + η)m+ d +2 −1φ′ +k(η) +�′ += −α(c) +km +� +1 − c2(η + 1) +8α(c) +km +� +(1 + η)m+ d +2 −1φk(η). + +BALL PROLATE SPHEROIDAL WAVE FUNCTIONS +17 +Then by integrating over [x, 1], one gets +φ′ +k(η) = +αk +(1 − η2)(1 + η)m+ d +2 −1 +� 1 +η +(1 + t)m+ d +2 −1� +1 − c2(t + 1) +8αk +� +φk(t)dt. +It has been shown that for αk > c2 +4 , q is decreasing over [am,d, 1], then +φ′ +k(η) ≤ +αk +1 − η2 +� +1 − c2(η + 1) +8αk +� +(1 − η)φ(1) = αk +� +1 − c2(η + 1) +8αk +� +φk(1). +Then, +φk(1) − φk(η) ≤ αkφk(1) +� +1 − c2(η + 1) +8αk +� +(1 − η). +Let xk be in a neighbourhood of 1 such that +� +1 − c2(xk+1) +8α(c) +km +� +(1 − xk) = +A +α(c) +km +. Consequently, +φk(1) ≤ +1 +1 − A +� +2m+ d +2 +1 � +m + d +2 − 1 +� +� +(1 − xn)(1 + xn)m+d/2 ≤ +� +2m+ d +2 +1 � +m + d +2 − 1 +� +(1 − A)A1/2 +(αk)1/2. +One concludes for the proof by noticing that min +A +1 +A1/2(1 − A) = 3 +√ +3 +2 . +□ +Theorem 3.11. Let c > 0 and α(m) +k +> max{ +c2+8 +4(2m+d), (2/3)6 +� +π +m+ d +2 −1 +�2 ++ Cm,d(c)} then +max +x∈Bd +���ψ(m,c) +k,ℓ +(x) +��� ≤ 3 +√ +3 +2 +� +2m+ d +2 +1(m + d +2 − 1) +� +N(d, m) +Ωd−1 +. +� +χ(m) +k +(c), +with Cm,d(c) = c2 +4 + (m + d +2)(m + d +2 − 1) − 1. +Proof. We start by recalling the Butlowski theorem concerning the behaviour of the local +extrema of the solution of a second order differential equation : +Theorem 3.12 (Butlowski see [1] p238). If φ is a solution of the differential equation +(p(t)y′(t))′ + q(t)y(t) = 0 +t ∈ (a, b) +p, q > 0 and p, q ∈ C1(a, b), +then the local maximums of |φ| is increasing or decreasing according to the condition that +p(t)q(t) is decreasing or increasing +In our case, we recall that +� +p(η)qk(η) +�′ += (1 + η)2m+d−2 +�� +α(m) +k +− c2(1 + η) +8 +�� +2m + d − 2 − η(2m + d) +� +− c2 +8 (1 − η2) +� +, +then it is easy to see that there exists a unique real number ηn so that pq is increasing in +[0, ηn] and decreasing in [ηn, 1]. Hence, the local maxima of φk are decreasing in [0, ηn] and +increasing in [ηn, 1]. Let η′ +1,n denote the first zero of φ′ +k. the next step is to locate η′ +1,n. For + +18 +AHMED SOUABNI +this claim, we start by using the change of function U(η) := p1/2(η)φk(η) which transform +(3.38) to the following equation on U, +U ′′ + +�p′(η)2 − 2p′′(η)p(η) +4p2(η) ++ qk(η) +p(η) +� +U = 0, +η ∈ (0, 1). +We should mention that U and φk have the same zeros on (0, 1). +Straightforward computations shows that +p′(η)2 − 2p′′(η)p(η) +p2(η) += 1 − +� +m + d +2 +�2 +(1 + η)2 ++ 4 +� +m + d +2 − 1 +� +(1 + η)2(1 − η) + +4 +(1 − η2)2 ≥ (m+ d +2)(1−m− d +2)+1. +Since, qk(η) +p(η) = αk − c2 +8 (1 + η) +1 − η2 +≥ αk − c2 +4 thus, +p′(η)2 − 2p′′(η)p(η) +4p2(η) ++ qk(η) +p(η) ≥ αk − Cm,d(c); +Cm,d(c) = c2 +4 + (m + d +2)(m + d +2 − 1) − 1. +then, by Sturm comparison theorem, between two zeros of sin( +� +αk − c2 +4 η) viewed as solution +of V ′′ + +� +αk − c2 +4 +� +V = 0, there exists a zero of U. Consequently, ηk,1 ≤ +π +� +αk − c2 +4 +and +η′ +k,1 ≤ +2π +� +αk − c2 +4 += bk. +Let η ∈ (0, η′ +n,1) and recall that +� +(1 − η2)(1 + η)m+ d +2 −1φ′ +k(η) +�′ += −αk +� +1 − c2(η + 1) +8αk +� +(1 + η)m+ d +2 −1φk(η). +Then by integrating over [η, η′ +n,1], one gets +φ′ +k(η) = − +αk +(1 − η2)(1 + η)m+ d +2 −1 +� η′ +n,1 +η +(1 + t)m+ d +2 −1� +1 − c2(t + 1) +8α(c) +km +� +φ(t)dt. +Using H¨older inequality and the normalization of φk, one gets +|φ′ +k(η)| +≤ +αk +(1 − η2)(1 + η)m+ d +2 −1 +� +2m+ d +2 −1 +����� +� η′ +n,1 +η +(1 + t) +m+ d +2 −1 +2 +φ(t)dt +����� +≤ +� +2m+ d +2 −1 +� +η′ +k,1 +1 − η′ +k,1 +αk ≤ +� +2m+ d +2 −1 +√bk +1 − bk +αk +(3.41) +Then, +|φk(η′ +n,1)| ≤ +� +2m+ d +2 −1 (bk)3/2 +1 − bk +αk ≤ +3 +� +3(m + d +2 − 1) +2 +(αk)1/2 . + +BALL PROLATE SPHEROIDAL WAVE FUNCTIONS +19 +It may be useful to note that the last inequality follows from the fact that αk ≥ (2/3)6 +� +π +m + d +2 − 1 +�2 ++ +c2 +4 . To conclude for the proof, it suffices to combine the previous analysis with (2.12). +□ +3.5. Computation of ball spheroidal wave functions. Note that ψ(m,c) +k,ℓ +∈ L2(Bd), then +it can be expanded with respect of the ball polynomials basis as +ψ(m,c) +k,ℓ +(x) = +� +j +β(k,m) +j +P (m) +j,l (x). +In [24], authors have used the Bouwkamp method to compute ball PSWFs where it has +been shown that the computation of these functions and their eigenvalues amounts to the +determination of the eigenvectors and associate eigenvalues of a tridiagonal matrix. Then +it is interesting to study the spectral decay rate of the ball PSWFs expansion coefficients +(β(k,m) +j +)j. For this purpose, we recall the finite Fourier transform of ball polynomials given +by (3.21). +� +Bd e−icP (m) +j,ℓ (x)dx = (2π)d/2(−i)m(−1)j +2(cτ) +d−1 +2 +J2j+m+d/2(cτ) +√cτ +Y (m) +ℓ +(ˆy). +Then, +β(k,m) +j += +� +Bd ψ(m,c) +k,ℓ +(x)P (m) +j,ℓ (x)dx = +1 +µ(m) +k +(c) +� +Bd +� � +Bd e−icP (m) +j,ℓ (x)dx +� +ψ(m,c) +k,ℓ +(y)dy += +(2π)d/2(−i)m(−1)j +µ(m) +k +(c).c +d−1 +2 +� 1 +0 +τ m+ d +2 − 1 +2 φ(m,c) +k +(2τ 2 − 1)J2j+m+d/2(cτ) +√cτ +dτ. +Consequently, by Cauchy-Schwartz inequality together with the normalization of the radial +part of ball spheroidal wave functions, one gets +|β(k,m) +j +| ≤ +(2π)d/2 × c4j+2m+ d−1 +2 +24j+2m+d(4j + 2m + d)|µ(m) +k +(c)| +1 +Γ2(2j + m + d +2 + 1). +Recall that form [2], one has +(3.42) +√ +2e +�x + 1/2 +e +�1/2 +≤ Γ(x + 1) ≤ +√ +2π +�x + 1/2 +e +�1/2 +. +Then, by combining the two last inequalities, one gets +(3.43) +|β(k,m) +j +| ≤ +(2π)d/2 × c2j+m +24j+2m+d(4j + 2m + d)|µ(m) +k +(c)| +� +ec +2j + m + d+1 +2 +�2j+m+ d+1 +2 +. +We may therefore summarise this calculations in the following +Proposition 3.13. For given real number c > 0, let +β(k,m) +j += +� +Bd ψ(m,c) +k,ℓ +(x)P (m) +j,ℓ (x)dx. + +20 +AHMED SOUABNI +Then we have +|β(k,m) +j +| ≤ +(2π)d/2 × c2j+m +24j+2m+d(4j + 2m + d)|µ(m) +k +(c)| +� +ec +2j + m + d+1 +2 +�2j+m+ d+1 +2 +. +4. Approximation of almost band-limited functions over the d-dimensional +unit ball +The aim of this section is to study the quality of approximation in the framework of the d- +dimensional ball prolate spheroidal wave functions and the ball polynomials series expansion. +4.1. Approximation by ball prolate spheroidal wave functions. In this paragraph, we +show that the ball prolate spheroidal wave functions are well adapted for the approximation +of almost band-limited functions. For this claim, we start by proving that ball PSWFs are +also well adapted to approach functions from the Paley-Wiener space +Bc = {f ∈ L2(Rd) : F.f(u) = 0 +∀u ̸∈ Bd(0, c)}. +Definition 4.1. Let c > 0 and ǫΩ. A function f ∈ L2(Rd) is said to be ǫΩ-band-limited +function if +� +x̸∈Ω +| ˆf(x)|dx ≤ ǫΩ∥f∥2 +L2(Rd). +Let us denote by SN.f := +N +� +k=0 +< f, ψ(m,c) +k,ℓ +> ψ(m,c) +k,ℓ +(x) the orthogonal projection of a +function f ∈ L2(Bd) on the span of the N + 1 first ball prolate functions, then we may now +state our first main theorem. +Theorem 4.2. Let f ∈ L2(Rd) be an ǫΩ-band-limited function with Ω = B(0, c). Then for +any positive integer N ≥ ec/2, we have +(4.44) +∥f − SNf∥L2(Bd) ≤ +� +2ǫΩ + C|µN(c)|(χN(c))1/2� +∥f∥L2(Rd), +where C = +� +c +√ +2π +�d/2 +πd/4 +� +Γ(d +2 + 1) +3 +� +3(m + d +2 − 1) +2 +� +N(d, m) +Ωd−1 +.. +Proof. Let us first study the case where f ∈ Bc. By Parseval’s inequality, +(4.45) +∥f − SN.f∥2 +2 = +� +k>N +| < f, ψ(m,c) +k,ℓ +> |2. +Then the main step in this proof is how to estimate | < f, ψ(m,c) +k,ℓ +> |. +Recall that from the Fourier inversion formula, one has f(x) = +� c +2π +�d � +Bd eic ˆf(y)dy. + +BALL PROLATE SPHEROIDAL WAVE FUNCTIONS +21 +Consequently, for any positive integer k, we have +< f, ψ(m,c) +k,ℓ +> += +� +Bd f(x)ψ(m,c) +k,ℓ +(x)dx = +� c +2π +�d � +Bd ψ(m,c) +k,ℓ +(x) +� +Bd eic ˆf(y)dydx += +� c +2π +�d � +Bd +ˆf(y) +� +Bd eicψ(m,c) +k,ℓ +(x)dxdy += +� c +2π +�d +|µk(c)| +� +Bd +ˆf(y)ψ(m,c) +k,ℓ +(y)dy +≤ +� c +2π +�d +πd/4 +� +Γ(d/2 + 1) +|µk(c)| sup +y∈Bd |ψ(m,c) +k,ℓ +(y)|. +(4.46) +By combining (4.45), (4.46) and theorem 3.11, one gets +∥f − SNf∥L2(Bd) ≤ +� +C|µN(c)|(χN(c))1/2� +∥f∥L2(Rd). +Now, let f be an ǫΩ-band-limited function. +We have +∥f − SNf∥L2(Bd) ≤ ∥f − Bcf∥L2(Bd) + ∥Bcf − SNBcf∥L2(Bd) + ∥SN[Bcf − f]∥L2(Bd). +On the other hand, since f is ǫΩ-band-limited then ∥Bcf − f∥L2(Bd) ≤ ǫΩ∥f∥L2(Rd). Using +the fact that SN is a contraction one gets +∥SN[Bcf − f]∥L2(Bd) ≤ ∥Bcf − f∥L2(Bd) ≤ ǫΩ∥f∥L2(Rd). +Finally, we apply the previous proposition to the second term by noticing that Bcf is a +band-limited function. +□ +4.2. Approximation of almost band limited functions by ball polynomials. We +study in this section the quality of approximation of almost band limited functions by their +expansion in the basis of ball polynomials. To do this, we start by the following technical +lemma controlling the general term of the associated projection series. +Lemma 4.3. Let c > 0, then for every f ∈ Bc and any k ≥ ec +2 +(4.47) +���< f, P (m) +k,ℓ >L2(Bd) +��� ≤ +1 +22k+m+ d +2+1� +2ec(4k + 3m + d) +� +ec +2k + m + d+1 +2 +�2k+m+ d+1 +2 +. +Proof. Let f ∈ Bc be a band-limited function, then by the Fourier inversion formula, we have +f(x) = +1 +(2π)d/2 +� +Bd(0,c) +ˆf(y)eidy = +� c2 +2π +�d/2 � +Bd +ˆf(cy)eicdy. +Recall that the finite Fourier transform of the ball polynomials is given by +� +Bd eicP (m) +k,ℓ (y)dy = (2π)d/2 (−i)m(−1)j +2 +J2k+m+d/2(cρ) +(cρ)d/2 +Y m +ℓ (ˆx) +x = ρˆx. + +22 +AHMED SOUABNI +Then, by this last two relations, +< f, P (m) +k,ℓ > += +� +Bd f(x)P (m) +k,ℓ (x)dx = +� c2 +2π +�d/2 � +Bd +� +Bd +ˆf(cy)eicdyP (m) +k,ℓ (x)dx += +cd (−i)m(−1)j +2 +� 1 +0 +ρm+d−1 +� +Sd−1 +ˆf(cρˆy)J2k+m+d/2(cρ) +(cρ)d/2 +Y m +ℓ (ˆx)dσ(ˆy)dρ. +(4.48) +By (2.3), one gets +���� +J2k+m+d/2(cρ) +(cρ)d/2 +���� ≤ +(cρ)2k+m +22k+m+ d +2 Γ(2k + m + d +2 + 1) +, +then +(4.49) +���< f, P (m) +k,ℓ > +��� ≤ +c2k+2m+d−1 +22k+m+1+d/2Γ(2k + m + 1 + d/2) +� 1 +0 +ρ2k+2m+d−1 +���� +� +Sd−1 +ˆf(cρˆy)Y m +ℓ (ˆy)dσ(ˆy) +���� dρ. +On the other hand���� +� +Sd−1 +ˆf(cρˆy)Y m +ℓ (ˆy)dσ(ˆy) +���� ≤ +�� +Sd−1 +��� ˆf(cρˆy) +��� +2 +dσ(ˆy) +�1/2 +Now let g(ρ) := ρ +m+d−1 +2 +�� +Sd−1 +��� ˆf(cρˆy) +��� +2 +dσ(ˆy) +�1/2 +. We remark that +(4.50) +� 1 +0 +|g(ρ)|2 dρ = +� +Bd +��� ˆf(cy) +��� +2 +dy = 1 +cd ∥f∥2 +2. +Therefore +���< f, P (m) +k,ℓ > +��� ≤ +c2k+m+d/2 +22k+m+1+ d +2 √ +4k + 3m + dΓ(2k + m + 1 + d +2) +∥f∥2. +To conclude for the proof of this lemma, it remains to estimate Γ(2k + m + 1 + d +2) via Batir’s +inequality (3.42). +□ +In analogy to the previous case, we will define ΠN.f := +N +� +k=0 +< f, P (m) +k,ℓ +> P (m) +k,ℓ +the orthog- +onal projection on the span of the N + 1 first ball polynomials. We may have now all the +ingredients to state our second main theorem. +Theorem 4.4. Let f ∈ L2(Rd) be an ǫΩ band-limited function with Ω = B(0, c). Then for +any positive integer N ≥ +ec−m− d+1 +2 +2 +, we have +(4.51) +∥f − ΠNf∥L2(Bd) ≤ + +2ǫΩ + CN +� +ec +2(N + 1) + m + d+1 +2 +�2(N+1)+m+ d+1 +2 + + ∥f∥L2(Rd). +Here CN := +1 +22N+m+ d +2 +3√ +ec(4N+3m+d+4) + +1 + +1 +4 ln +� +ec +2N+m+2+ d+1 +2 +� + + +1/2 +. + +BALL PROLATE SPHEROIDAL WAVE FUNCTIONS +23 +Proof. Let f ∈ Bb. Note that for x ∈ Bd, f(x) − ΠN.f(x) = +� +k>N +< f, P (m) +k,ℓ > P (m) +k,ℓ (x), then +∥f − ΠN.f∥2 +L2(Bd) += +∞ +� +k=N+1 +| < f, P (m) +k,ℓ > |2 = +1 +24k+2m+d+2ec(4k + 3m + d) +� +ec +2k + m + d+1 +2 +�4k+2m+d+1 +≤ +1 +24N+2m+d+6ec(4N + 3m + d + 4) +∞ +� +k=N+1 +� +ec +2k + m + d+1 +2 +�4k+2m+d+1 +. +(4.52) +Let us treat the main factor separately, +∞ +� +k=N+1 +� +ec +2k + m + d+1 +2 +�4k+2m+d+1 += +� +ec +2N + m + 2 + d+1 +2 +�4N+2m+d+5 ++ +∞ +� +k=N+2 +� +ec +2k + m + d+1 +2 +�4k+2m+d+1 +≤ +� +ec +2N + m + 2 + d+1 +2 +�4N+2m+d+5 ++ +� ∞ +N+1 +� +ec +2(N + 2) + m + d+1 +2 +�4x+2m+d+1 +dx += +� +ec +2N + m + 2 + d+1 +2 +�4N+2m+d+5 ++ +exp +� +(2m + d + 1) ln +� +ec +2N + m + 2 + d+1 +2 +�� � ∞ +N+1 +exp +� +4x ln +� +ec +2N + m + 2 + d+1 +2 +�� +dx +≤ + +1 + +1 +4 ln +� +ec +2N+m+2+ d+1 +2 +� + + +� +ec +2N + m + 2 + d+1 +2 +�4N+2m+d+5 +. +Thus, for all f ∈ Bc +(4.53) +∥f − ΠNf∥L2(Bd) ≤ CN +� +ec +2(N + 1) + m + d+1 +2 +�2(N+1)+m+ d+1 +2 +∥f∥L2(Rd). +Now, let f be an ǫΩ-band-limited function. +∥f − SNf∥L2(Bd) ≤ ∥f − Bcf∥L2(Bd) + ∥Bcf − SNBcf∥L2(Bd) + ∥SN[Bcf − f]∥L2(Bd). +On the other hand, since f is ǫΩ-band-limited then ∥Bcf − f∥L2(Bd) ≤ ǫΩ∥f∥L2(Rd). Using +the fact that SN is a contraction one gets +∥SN[Bcf − f]∥L2(Bd) ≤ ∥Bcf − f∥L2(Bd) ≤ ǫΩ∥f∥L2(Rd). +By the previous analysis and (4.53), (4.51) follows at once. +□ + +24 +AHMED SOUABNI +References +[1] G. E. Andrews, R. Asqey and R. Roy. Special Functions. Cambridge University Press ,Cambridge , +New York , 1999. +[2] N.Batir. Inequality for the Gamma Function.Arch.91, (2008), 554–563. +[3] M.Boulsane. Non Asymptotic behavior and the distribution of the spectrum of the finite Hankel transform +operator. Integral transform and special functions, 32(12),(2021), 948-968. +[4] M. Boulsane and A. Karoui. The Finite Hankel Transform Operator: Some Explicit and Local Esti- +mates of the Eigenfunctions and Eigenvalues Decay Rates. J. Four. Anal. Appl, 24, (2018), 1554-1578. +[5] A.Elbert and A.Laforgia. A lower bound for Jµ(µ). Appl.Anal, 19, (1985), 137-145. +[6] J.A.Hogan and J.D.Lackey. 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J. 41 (1962) +1295–1336. +[12] J.Marzo. Marcinkiewicz-Zygmund inequalities and interpolation by spherical harmonics. J. Func. Anal., +250(2) , (2005), 559-587. +[13] T. Moumni and A. Karoui. Spectral Analysis of the Finite Hankel Transform Operator and Circular +Prolate Spheroidal Wave Functions. J. Comput. Appl. Math, 233 (2), (2009), 315-333. +[14] F.Dai and Y.Xu. Approximation theory and harmonic analysis on spheres and balls. Springer Mono- +graph in mathematics, (2013). +[15] A. YA. Olenko. Upper bound on √xJµ(x) and its applications. Integral Transforms and Special Func- +tions. 17 (2006), 455–467. +[16] F. W. Olver, Daniel W. Lozier, Ronald F. Boisvert, and Charles W.Clark. NIST Handbook +of Mathematical Functions Cambridge University Press, New York, NY, USA, 1st edition (2010). +[17] R.B. Paris. An inequality for the Bessel function Jν(νx). SIAM J.Math.Anal 15 1 (1984) 203–205. +[18] Y.Shkolnisky. Prolate spheroidal wave functions on a disk integration and approximation of two- +dimensional band limited function.. Appl.Comput.Harmon.Anal, 22(2) , (2007), 235–256. +[19] F.J.Simons,F.A.Dahlen and M.A.Wieczorek. Spatio-spectral concentration on a sphere . SIAM Rev, +48(3) , (2006), 504–536. +[20] D. Slepian. +Prolate spheroidal wave functions, Fourier analysis and uncertainity. IV. Extensions to +many dimensions; generalized prolate spheroidal functions. Bell System Tech. J. 43 (1964) 3009–3057. +[21] D.Slepian and H.O.Pollak. +Prolate spheroidal wave functions, Fourier analysis and uncertainity. I. +Bell System Tech. J. 43 (1964) 3009–3058. +[22] G. N. Watson. A treatise on the theory of Bessel functions.second edition. Cambridge University +Press.1966. +[23] M.A Taylor and B.A Wingate. +A generalization of prolate spheroidal wave functions with more +uniform resolutions to the triangle. J.Engrg.Math 56 (3)(2006) 221–235 . +[24] J.Zhang, H.Li, L.L.Wang and Z.Zhang. Ball prolate spheroidal wave functions in arbitrary dimen- +sions. Appl.Comput.Harmon.Anal. 1276 (2018) . +Ahmed Souabni Address: University of Carthage, Department of Mathematics, Faculty of +Sciences of Bizerte, Bizerte, Tunisia. +Email address: souabniahmed@yahoo.fr + diff --git a/atE_T4oBgHgl3EQfyxzf/content/tmp_files/load_file.txt b/atE_T4oBgHgl3EQfyxzf/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1bba97cdcf0f83a6922215e06125394b5e42236b --- /dev/null +++ b/atE_T4oBgHgl3EQfyxzf/content/tmp_files/load_file.txt @@ -0,0 +1,782 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf,len=781 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content='08320v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content='CA] 19 Jan 2023 FURTHER PROPERTIES OF BALL PROLATES AND APPROXIMATION OF RELATED ALMOST BAND-LIMITED FUNCTIONS AHMED SOUABNI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' In this paper we aim to investigate the approximation of almost band limited functions using their expansions in the base of ball polynomials and the base of ball prolate spheroidal wave functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' To do this, we start by giving some required properties on the ball prolate spheroidal wave functions for our proposed proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' Note that these functions are both an extension of the classical PSWFs (d = 1) and the ball polynomials (c = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' MSC : 42C10, 65L70, 41A10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' Keywords: Ball prolate spheroidal wave functions, Ball polynomials, Finite Fourier trans- form, Almost band-limited functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' Introduction Time-limited and band-limited functions are fundamental tools in signal processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' By Heisenburg’s uncertainty principle, a signal can not be time and band-limited simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' That is why a natural assumption is that a signal is almost band-limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' This issue has been initially carried thought Landau, Pollak and Slepian since their pioneer work in the 1960’s, where prolate spheroidal wave functions have been introduced as the optimal orthogonal system to represent almost band-limited functions [6] [10] [11] [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' From the investigation of the above problem, Slepian was the first to note that PSWFs are the eigenfunctions of the finite Fourier transform operator corresponding to the eigenvalue λ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content='e � 1 −1 eicxtψ(t)dt = λψ(x) x ∈ I = (−1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' Slepian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' [21] proved that this integral operator commutes with some Sturm-Liouville operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' Hence, PSWFs are also solutions of the second order differential equation � (1 − x2)ψ′(x) �′ + (χ − c2x2)ψ = 0 recovered also by separation of variables for solving the Helmholtz equation in spherical co- ordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' This point is fundamental because that it is a perturbation of the Legendre’s differential equation and in that way we link up PSWFs with orthogonal polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' We are interested in the theory of prolate spheroidal wave functions because they have a wide range of applications and remarkable properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' Many extensions of the time-frequency con- centration problem on the finite interval to other geometries like the disk, 3D ball, sphere, triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' have been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' The reader may consult for example [8] [18] [19] [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' This work was supported in part by the DGRST research grant LR21ES10 and the PHC-Utique research project 20G1503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' 1 2 AHMED SOUABNI We are interested in the extension given by Slepian in [20] where this problem has been extended to the d-dimensional case .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' In contrast with the one dimensional case, the prob- lem of time-frequency concentration over bounded higher dimension domain has not received enough attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' In the first part of this work, we will be interested in the prolate spheroidal wave functions in the multidimensional ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' Note that the first who studied this issue is Slepian in [20] by ex- tending the finite Fourier transform to the d-dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' Recently, in [24], authors have given a very important contribution consisting on writing the Sturm-Liouville operator defining ball prolate spheroidal wave functions in a suitable form allowing to preserve the key features of the one-dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' More precisely, they expressed the Sturm-Liouville operator of interest as a perturbation to the order c2∥x∥2 of the one defining the ball polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' Thus, we have all ingredients to develop spectral methods relative to the study of prolate spheroidal wave functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' We should mention here that whereas a more general context has been considered in [24], the aim of this work is to give some refined bounds of the eigenvalues and eigenfunctions of the integral operator and to establish some other properties of the ball PSWFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' The second purpose of this work is to study the quality of approximation of almost band- limited functions by ball prolate spheroidal wave functions series expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' In spite of their important properties, we can’t handle ball PSWFs in a straightforward way because there is no explicit formula to compute them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' That is why one classical scheme is to compute ex- plicitly their coefficients in terms of ball polynomials basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' Then, it is convenient to develop almost band-limited functions directly in the base of ball polynomials and see what happens with the quality of approximation in this framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' Let us now be a little more specific, ball prolate spheroidal wave functions (ψ(m,c) k,ℓ ) are defined as solutions of the following concentration problem Find f = arg max f∈Bc � Bd |f(x)|2dx � Rd |f(x)|2dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' Here Bc := {f ∈ L2(Rd) : ˆf(u) = 0 ∀u ̸∈ Bd(0, c)}, Bd(0, c) := {x ∈ Rd : ∥x∥ ≤ c} and Bd := Bd(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' The solutions of this problem are eigenfunctions of the finite Fourier transform given by Fc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content='f(x) = � Bd e−icf(y)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' and then, by commutativity, eigenfunctions of Lc,x = −∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' (1 − ∥x∥2)∇ − ∆0 + c2∥x∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' Namely, Fc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content='ψ(m,c) k,ℓ = µn(c)ψ(m,c) k,ℓ , Lc,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content='ψ(m,c) k,ℓ = χ(m) n (c)ψ(m,c) k,ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' Note that by the form under which this last differential operator is given, by Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' in [24], the ball PSWFs extend the orthogonal ball polynomials (c=0) P (m) k,ℓ (x) = �P (0,m+ d 2 −1) k (2∥x∥2−1)Y m l (ˆx), x ∈ Bd, 1 ≤ l ≤ 2m + d − 2 m �m + d − 3 m − 1 � , k, m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' BALL PROLATE SPHEROIDAL WAVE FUNCTIONS 3 and also provide a Bouwkamp spectral algorithm for the computation of ball PSWFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' Our first result is an estimation of ���ψ(m,c) k,ℓ ��� ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' Theorem A : Let c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' For any integer k such that χ(m) k > max{ c2 + 8 (2m + d), (2/3)6 � 2π m + d 2 − 1 �2 + 4(m + d 2)(m + d 2 − 1) − 4 + c2} + m(m + d), we have max x∈Bd ���ψ(m,c) k,ℓ (x) ��� ≤ 3 � 3 � m + d 2 − 1 � 2 � N(d, m) Ωd−1 � χ(m) k (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' As mentioned before, and as application of this first part, we will give the quality of approx- imation by ball PSWFs and by ball polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' We should mention here that this question has been solved in the one-dimensional case in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' At first, let us define the concept of almost-band-limited function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' Let c > 0, Ω = Bd(0, c) and ǫΩ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' A function f ∈ L2(Rd) is said to be ǫΩ-band-limited function if � ∥x∥>c | ˆf(x)|2dx ≤ ǫ2 Ω∥f∥2 L2(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' The approximation of almost band-limited functions by ball prolate spheroidal functions and by ball polynomials are given by the two following theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' Theorem B : Let f ∈ L2(Rd) be an ǫΩ-band-limited function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' Then for any positive integer N ≥ ec 4 , we have ∥f − SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content='f∥L2(Bd) ≤ � 2ǫΩ + Cm,d|µN(c)| � χ(m) N (c) �1/2� ∥f∥L2(Rd), where Cm,d = 3 2 � c (4π)1/4 �d � 3(m + d 2 − 1) d 2 + 1 and SNf is the orthogonal projection of a func- tion f ∈ L2(Rd) on the space spanned by the N +1 first ball prolate spheroidal wave functions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' Theorem C : Let f ∈ L2(Rd) be an ǫΩ band-limited function with Ω = B(0, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' Then, for any positive integer N ≥ ec−m− d+1 2 2 , we have ∥f − ΠN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content='f∥L2(Bd) ≤ \uf8eb \uf8ed2ǫΩ + CN � ec 2(N + 1) + m + d+1 2 �2(N+1)+m+ d+1 2 \uf8f6 \uf8f8 ∥f∥L2(Rd), where CN,m,d = 1 22N+m+ d 2 +3� ec(4N + 3m + d + 4) \uf8ee \uf8ef\uf8ef\uf8f01 + 1 4 ln � ec 2N+m+2+ d+1 2 � \uf8f9 \uf8fa\uf8fa\uf8fb 1/2 and ΠNf is the orthogonal projection of a function f ∈ L2(Rd) on the space spanned by the N +1 first 4 AHMED SOUABNI ball polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' The remainder of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' Section 2 is devoted to some preliminary results that will be useful afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' In section 3, we give some spectral properties of ball prolate spheroidal wave functions, namely the behaviour of the eigenvalues of the associated integral operator and some local estimates giving an upper bound of ���ψ(m,c) k,ℓ ��� ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' We conclude, in section 4, by the quality of approximation in the ball PSWFs basis comparing with the ball polynomials one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' Mathematical preliminaries about some special functions In this section, we recall some important properties about some special functions, mainly, the ball polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' For this purpose, we introduce some preliminaries about spherical harmonics which appear in the definition of ball polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' Furthermore, we recall some properties about Bessel functions which will be frequently used throughout the forthcoming sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' Bessel functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' For α > − 1 2, the Bessel functions Jα are the bounded solutions of the ordinary differential equation given by, (see for example [22]), x2y′′ + xy′ + (x2 − α2)y = 0, x > 0, which is equivalent to : (xy′)′ + � x − α2 x � y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' Many recurrence relations for Bessel function of the first kind exist in literature (we refer reader for example to [22]), among which we cite (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content='1) d dx[xαJα(x)] = xαJα−1(x), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content='2) d dx[x−αJα] = −x−αJα−1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' Bounds and local estimates of Jα are frequently used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' A first simple and useful local estimate is given by, see [15] sup x≥0 √x|Jα(x)| ≤ cα, with cα = � � 2/π if |α| ≤ 1/2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content='675 � α1/3 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content='9 α1/3 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content='1 α if α > 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' A second well known estimate of the Bessel function is given in [[16] p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content='227] by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content='3) ���Jν(x) xν ��� ≤ 1 2νΓ(ν + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' An other estimate of the Bessel function, when the argument is less than the order, is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content='4) 1 ≤ Jν(νx) xνJν(ν) ≤ eν(1−x) ν > 0 and 0 < x ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' BALL PROLATE SPHEROIDAL WAVE FUNCTIONS 5 In [17], author has given a more precise inequality where it has been shown that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content='5) exp �ν2(1 − x2) 4ν + 4 � ≤ Jν(νx) xνJν(ν) ≤ exp �ν2(1 − x2) 2ν + 4 � ν > 0 and 0 < x ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' The following inequality gives us a lower bound of Jν(ν) (we refer reader to [5]) Jν(ν) ≥ Γ(1/3) 22/331/6π(ν + α0)1/3 α0 ∼= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content='0943498 Thus, by combining the last two inequalities, one gets (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content='6) Jν(νx) ≥ Γ(1/3) 22/331/6π(ν + α0)1/3 xν exp �ν2(1 − x2) 4ν + 4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' (ν > 0 and 0 < x ≤ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' The spherical Bessel functions are defined as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content='7) j(α) n,c (x) = � 2(2n + α + 1)J2n+α+1(cx) √cx , x ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' This later set of functions satisfy the orthogonality relation, � +∞ 0 j(α) n,c (x)j(α) m,c(x)x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' = δn,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' Recall that the Hankel transform of a function f ∈ L2(0, ∞) is given by Hα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content='f(x) := � ∞ 0 √xyJα(xy)f(y)dy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' α > −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' The Hankel transform of the spherical Bessel functions are given by, see for example [20] Hα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content='j(α) n,c (x) = � 2(2n + α + 1) c �x c �α+ 1 2 P (α,0) n � 1 − 2 �x c �2� χ[0,c](x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content='8) By noticing that the Hankel transform is an involution, one can write (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content='8) in a more suitable form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content='9) � 1 0 yα+1Jα(cxy)P (0,α) n (2y2 − 1)dy = (−1)n J2n+α+1(cx) cx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' Spherical harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' Let Rd be the d-dimensional Euclidean space, x will denote the column vector (x1, · · · xd)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' We will denote the inner product over Rd, for x, y ∈ Rd, by < x, y >:= d � i=1 xiyi and ∥x∥ will denote the Euclidean associated norm ∥x∥ := √< x, x > = � x2 1 + · · · + x2 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' We also introduce its polar spherical coordinates (r := ∥x∥, ˆx := x r ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' The unit sphere Sd−1 and the unit ball Bd of Rd are denoted respectively by Sd−1 := {ˆx ∈ Rd : ∥ˆx∥ = 1}, Bd := {x ∈ Rd : ∥x∥ ≤ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' The inner product of L2(Sd−1) is defined as < f, g >Sd−1:= � Sd−1 f(ˆx)g(ˆx)dσ(ˆx), 6 AHMED SOUABNI where dσ is the surface measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' Let Hd n be the space of harmonic homogeneous polynomials of degree n and N(d, n) := dim Hd n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' It is well known that N(d, n) = 2n + d − 2 n �n + d − 3 n − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' Note that the radial and the angular dependence of a function Hn ∈ Hd n can be separated : Hn(x) = Hn(rˆx) = rnHn(ˆx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' A spherical harmonic of degree n denoted Yn(ˆx) is a harmonic homogeneous polynomial of degree n in d variables restricted to the unit (d − 1)-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' It is well known that the spherical harmonics satisfy ∆0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content='Yn = −n(n + d − 2)Yn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE_T4oBgHgl3EQfyxzf/content/2301.08320v1.pdf'} +page_content=' In other words, Yn are eigenfunctions of the angular part of the Laplace operator given by ∆0 = � 1≤j 0 is an arbitrary time horizon and 𝑄𝑇 := (0,𝑇) × Ω as well as Σ𝑇 = (0,𝑇) × 𝜕Ω denote the +space-time cylinder and its lateral boundary, respectively. The boundary is decomposed into two +parts: 𝜕Ω𝐷 representing the exits and 𝜕Ω𝑊 the part where the domain is constrained by walls. For +theoretical purposes (regularity of solutions) we assume 𝜕Ω𝐷 ∩ 𝜕Ω𝑊 = ∅ meaning that both boundary +parts are separated from each other, see Figure 4.1. In a similar way we defne Σ𝐷 = (0,𝑇) × 𝜕Ω𝐷 and +Σ𝑊 = (0,𝑇) × 𝜕Ω𝑊 . +The unknown variables in our system of equations are the density of the crowd 𝜌 : 𝑄𝑇 → R+, a +potential specifying the current time to escape 𝜙 : 𝑄𝑇 → R. In addition, there are 𝑀 agents which may +infuence the motion of the crowd via attractive forces. Their positions are denoted by 𝑥𝑖 : (0,𝑇) → R2, +𝑖 = 1, . . . , 𝑀. In addition, each agent is able to regulate the strength by which it acts on the crowd. This +is encoded in the intensities 𝑐𝑖 : (0,𝑇) → R+, 𝑖 = 1, . . . , 𝑀. Both the agent trajectories and interaction +strength are summarized in a vector 𝒙 = (𝑥1, . . . ,𝑥𝑛)⊤ and 𝒄 = (𝑐1, . . . ,𝑐𝑛)⊤, respectively. +The mathematical equations describing the movement of a pedestrian crowd infuenced by agents +then read as follows. For given agent movement directions 𝒖 = (𝑢1, . . . ,𝑢𝑀)⊤ with 𝑢𝑖 ∈ 𝐿∞(0,𝑇; R2) +the unknowns 𝜌,𝜙, 𝒙 are related to each other by means of +𝜕𝑡𝜌 − ∇ · �𝜌 𝛽(𝜌,𝜙, 𝒙, 𝒄)� = 𝜀 Δ𝜌 +in 𝑄𝑇, +(1.1a) +−𝛿1 Δ𝜙 + |∇𝜙|2 = +1 +𝑓 (𝜌)2 + 𝛿2 +in 𝑄𝑇, +(1.1b) +�𝑥𝑖(𝑡) = 𝑓 �𝜌(𝑡,𝑥𝑖(𝑡))� 𝑢𝑖(𝑡) +for 𝑡 ∈ (0,𝑇), +𝑖 = 1, . . . , 𝑀. +(1.1c) +Moreover, we impose the boundary conditions +−�𝜀 ∇𝜌 + 𝜌 𝛽(𝜌,𝜙, 𝒙, 𝒄)� · 𝑛 = 𝛾 𝜌, +𝜙 = 0 +on ΣD, +�𝜀 ∇𝜌 + 𝜌 𝛽(𝜌,𝜙, 𝒙, 𝒄)� · 𝑛 = 0, +∇𝜙 · 𝑛 = 0 +on ΣW, +(1.2) +as well as the initial conditions +𝜌(0, ·) = 𝜌0 +in Ω, +𝑥𝑖(0) = 𝑥𝑖,0 +for 𝑖 = 1, . . . , 𝑀. +(1.3) +Here, 𝜀,𝛿1,𝛿2 > 0 are regularization parameters and the corresponding terms in the system are needed +to guarantee a certain regularity for the solution, see Theorem 2.1. The domain Ω is sufciently large +such that 𝑥𝑖(𝑡) ∈ Ω on [0,𝑇] for 𝑖 = 1, . . . , 𝑀 and 𝑡 ∈ [0,𝑇] if |𝑢𝑖(𝑡)| ≤ 1. +Let us briefy discuss the meaning of the respective terms: Equation (1.1a) states that pedestrians are +transported according to the velocity feld 𝛽, see (2.1) below, while also performing (little) random +motion encoded by the Laplacian of 𝜌. The second equation (1.1b) is a modifed and regularized Eikonal +equation whose solution is the distance to the closest exit, mitigating areas of high density via the +term on the right-hand side. Here, the additional difusion accounts for the fact that pedestrians do not +know their environment exactly. Then, (1.1c) governs the motion of the agents, whose speed is also +infuenced by the surrounding pedestrian density. The function 𝑓 : [0, 1] → [0, 1] is a density-velocity +rule, chosen in such a way that 𝑓 (𝜌) determines the maximum velocity an individual can move if the +density in its current position is 𝜌. We choose 𝑓 to be monotonically decreasing meaning that higher +2023-01-09 +cbna +page 2 of 23 + +J. Pietschmann, A. Stötzner and M. Winkler +Pedestrian dynamics +densities lead to slower movements. The velocity feld 𝛽 will refect the fact that pedestrians are, on +the one hand, trying the minimize their exit time which amounts to a drift term in the direction of +∇𝜙 and on the other hand, they are attracted by the agents which is realized by additional attractive +potentials whose center depends on the agents’ positions 𝒙. This results in a velocity which is the sum +of two terms. Furthermore, to account for the efect that the velocity will deteriorate in regions of +high density, it will be modifed by an additional multiplicative factor 𝑓 (𝜌). As in the equations for the +motion of the agents, 𝑓 is monotonically decreasing and becomes zero at a given maximal density. +The boundary conditions (1.2) allow for an outfow with velocity 𝛾 on parts of the boundary (Σ𝐷) while +no-fux conditions on the remaining parts are to be interpreted as walls (Σ𝑊 ). A detailed description of +the involved non-linearities will be given in the next section, but we also refer to Herzog, Pietschmann, +Winkler, 2020 for more details on the model and the regularizing terms. +Analytical properties of the unregularized Hughes’ model introduced in Hughes, 2002 (i.e. 𝜀 = 𝛿1 = +𝛿2 = 0 in (1.1)), without control, are difcult because of the low regularity of ∇𝜙 that appears on a set +depending on the solution 𝜌 of the frst equation, but see Amadori, Goatin, Rosini, 2014; Amadori, +Di Francesco, 2012; El-Khatib, Goatin, Rosini, 2013. Thus, regularized variants have been considered, +see Di Francesco, Markowich, et al., 2011 for an instance where 𝜀 = 0 but 𝛿1, 𝛿2 ≠ 0. In fact, the result +there is obtained as a vanishing viscosity limit 𝜀 → 0. There is also a number of extensions and +variants of the model, aiming to understand additional properties, make it more realistic, or consider +diferent settings like graphs, see Burger, Di Francesco, et al., 2014; Carrillo, Martin, Wolfram, 2016; +Carlini et al., 2016; Di Francesco, Fagioli, et al., 2017; Colombo, Gokieli, Rosini, 2018. +Control of systems by means of a small number of agents has received lots of interest recently, both +on a discrete level (i.e. one considers a large system of ODEs for the motion of individuals coupled to a +small number of equations for the agents), see Caponigro et al., 2013; Himakalasa, Wongkaew, 2021, +but also for coupled PDE-ODE systems, Albi, Bongini, et al., 2016; Albi, Fornasier, Kalise, 2017. Let us +emphasize that whenever PDEs are coupled to ODEs in such a fashion that the solution of the PDE +needs to evaluated at the solution of the ODE (as in (1.1c)), regularity is needed. While in our case, this +is obtained by the additional difusion in (1.1a), when hyperbolic models for the transport of pedestrian +are considered, an additional regularization in the ODE is needed, see e.g. Borsche, Colombo, et al., +2015; Borsche, Klar, et al., 2014; Borsche, Meurer, 2019. Finally, let us mention that the ODE-ODE and +PDE-ODE perspective are closely related by means of so-called mean feld limits when the number +of agents tends to infnity, see Burger, Pinnau, et al., 2016; Pinnau, Totzeck, 2018 and also the recent +overview on control of crowds, Banda, Herty, Trimborn, 2020. +For the numerical discretization of (1.1a), we employ, as we think of the parameter 𝜀 being small, a +fnite volume scheme for the spatial discretization, which may also be interpreted as a discontinuous +Galerkin scheme. In combination with the Lax-Friedrichs numerical fuxes the scheme is stable and +preserves the bounds 0 ≤ 𝜌 ≤ 1 inherent in our model. Such structure-preserving discretizations of +PDEs gained much attention, e.g., in the context of chemotaxis problems Epshteyn, Kurganov, 2009; +Li, Shu, Yang, 2017; Strehl et al., 2010; Guo, Li, Yang, 2018; Ibrahim, Saad, 2014; Filbet, 2006. The +previously mentioned articles difer also in the choice of the time-stepping scheme. For the treatment +of (1.1) we will use an implicit-explicit fnite diference scheme, whereas the difusion-related terms +are established implicitly and the convection-related terms explicitly. The equation (1.1b) is discretized +with standard linear fnite elements and for (1.1c) we employ a backward Euler scheme. +The paper is organized as follows: In Section 2 we give a precise defnition of our problem and recall +the analytical results from Herzog, Pietschmann, Winkler, 2020. In Section 3 we provide a numerical +2023-01-09 +cbna +page 3 of 23 + +J. Pietschmann, A. Stötzner and M. Winkler +Pedestrian dynamics +discretization scheme in space and time and analyse some of its properties, in particular, we show that +it preserves physical bounds of the density of pedestrians. Corresponding optimization algorithms are +discussed in Section 3.5 and Section 4 fnally provides the results of our numerical experiments. +2 The continuous optimal control problem +Let us motivate the remaining quantities arising in the system of equations. The function 𝑓 : [0, 𝜌max] → +R is a density-velocity relation and is assumed to be 𝑊 3,∞(R) ∩ 𝐶𝑐(R) with 𝑓 (0) = 1 and 𝑓 (𝜌max) = 0 +with 𝜌max denoting the maximal density. A usual choice is +𝑓 (𝜌) = 𝜉 +� +1 − +𝜌 +𝜌max +� +with a cut-of function 𝜉 ∈ 𝐶∞ +𝑐 (−1, 2) satisfying 𝜉 ≡ 1 on (0, 1). Obviously, a higher density leads to a +lower velocity. Throughout the article we set 𝜌max = 1. The movement direction of the crowd described +by the function −𝛽(𝜌,𝜙, 𝒙, 𝒄) is modelled as follows. The primary interest of the crowd is to move +either towards the closest emergency exit, this is the direction −∇𝜙. This is mitigated by the attraction +of close by agents which is the direction −∇𝜙𝐾, where is an agent potential defned by +𝜙𝐾 (𝒙, 𝒄;𝑡,𝑥) � +𝑀 +∑︁ +𝑖=1 +𝑐𝑖(𝑡) 𝐾 �𝑥 − 𝑥𝑖(𝑡)�. +Here, 𝐾(𝑥) = 𝑘(|𝑥|), 𝑘 ∈ 𝑊 3,∞(R) is a radially symmetric function and 𝑐𝑖 ∈ 𝐻 1(0,𝑇), 𝑖 = 1, . . . , 𝑀, are +time-dependent intensity functions. Typical choices for attractive agent potentials are either the bumb +function +𝑘(𝑟) = +� +exp +� +− +𝑅2 +𝑅2−𝑟 2 +� +, +if 𝑟 < 𝑅, +0, +otherwise +with attraction radius 𝑅 > 0, or the Morse potential +𝑘(𝑟) = e−2𝑎 (𝑟−𝑟𝑎) − 2 e−𝑎 (𝑟−𝑟𝑎) +with certain parameters 𝑎,𝑟𝑎 > 0, realizing a repulsion in the near and an attraction in the far range of +the agents. This is useful to avoid a high density very close to the respective agent. We refer to Carillo, +Huang, Martin, 2014 for a more detailed discussion on potentials in the context of focking problems. +These considerations yield a velocity feld defned as follows +𝛽(𝜌,𝜙, 𝒙, 𝒄) = 𝑣0𝑓 (𝜌) ℎ(∇(𝜙 + 𝜙𝐾 (𝒙, 𝒄))) +with +ℎ(𝑥) = min 𝜀{1, |𝑥|} 𝑥 +|𝑥|, +(2.1) +where ℎ is a smoothed projection into the unit ball in R2 and the factor 𝑓 (𝜌) again links the allowed +movement speed to the current density, this is, |𝛽(𝜌,𝜙, 𝒙, 𝒄)| ≤ 𝑣0𝑓 (𝜌) in 𝑄𝑇 . +We briefy introduce the function spaces used in the sequel, see also Denk, Hieber, Prüss, 2007. For a +domain Ω ⊂ R2 we denote by𝑊 𝑘,𝑝(Ω), 𝑘 ∈ N0, 𝑝 ∈ [1, ∞], the usual Sobolev spaces and by𝑊 𝑘−1/𝑝,𝑝(Γ) +for 𝑘 ≥ 1 the corresponding trace spaces which may be equipped with the Sobolev-Slobodetskij norm. +2023-01-09 +cbna +page 4 of 23 + +J. Pietschmann, A. Stötzner and M. Winkler +Pedestrian dynamics +Furthermore, we write 𝐻𝑘 (Ω) = 𝑊 𝑘,2(Ω). Special spaces incorporating already boundary conditions +are 𝐻 1 +𝐷 (Ω) = {𝑣 ∈ 𝐻 1(Ω) : 𝑣|𝜕ΩD ≡ 0} and 𝑊 2,𝑝 +DN (Ω) := {𝑣 ∈ 𝑊 2,𝑝(Ω) : 𝑣|𝜕Ω𝐷 = 0, 𝜕𝑛𝑣|𝜕Ω𝑊 = 0}. For +time-dependent functions 𝑣 : [0,𝑇] → 𝑋 for some Banach space 𝑋 we defne +𝐿𝑝(0,𝑇;𝑋) := {𝑣 : (0,𝑇) → 𝑋 | +∫ 𝑇 +0 +∥𝑣(𝑡)∥𝑝 +𝑋 d𝑡 < ∞}, +𝑝 ∈ [1, ∞), +as well as +𝑊 𝑠,𝑝(0,𝑇;𝑋) := {𝑣 : (0,𝑇) → 𝑋 | 𝜕ℓ +𝑡𝑣 ∈ 𝐿𝑝(0,𝑇;𝑋), 0 ≤ ℓ ≤ 𝑠}, +𝑠 ∈ N0, 𝑝 ∈ [1, ∞). +For the application we have in mind the following spaces +𝑊 𝑟,𝑠 +𝑝 (𝑄𝑇 ) � 𝐿𝑝(0,𝑇;𝑊 𝑟,𝑝(Ω)) ∩𝑊 𝑠,𝑝(0,𝑇; 𝐿𝑝(Ω)), +𝑝 ∈ [1, ∞), +𝑟,𝑠 ∈ N0, +are of interest which are equipped with the natural norms +∥𝑣∥𝑊 𝑟,𝑠 +𝑝 +(𝑄𝑇 ) := +� +∥𝑣∥𝑝 +𝐿𝑝 (0,𝑇;𝑊 𝑟,𝑝 (Ω)) + ∥𝑣∥𝑝 +𝑊 𝑠,𝑝 (0,𝑇;𝐿𝑝 (Ω)) +�1/𝑝 +. +Spaces with non-integral 𝑟 and 𝑠 are defned as interpolation spaces. +In a previous work, Herzog, Pietschmann, Winkler, 2020, a global (in time) well-posedness and +regularity result for (1.1)–(1.3) was established. Furthermore, optimality conditions for related optimal +control problems where this system occurs as a constraint were derived. For convenience of the reader +we briefy summarize the most important results needed in the present article. +First, there holds the following existence and regularity result: +Theorem 2.1. Assume that 𝜌0 ∈ 𝑊 3/2,4(Ω) and fx 𝑇 > 0. Given arbitrary agent movement directions +𝒖 = (𝑢1, . . . ,𝑢𝑀)ᵀ ∈ 𝐿∞(0,𝑇; R2)𝑀 and intensities 𝒄 = (𝑐1, . . . ,𝑐𝑀) ∈ 𝐻 1(0,𝑇)𝑀, there exists a unique +strong solution (𝜌,𝜙, 𝒙) to (1.1)–(1.3) which satisfes, for any 2 < 𝑝 < ∞, 𝜌 ∈ 𝑊 2,1 +𝑝 (𝑄𝑇 ) and 𝜙 ∈ +𝐿∞(0,𝑇;𝑊 2,𝑝(Ω)). The agent trajectories 𝑥𝑖, 𝑖 = 1, . . . , 𝑀 are absolutely continuous. Moreover, the +a priori estimate +∥𝜌∥𝑊 2,1 +𝑝 (𝑄𝑇 ) + ∥𝜙∥𝐿∞(0,𝑇;𝑊 2,𝑝 (Ω)) ≤ 𝐶∥𝜌0∥𝑊 1,𝑝 (Ω), +holds with 𝐶 > 0 depending only on the domain, the bounds for the coefcients and the respective kernel. +The previous result allows to defne an operator, the so-called control–to–state operator, +𝑆 : Q → Y, +𝒒 ≔ (𝒖, 𝒄) ↦→ 𝑆(𝒒) = 𝒚 ≔ (𝜌,𝜙, 𝒙) +with control and state spaces +Q � U × C := 𝐿∞(0,𝑇; R2)𝑀 × 𝐻 1(0,𝑇)𝑀, +Y � 𝑊 2,1 +𝑝 (𝑄𝑇 ) × +� +𝐿∞(0,𝑇;𝑊 2,𝑝 +DN (Ω)) ∩𝑊 1,𝑝(0,𝑇;𝑊 1,𝑝(Ω)) +� +×𝑊 1,𝑠(0,𝑇; R2)𝑀 +for 𝑠 = +� +1 +2 − 1 +𝑝 +�−1 +. Furthermore we defne the set of admissible controls +Qad := {(𝒖, 𝒄) ∈ Q : |𝑢𝑖(𝑡)| ≤ 1, 0 ≤ 𝑐𝑖(𝑡) ≤ 1 f.a.a. 𝑡 ∈ (0,𝑇) and all 𝑖 = 1, . . . , 𝑀}. +2023-01-09 +cbna +page 5 of 23 + +J. Pietschmann, A. Stötzner and M. Winkler +Pedestrian dynamics +The optimal control problem we study in this article reads +Minimize +J (𝒚; 𝒒) � +∫ +� +𝑄𝑇 +𝑒𝜈 𝑡 𝜌(𝑡,𝑥) d𝑥 d𝑡 − 𝜇 +𝑀 +∑︁ +𝑖=1 +∫ 𝑇 +0 +ln(𝜉(𝑥𝑖(𝑡))) d𝑡 ++ 𝛼1 +2𝑇 +𝑀 +∑︁ +𝑖=1 +∥𝑢𝑖∥2 +𝐻 1(0,𝑇;R2) + 𝛼2 +2𝑇 +𝑀 +∑︁ +𝑖=1 +∥𝑐𝑖∥2 +𝐻 1(0,𝑇) +(2.2a) +subject to +𝒚 := (𝜌,𝜙, 𝒙) = 𝑆(𝒒), +(2.2b) +𝒒 := (𝒖, 𝒄) ∈ Qad. +(2.2c) +The objective functional J aims at a fast evacuation of the crowd. By the factor 𝑒𝜈 𝑡 higher densities +at a later time are penalized more. We observe the density in a subregion � +𝑄𝑇 = 𝐼 × �Ω where �Ω ⊂ Ω is +a subregion which the pedestrians must leave. We use the temporal 𝐻 1-norm of the agent movement +directions and the intensities as a regularization to guarantee the smoothness required in Theorem 2.1. +The regularization parameters 𝛼1, 𝛼2 > 0 are arbitrary but positive. +The fourth term in the objective is a barrier used to avoid that the agents walk through walls. The +barrier function 𝜉 ∈ 𝐻 1 +𝐷 (Ω) is the weak solution of the singularly perturbed problem +−𝛿4Δ𝜉 + 𝜉 = 1 +in Ω, +(2.3a) +𝜉 = 0 +on 𝜕Ω. +(2.3b) +The barrier function − ln(𝜉(𝑥𝑖(𝑡))) tends to infnity if dist(𝑥𝑖(𝑡), 𝜕Ω) → 0 for some 𝑡 ∈ [0,𝑇]. For +𝑥𝑖(𝑡) ∈ int Ω there holds lim𝜇→0 𝜇 ln(𝜉(𝑥𝑖(𝑡))) = 0. Here we choose 𝜇 > 0 to be fxed but small. +The control constraint (𝒖, 𝒄) ∈ Qad guarantees that the agents do not move faster than the density in +their current position allows and that the intensity is bounded by reasonable values. +We have the following well-posedness result and necessary optimality condition. +Theorem 2.2. There exists at least one global solution (𝒚, 𝒒) ∈ Y × Qad of (2.2). +Furthermore, each local minimizer (𝒚, 𝒒) ∈ Y × Qad, 𝒚 = (𝜌,𝜙, 𝒙), 𝒒 = (𝒖, 𝒄), of (2.2) fulflls for all +directions in the tangential cone at (𝒖, 𝒄), namely 𝛿𝒒 = (𝛿𝒖,𝛿𝒄) ∈ TQad(𝒖, 𝒄), +∫ +� +𝑄𝑇 +𝑒𝜈 𝑡 𝛿𝜌(𝑡,𝑥) d𝑥 d𝑡 + 𝛼1 +𝑇 (𝒖 , 𝛿𝒖)𝐻 1(0,𝑇;R2)𝑀 + 𝛼2 +𝑇 (𝒄 , 𝛿𝒄)𝐻 1(0,𝑇)𝑀 +− 𝜇 +𝑀 +∑︁ +𝑖=1 +∫ 𝑇 +0 +∇𝜉(𝑥𝑖(𝑡))⊤ 𝛿𝑥𝑖(𝑡) +𝜉(𝑥𝑖(𝑡)) +d𝑡 ≥ 0, +with 𝒚 = 𝑆(𝒖, 𝒄) and 𝛿𝒚 = (𝛿𝜌,𝛿𝜙,𝛿𝒙) = 𝑆 ′(𝒖, 𝒄) (𝛿𝒖,𝛿𝒄) characterized by the system +𝜕𝑡𝛿𝜌 − 𝜀Δ𝛿𝜌 − ∇ · +� +𝛿𝜌 𝛽(𝜌,𝜙, 𝒙, 𝒄) + 𝜌 +� 𝜕𝛽(𝜌,𝜙, 𝒙, 𝒄) +𝜕(𝜌,𝜙, 𝒙, 𝒄) (𝛿𝜌,𝛿𝜙,𝛿𝒙,𝛿𝒄) +�� += 0, +(2.4a) +2023-01-09 +cbna +page 6 of 23 + +J. Pietschmann, A. Stötzner and M. Winkler +Pedestrian dynamics +−𝛿1 Δ𝛿𝜙 + 2∇𝜙 · ∇𝛿𝜙 + 2𝑓 (𝜌) 𝑓 ′(𝜌) +(𝑓 2(𝜌) + 𝛿2)2𝛿𝜌 = 0, +(2.4b) +�𝛿𝑥𝑖 − 𝑣0 𝑓 ′(𝜌(·,𝑥𝑖)) �∇𝜌(·,𝑥𝑖)ᵀ𝛿𝑥𝑖 + 𝛿𝜌(·,𝑥𝑖)� 𝑢𝑖 = 𝑣0 𝑓 (𝜌(·,𝑥𝑖)) 𝛿𝑢𝑖, +(2.4c) +for 𝑖 = 1, . . . , 𝑀, together with the boundary conditions (1.2) and homogeneous initial conditions +𝛿𝜌(0, ·) = 0 +and +𝛿𝑥𝑖(0) = 0, +𝑖 = 1, . . . , 𝑀. +(2.5) +The proof of the theorem above is very close to those of Theorem 3.8 and Theorem 4.4 Herzog, +Pietschmann, Winkler, 2020. The main diference is that the model in Herzog, Pietschmann, Winkler, +2020 only allows to control the velocity 𝒖 of the agents, yet not their strength 𝒄. As for the existence +proof, this does not impose any additional difculty due to the uniform 𝐿∞-boundedness of 𝒄 as a +consequence of the embedding 𝐻 1 ↩→ 𝐿∞ in one spatial dimension. For the diferentiability result, one +has to add the derivatives with respect to 𝒄, yielding an additional term in (2.4a) that, however, can be +estimated similarly to the remaining terms. +3 Discretization of the state equation +In this section we introduce the numerical scheme used to compute approximate solutions of the +forward system (1.1)–(1.3). To this end, we introduce a semi-implicit time-stepping scheme and use a +fnite volume discretization for the density function 𝜌 and continuous Lagrange fnite elements for the +potential function. +3.1 Space discretization +For the spatial discretization of the system (1.1)–(1.3) we defne a family of geometrically conforming +triangular meshes {Tℎ}ℎ>0. For each 𝑇 ∈ Tℎ we denote by ℎ𝑇 = diam(𝑇) the element diameter and +by 𝜌𝑇 the diameter of the largest inscribed ball in 𝑇. The mesh parameter is then ℎ = max𝑇 ∈Tℎ ℎ𝑇 . +The mesh family is assumed to be shape regular meaning that there is a constant 𝜅 > 0 such that +ℎ𝑇 /𝜌𝑇 ≤ 𝜅 for all 𝑇 ∈ Tℎ and all ℎ > 0. By F i +ℎ we denote the set of interior element edges, by F bd +ℎ +the +boundary edges and write Fℎ := F i +ℎ ∪ F bd +ℎ . Furthermore, to each edge 𝐹 ∈ Fℎ we associate the normal +vector 𝑛𝐹 which is pointing outward in case of a boundary edge and has a fxed orientation in case of +an interior edge. +We propose a fnite volume scheme for the transport equation. As we use the fnite element package +FEniCS for our implementation, we use a notation which is rather usual for discontinuous Galerkin +discretizations, see Di Pietro, Ern, 2012 for an overview. The fnite-dimensional function spaces are +defned by +𝑉ℎ = {𝑣 ∈ 𝐿∞(Ω) : 𝑣|𝑇 ∈ P0(𝑇) for all 𝑇 ∈ Tℎ, }, +𝑊ℎ = {𝑣 ∈ 𝐶(Ω) : 𝑣|𝑇 ∈ P1(𝑇) for all 𝑇 ∈ Tℎ}, +𝑊ℎ,D := 𝑊ℎ ∩ 𝐻 1 +𝐷 (Ω), +2023-01-09 +cbna +page 7 of 23 + +J. Pietschmann, A. Stötzner and M. Winkler +Pedestrian dynamics +where P𝑘 (𝑇) denotes the space of polynomials on 𝑇 of degree not larger than 𝑘 ∈ N0. For a function +𝑣 : Ω → R, we defne interface averages and jumps in the following way +{𝑣}𝐹 := 1 +2 (𝑣|𝑇1 + 𝑣|𝑇2), +⟦𝑣⟧𝐹 := 𝑣|𝑇1 − 𝑣|𝑇2 , +∀𝐹 ∈ F i +ℎ, +where 𝑇1,𝑇2 ∈ Tℎ are chosen in such a way that 𝑛𝐹 = 𝑛𝜕𝑇1 |𝐹 = −𝑛𝜕𝑇2 |𝐹. +In order to discretize the system (1.1) we use discontinuous approximations for 𝜌 and continuous ones +for 𝜙. The unknowns in our semi-discrete scheme are +𝜌ℎ(𝑡) ∈ 𝑉ℎ, +𝜙ℎ(𝑡) ∈ 𝑊ℎ,D, +𝑥1(𝑡), . . . ,𝑥𝑀 (𝑡) ∈ R2, +𝑐1(𝑡), . . . ,𝑐𝑀 (𝑡) ∈ R +for all 𝑡 ∈ [0,𝑇]. The approximate transport direction is then given by +𝛽ℎ(𝜌ℎ,𝜙ℎ, 𝒙, 𝒄) ≔ 𝑓 (𝜌ℎ) ℎ(∇𝜙ℎ + 𝜙𝐾 (𝒙, 𝒄)) +with +𝜙𝐾 (𝒙, 𝒄;𝑡,𝑥) := +𝑀 +∑︁ +𝑗=1 +𝑐 𝑗 (𝑡) 𝐾(𝑥 − 𝑥𝑖(𝑡)). +The semi-discretization of (1.1a) then reads +Find 𝜌ℎ : [0,𝑇] → 𝑉ℎ with 𝜌ℎ(0) = proj𝑉ℎ (𝜌0) and +(𝜕𝑡𝜌ℎ(𝑡), 𝑣ℎ)Ω + 𝑎(𝜌ℎ(𝑡), 𝑣ℎ) + 𝑏(𝛽ℎ)(𝜌ℎ(𝑡), 𝑣ℎ) = 0 +∀𝑣ℎ ∈ 𝑉ℎ,𝑡 ∈ (0,𝑇). +(3.1) +Here, proj𝑉ℎ : 𝐿2(Ω) → 𝑉ℎ is some projection operator, (·, ·)Ω is the standard 𝐿2(Ω)-inner product and +the bilinear forms are defned by +𝑎(𝜌ℎ, 𝑣ℎ) = 𝜀 +∑︁ +𝐹 ∈Fi +ℎ +∫ +𝐹 +𝜏𝐹 ⟦𝜌ℎ⟧⟦𝑣ℎ⟧ d𝑠 + +∑︁ +𝐹 ∈Fbd +ℎ +𝜒𝜕Ω𝐷𝛾 +∫ +𝐹 +𝜌ℎ 𝑣ℎ d𝑠 +(3.2a) +𝑏(𝛽ℎ)(𝜌ℎ, 𝑣ℎ) = − +∑︁ +𝐹 ∈Fi +ℎ +∫ +𝐹 +(𝜌ℎ 𝛽ℎ)∗ +𝐹 ⟦𝑣ℎ⟧ d𝑠. +(3.2b) +The parameter 𝜏𝐹 is defned by +𝜏𝐹 ≔ +1 +|𝑥𝑇1 − 𝑥𝑇2|, +where 𝑥𝑇 is the intersection of the orthogonal edge bisectors of 𝑇 ∈ Tℎ. The term 𝜏𝐹 ⟦𝜌ℎ⟧⟦𝑣ℎ⟧ with +𝑣ℎ = 𝜒𝑇 for some 𝑇 ∈ Tℎ approximates the difusive fux ∇𝜌ℎ · 𝑛𝜕𝑇 over the edge 𝐹 ⊂ 𝑇. The bilinear +form 𝑏 establishes the convective fux 𝛽 𝜌 · 𝑛𝜕𝑇 . As numerical fux function (·)∗, we choose the +Lax-Friedrichs fux, see Rider, Lowrie, 2002, defned by +(𝜌ℎ 𝛽)∗ +𝐹 = {𝜌ℎ 𝛽}𝐹 · 𝑛𝐹 − 𝜂 +2 ⟦𝜌ℎ⟧𝐹 . +(3.3) +The stabilization parameter 𝜂 ∈ R is specifed later. For the closely related chemotaxis model such an +approach has been sucessfully applied in Li, Shu, Yang, 2017; Guo, Li, Yang, 2018. Of course, also other +fux functions are possible, e.g., the central upwind fux, Epshteyn, Kurganov, 2009. +2023-01-09 +cbna +page 8 of 23 + +J. Pietschmann, A. Stötzner and M. Winkler +Pedestrian dynamics +The Eikonal equation (1.1b) is discretized in space using standard linear Lagrange elements which +yields +𝛿1 (∇𝜙ℎ(𝑡), ∇𝑤ℎ)Ω + �|∇𝜙ℎ(𝑡)|2,𝑤ℎ +� +Ω = +� +1 +𝑓 (𝜌ℎ(𝑡))2 + 𝛿2 +,𝑤ℎ +� +Ω +∀𝑤ℎ ∈ 𝑊ℎ,D. +(3.4) +In our numerical experiments we used the Newton solver integrated in FEniCS. The Jacobian is +established by automatic diferentiation. +The ordinary diferential equations for the agent trajectories (1.1c) depend on a point evaluation +𝜌(𝑡,𝑥𝑖(𝑡)) of a function which is discontinuous in the discrete setting. In particular, this term would +not be diferentiable with respect to 𝑥𝑖(𝑡). As a remedy, we use instead of a point evaluation, see also +Borsche, Colombo, et al., 2015, the following regularization +𝜌ℎ(𝑡,𝑥𝑖(𝑡)) ≈ 𝜂𝑥𝑖 (𝑡) ∗ 𝜌ℎ(𝑡), +with +𝜂𝑥𝑖 (𝑡) := +𝛿𝑥𝑖 (𝑡) +𝛿𝑥𝑖 (𝑡) ∗ 1, +where ∗ stands for the convolution integral 𝛿𝑥0 ∗ 𝑣 = +∫ +Ω 𝛿𝑥0 𝑣 d𝑥 of the functions 𝑣 ∈ 𝐿1(Ω) and some +kernel function 𝛿𝑥0 ∈ 𝐶∞(R2). An obvious choice is the regularized Dirac delta function +𝛿𝑥0(𝑥) := +1 +2 𝜋 𝜁 e− ∥𝑥−𝑥0∥2 +2𝜁 +with small locality parameter 𝜁 > 0. Note that for 𝜁 → 0 there holds 𝛿𝑥0 ∗ 𝑣 → 𝑣(𝑥0) for any 𝑣 ∈ 𝐶(Ω). +Furthermore, the regularized Dirac delta fulflls +∫ +R2 𝛿𝑥0 d𝑥 = 1 for arbitrary 𝑥0 ∈ R2. The discretized +ordinary diferential equation then reads +�𝑥𝑖(𝑡) = 𝑣0 𝑓 �𝜂𝑥𝑖 (𝑡) ∗ 𝜌ℎ(𝑡)� 𝑢𝑖(𝑡), +𝑛 = 1, . . . , 𝑁 +(3.5) +and initial conditions 𝑥𝑖(0) = 𝑥𝑖,0. +3.2 Time discretization +For the temproal discretization we cover the time interval [0,𝑇] by an equidistant grid 𝐼𝜏 � {𝑡𝑛}𝑁 +𝑛=0 +with grid points 𝑡𝑛 := 𝑛 𝜏 and grid size 𝜏 := 𝑇/𝑁. The spatial and temporal discretization parameters +are summarized in 𝜎 = (ℎ,𝜏). Moreover, we defne the space of grid functions +𝐻𝜏 (𝑉 ) = {𝑣 : 𝐼𝜏 → 𝑉 }, +with 𝑉 an arbitrary linear space. If 𝑉 is again a function space containing functions 𝑣 : Ω → R we +write 𝑣(𝑡𝑛) = 𝑣(𝑡𝑛, ·). The functions 𝜌ℎ, 𝜙ℎ, 𝑥𝑖, 𝑢𝑖 and 𝑐𝑖 arising in the semidiscrete equations (3.1), (3.4) +and (3.5) are approximated by grid functions +𝜌𝜎 ∈ 𝐻𝜏 (𝑉ℎ), +𝜙𝜎 ∈ 𝐻𝜏 (𝑊ℎ,D), +𝑢𝑖,𝜎,𝑥𝑖,𝜎 ∈ 𝐻𝜏 (R2), +𝑐𝑖,𝜎 ∈ 𝐻𝜏 (R) +for 𝑖 = 1, . . . , 𝑀. For brevity we write for 𝑛 = 0, . . . , 𝑁 +𝜌𝑛 +ℎ ≔ 𝜌𝜎 (𝑡𝑛, ·), +𝜙𝑛 +ℎ ≔ 𝜙𝜎 (𝑡𝑛, ·), +𝑥𝑛 +𝑖 ≔ 𝑥𝑖,𝜎 (𝑡𝑛), +𝑢𝑛 +𝑖 ≔ 𝑢𝑖,𝜎 (𝑡𝑛) +2023-01-09 +cbna +page 9 of 23 + +J. Pietschmann, A. Stötzner and M. Winkler +Pedestrian dynamics +and for the transport vector we use +𝛽𝑛 +ℎ = 𝛽ℎ(𝜌𝑛 +ℎ,𝜙𝑛 +ℎ, 𝒙𝑛, 𝒄𝑛). +We replace the temporal derivative by a diference quotient and use a semi-implicit time-stepping +scheme, more precisely, the difusion-related terms are evaluated implicitly and the convection-related +terms explicitly. This yields the fully-discrete system +(𝜌𝑛+1 +ℎ , 𝑣ℎ)Ω + 𝜏 𝑎(𝜌𝑛+1 +ℎ , 𝑣ℎ) = (𝜌𝑛 +ℎ, 𝑣ℎ)Ω − 𝜏 𝑏(𝛽𝑛 +ℎ)(𝜌𝑛 +ℎ, 𝑣ℎ), +(3.6a) +𝛿1 +�∇𝜙𝑛 +ℎ, ∇𝑤ℎ +� +Ω + �|∇𝜙𝑛 +ℎ |2,𝑤ℎ +� +Ω = +� +1 +𝑓 (𝜌𝑛 +ℎ)2 + 𝛿2 +,𝑤ℎ +� +Ω +, +(3.6b) +𝑥𝑛+1 +𝑖 +− 𝑥𝑛 +𝑖 = 𝜏 𝑣0 𝑓 +� +𝜂𝑥𝑛+1 +𝑖 +∗ 𝜌𝑛+1 +ℎ +� +𝑢𝑛+1 +𝑖 +, +(3.6c) +for all test functions 𝑣ℎ ∈ 𝑉ℎ, 𝑤ℎ ∈ 𝑊ℎ,D and indices 𝑖 = 1, . . . , 𝑀, 𝑛 = 0, . . . , 𝑁 − 1. Furthermore, the +initial conditions are established by means of: +𝜌0 +ℎ = proj𝑉ℎ (𝜌0), +𝑥0 +𝑖 = 𝑥𝑖,0, 𝑖 = 1, . . . , 𝑀. +Note that the system of equations (3.6) completely decouples and we can compute each variable after +the other, in the following order +𝜌0 +ℎ, 𝒙0 ↦→ 𝜙0 +ℎ ↦→ 𝜌1 +ℎ ↦→ 𝒙1 ↦→ 𝜙1 +ℎ ↦→ . . . ↦→ 𝜌𝑁−1 +ℎ +↦→ 𝒙𝑁−1 ↦→ 𝜙𝑁−1 +ℎ +↦→ 𝜌𝑁 +ℎ ↦→ 𝒙𝑁 . +(3.7) +3.3 Quality of discrete solutions +In this section we study some basic properties for the solutions of (3.6). In particular, it is of inter- +est whether the physical bounds observed for the solution of the continuous system (1.1)–(1.3) are +transferred to the discrete setting. +The basis functions of the fnite element space 𝑉ℎ are denoted by {𝜒𝑇 }𝑇 ∈Tℎ defned by 𝜒𝑇 |𝑇 ′ ≡ 𝛿𝑇,𝑇 ′ +for all 𝑇,𝑇 ′ ∈ Tℎ. Note that by a slight abuse of notation we use the elements of Tℎ as indices here. +Introducing the matrices 𝑀 = (𝑚𝑇,𝑇 ′)𝑇,𝑇 ′∈Tℎ, 𝐴 = (𝑎𝑇,𝑇 ′)𝑇,𝑇 ′∈Tℎ and 𝐵𝑛 = (𝑏𝑛 +𝑇,𝑇 ′)𝑇,𝑇 ′∈Tℎ with entries +𝑚𝑇,𝑇 ′ = +� +|𝑇 |, +if 𝑇 = 𝑇 ′, +0, +otherwise, +(3.8a) +𝑎𝑇,𝑇 ′ = +������ +������ +𝜀 +∑︁ +𝐹 ∈F𝑇 ∩Fi +ℎ +𝜏𝐹 |𝐹 | + 𝛾 +∑︁ +𝐹 ∈F𝑇 ∩Fbd +ℎ +|𝐹 |, +if 𝑇 = 𝑇 ′, +−𝜀 𝜏𝐹 |𝐹 |, +if 𝑇 ≠ 𝑇 ′ and 𝐹 := 𝜕𝑇 ∩ 𝜕𝑇 ′ ≠ ∅, +0, +otherwise, +(3.8b) +𝑏𝑛 +𝑇,𝑇 ′ = +�������� +�������� +− 1 +2 +∑︁ +𝐹 ∈F𝑇 ∩Fi +ℎ +�∫ +𝐹 +𝛽𝑛 +ℎ |𝑇 · 𝑛𝜕𝑇 d𝑠 − 𝜂 |𝐹 | +� +, +if 𝑇 = 𝑇 ′, +− 1 +2 +�∫ +𝐹 +𝛽𝑛 +ℎ |𝑇 ′ · 𝑛𝜕𝑇 d𝑠 + 𝜂 |𝐹 | +� +, +if 𝑇 ≠ 𝑇 ′ and 𝐹 = 𝜕𝑇 ∩ 𝜕𝑇 ′ ≠ ∅, +0, +otherwise, +(3.8c) +2023-01-09 +cbna +page 10 of 23 + +J. Pietschmann, A. Stötzner and M. Winkler +Pedestrian dynamics +allows to rewrite the system of equations (3.6a) as +(𝑀 + 𝜏 𝐴) �𝜌𝑛+1 = (𝑀 − 𝜏 𝐵𝑛) �𝜌𝑛. +(3.9) +Here, �𝜌𝑛, 𝑛 = 0, . . . , 𝑁, are the vector representations of 𝜌𝑛 +ℎ with respect to the basis {𝜒𝑇 }𝑇 ∈Tℎ. Note +that the matrix 𝐵𝑛 depends also on �𝜌𝑛. +Theorem 3.1. The numerical scheme (3.6a) is mass conserving in the following sense. Assuming that +𝛾 = 0 holds, i. e., there are no-fux boundary conditions present at all boundary parts 𝜕ΩD and 𝜕ΩW, the +solution 𝜌𝜎 fulflls +∫ +Ω +𝜌𝑛 +ℎ d𝑥 = +∫ +Ω +proj𝑉ℎ (𝜌0) d𝑥 +∀𝑛 = 0, 1, . . . , 𝑁 . +Proof. The assertion is trivially fulflled for 𝑛 = 0 as the initial condition is established by 𝜌0 +ℎ = +proj𝑉ℎ (𝜌0). In matrix-vector notation the assertion is equivalent to �1⊤𝑀 �𝜌𝑛+1 = �1⊤𝑀 �𝜌𝑛. This follows +from (3.9) after using +�1⊤𝐴 �𝜌𝑛+1 = +∑︁ +𝑇 ∈Tℎ +∑︁ +𝑇 ′∈Tℎ +𝑎𝑇,𝑇 ′𝜌𝑛+1 +𝑇 ′ = 0 +and +�1⊤𝐵𝑛 �𝜌𝑛 = +∑︁ +𝑇 ∈Tℎ +∑︁ +𝑇 ′∈Tℎ +𝑏𝑇,𝑇 ′𝜌𝑛 +𝑇 ′. +In this expression, the stabilization terms (the ones multiplied with 𝜂) cancel out. Furthermore, after +sorting terms in the sum by the edges 𝐹 ∈ F i +ℎ and denoting by 𝑇𝐹,1,𝑇𝐹,2 the two triangles meeting in 𝐹, +we obtain the terms +�1⊤𝐵𝑛 �𝜌𝑛 = − 1 +2 +∑︁ +𝐹 ∈Fi +ℎ +∫ +𝐹 +� +𝛽ℎ|𝑇𝐹,1 𝑛𝜕𝑇𝐹,1 𝜌𝑇𝐹,1 + 𝛽ℎ|𝑇𝐹,2 𝑛𝜕𝑇𝐹,1 𝜌𝑇𝐹,2 ++𝛽ℎ|𝑇𝐹,2 𝑛𝜕𝑇𝐹,2 𝜌𝑇𝐹,2 + 𝛽ℎ|𝑇𝐹,1 𝑛𝜕𝑇𝐹,2 𝜌𝑇𝐹,1 +� +d𝑠 = 0. +The last step follows due to 𝑛𝜕𝑇𝐹,2 = −𝑛𝜕𝑇𝐹,2 which implies �1⊤𝐵𝑛 �𝜌𝑛 = 0. +□ +Theorem 3.2. Choose 𝜂 = 1 in (3.3) and denote by 𝜅 > 0 the maximal aspect ratio of the mesh family +Tℎ, see Section 3.1. Let 𝜏 be chosen to satisfy the CFL condition +𝜏 ≤ +𝜋 +3𝜅2 min +𝑇 ∈Tℎ ℎ𝑇 . +(3.10) +If furthermore, there holds proj𝑉ℎ (𝜌0) ∈ [0, 1] a. e. in Ω and 𝛽𝑛 +ℎ = (1 − 𝜌𝑛 +ℎ) Φ𝑛 with |Φ𝑛| ≤ 1, 𝑛 = 0, . . . , 𝑁, +the solutions of (3.6) fulfll for all 𝑛 = 0, . . . , 𝑁 +𝜌𝑛 +ℎ (𝑥) ∈ [0, 1] +f.a.a. 𝑥 ∈ Ω. +Proof. The diagonal entries of (𝑀 + 𝜏 𝐴) are all positive and the of-diagonal entries are negative. +Moreover, one easily concludes the strict diagonal dominance, this is, +∑︁ +𝑇′∈Tℎ +𝑇′≠𝑇 +|𝑚𝑇,𝑇 ′ + 𝜏 𝑎𝑇,𝑇 ′| < 𝑚𝑇,𝑇 + 𝜏 𝑎𝑇,𝑇 . +2023-01-09 +cbna +page 11 of 23 + +J. Pietschmann, A. Stötzner and M. Winkler +Pedestrian dynamics +This implies that (𝑀 +𝜏 𝐴) is an M-matrix and consequently, the inverse (𝑀 +𝜏 𝐴)−1 exists and fulflls +(𝑀 + 𝜏 𝐴)−1 ≥ 0. +Let 𝑛 ∈ N0 be fxed and assume that 𝜌𝑛 +ℎ (𝑥) ∈ [0, 1] for almost all 𝑥 ∈ Ω. We show 𝜌𝑛+1 +ℎ +≥ 0 by +confrming that the right-hand side of (3.9) has non-negative entries only. Assuming that 𝜌𝑛 +𝑇 ≥ 0, +𝑇 ∈ Tℎ, we show that the entries of (𝑀 − 𝜏 𝐵𝑛) are non-negative as well. The entries on the diagnoal +have the form +|𝑇 | + 𝜏 +2 +∑︁ +𝐹 ∈F𝑇 ∩Fi +ℎ +∫ +𝐹 +(𝛽𝑛 +ℎ |𝑇 · 𝑛𝜕𝑇 − 1) d𝑠 ≥ |𝑇 | − 𝜏 |𝜕𝑇 |, +where the frst step follows from the assumption (2.1) implying |𝛽𝑛 +ℎ | ≤ 1. To estimate further we take +the geometric mesh quantities and the shape regularity ℎ𝑇 ≤ 𝜅 𝜌𝑇 into account and arrive at +|𝑇 | +|𝜕𝑇 | ≥ 𝜋 𝜌2 +𝑇 +3ℎ𝑇 +≥ 𝜋 ℎ𝑇 +3𝜅2 ≥ 𝜏, +where the last step is the CFL condition (3.10). The previous two estimates confrm (𝑚𝑇,𝑇 −𝜏 𝑏𝑛 +𝑇,𝑇 ) ≥ 0 +for all 𝑇 ∈ Tℎ. The remaining entries in the matrix, namely (𝑚𝑇,𝑇 ′ − 𝜏 𝑏𝑛 +𝑇,𝑇 ′) with 𝐹 = 𝜕𝑇 ∩ 𝜕𝑇 ′ ≠ ∅, +have the form +𝜏 +2 +∫ +𝐹 +(𝛽𝑛 +ℎ |𝑇 ′ · 𝑛𝜕𝑇 + 1) d𝑠 ≥ 0. +The non-negativity follows again from |𝛽𝑛 +ℎ | ≤ 1. Combining the previous arguments provides the +lower bound 𝜌𝑛+1 +ℎ +≥ 0. +To show the upper bound we rearrange the equation system (3.9) in the form +(𝑀 + 𝜏 𝐴) (�1 − �𝜌𝑛+1) = 𝑀 (�1 − �𝜌𝑛) + 𝜏 𝐵𝑛 �𝜌𝑛 + 𝜏 𝐴�1. +(3.11) +We may rewrite the transport term using 𝜌𝑛 +ℎ 𝛽𝑛 +ℎ = (1 − 𝜌𝑛 +ℎ) �𝛽𝑛 +ℎ with �𝛽𝑛 +ℎ = 𝜌𝑛 +ℎ Φ. In (3.11) we reformulate +the expression involving 𝐵𝑛 by means of +[𝐵𝑛 �𝜌𝑛]𝑇 = − +∑︁ +𝐹 ∈F𝑇 ∩Fi +ℎ +∫ +𝐹 +� +{(1 − 𝜌𝑛 +𝑇 ) �𝛽𝑛 +ℎ}𝐹 · 𝑛𝐹 + 1 +2 ⟦1 − 𝜌𝑛 +ℎ⟧ +� +𝐹 +⟦𝜒𝑇 ⟧𝐹 d𝑠 += − 1 +2 +∑︁ +𝐹 ∈F𝑇 ∩Fi +ℎ +∫ +𝐹 +(�𝛽𝑛 +ℎ |𝑇 · 𝑛𝜕𝑇 + 1) d𝑠 · (1 − 𝜌𝑛 +𝑇 ) +− 1 +2 +∑︁ +𝐹 ∈F𝑇 ∩Fi +ℎ +∫ +𝐹 +(�𝛽𝑛 +ℎ |𝑇𝐹 · 𝑛𝜕𝑇 − 1) d𝑠 · (1 − 𝜌𝑛 +𝑇𝐹 ) +=: [�𝐵𝑛(1 − �𝜌𝑛)]𝑇 . +with 𝑇𝐹 ∈ Tℎ \ {𝑇 }, 𝑇𝐹 ∩𝑇 = 𝐹. With the same arguments like above one can show that the entries of +𝑀 +𝜏 �𝐵𝑛 are non-negative and together with 1 − �𝜌𝑛 ≥ 0, 𝐴�1 ≥ 0 and the M-matrix property of 𝑀 +𝜏 𝐴 +we arrive at the desired bound 1 − �𝜌𝑛+1 ≥ 0. +□ +3.4 The discrete optimal control problem +Next, we study a discrete version of the optimal control problem (2.2). The control and state variables +are grid functions in time and thus, we introduce the discrete 𝐻 1(0,𝑇)-inner product for functions +2023-01-09 +cbna +page 12 of 23 + +J. Pietschmann, A. Stötzner and M. Winkler +Pedestrian dynamics +𝑢𝜏, 𝑣𝜏 : {𝑡𝑛}𝑁 +𝑛=0 → 𝑉 , with 𝑉 some Hilbert space, +(𝑢𝜏, 𝑣𝜏)𝐻 1(0,𝑇;𝑉 ),𝜏 := 𝜏 +𝑁 +∑︁ +𝑛=0 +(𝑢𝑛, 𝑣𝑛)𝑉 ×𝑉 + 𝜏−1 +𝑁−1 +∑︁ +𝑛=0 +�𝑢𝑛+1 − 𝑢𝑛, 𝑣𝑛+1 − 𝑣𝑛� +𝑉 ×𝑉 . +This induces the norm ∥𝑢𝜏 ∥2 +𝐻 1(0,𝑇;𝑉 ),ℎ := (𝑢𝜏,𝑢𝜏)𝐻 1(0,𝑇;𝑉 ),ℎ. For the discrete control space we obtain +U𝜎 := {𝒖𝜎 = (𝑢1,𝜎, . . . ,𝑢𝑀,𝜎) : 𝑢𝑖,𝜎 ∈ 𝐻𝜏 (R2) for 𝑖 = 1, . . . , 𝑀}} +C𝜎 := {𝒄𝜎 = (𝑐1,𝜎, . . . ,𝑐𝑀,𝜎) : 𝑐𝑖,𝜎 ∈ 𝐻𝜏 (R) for 𝑖 = 1, . . . , 𝑀}, +Q𝜎 := U𝜎 × C𝜎, +and the admissible set by +U𝜎,ad := {𝒖𝜎 ∈ U𝜎 : |𝑢𝑛 +𝑖 | ≤ 1, 𝑖 = 1, . . . , 𝑀, 𝑛 = 0, . . . , 𝑁 }, +C𝜎,ad := {𝒄𝜎 ∈ C𝜎 : 0 ≤ 𝑐𝑛 +𝑖 ≤ 1,𝑖 = 1, . . . , 𝑀,𝑛 = 0, . . . , 𝑁 }, +Q𝜎,ad := U𝜎,ad × C𝜎,ad. +The discrete state space is defned by +Y𝜎 := 𝐻𝜏 (𝑉ℎ) × 𝐻𝜏 (𝑊ℎ,D) × 𝐻𝜏 (R2)𝑀. +With these defnitions, the discrete optimal control problem related to (2.2) reads as +Minimize +J𝜎 (𝒚𝜎, 𝒒𝜎) = 𝜏 +𝑁 +∑︁ +𝑛=1 +𝑒𝜈 𝑡𝑛 +∫ +Ω +𝜌𝑛 +ℎ (𝑥) d𝑥 − 𝜇 𝜏 +𝑀 +∑︁ +𝑖=1 +𝑁 +∑︁ +𝑛=1 +ln(𝜂𝑥𝑛 +𝑖 ∗ 𝜉ℎ) ++ 𝛼1 +2𝑇 +𝑀 +∑︁ +𝑖=1 +∥𝑢𝑖,𝜎 ∥2 +𝐻 1(0,𝑇),𝜏 + 𝛼2 +2𝑇 +𝑀 +∑︁ +𝑖=1 +∥𝑐𝑖,𝜎 ∥2 +𝐻 1(0,𝑇),𝜏 +(3.12a) +subject to +𝒚𝜎 := (𝜌𝜎,𝜙𝜎, 𝒙𝜎) = 𝑆𝜎 (𝒒𝜎), +(3.12b) +𝒒𝜎 := (𝒖𝜎, 𝒄𝜎) ∈ Q𝜎,ad, +(3.12c) +where is 𝜉ℎ ∈ 𝑊ℎ the fnite element approximation of (2.3) with frst-order Lagrange elements. Fur- +thermore, 𝑆𝜎 is the solution operator of (3.6). Note that, in order to maintain the diferentiability of +the barrier term with respect to 𝑥𝑛 +𝑖 , we use a regularization of the point evaluation of the nonsmooth +function 𝜉ℎ, compare also (3.6c). +We may write the control problem (3.12) in the more compact reduced form +𝑗𝜎 (𝒒𝜎) := J𝜎 (𝑆𝜎 (𝒒𝜎), 𝒒𝜎) → min! +subject to +𝒒𝜎 ∈ Q𝜎,ad. +(3.13) +To deduce a necessary optimality condition we apply the Lagrange formalism. The Lagrange function +L𝜎 : Y𝜎 × Q𝜎 × Y𝜎 → R +2023-01-09 +cbna +page 13 of 23 + +J. Pietschmann, A. Stötzner and M. Winkler +Pedestrian dynamics +coupling the discrete state equation (3.6) reads +L𝜎 (𝜌𝜎,𝜙𝜎, 𝒙𝜎; 𝒖𝜎, 𝒄𝜎; 𝜆𝜌,𝜎, 𝜆𝜙,𝜎, 𝜆𝒙,𝜎) = J𝜎 (𝜌𝜎,𝜙𝜎, 𝒙𝜎; 𝒖𝜎, 𝒄𝜎) +− +∫ +Ω +(𝜌0 +ℎ − proj𝑉ℎ (𝜌0)) 𝜆0 +𝜌,ℎ d𝑥 − +𝑁−1 +∑︁ +𝑛=0 +�∫ +Ω +(𝜌𝑛+1 +ℎ +− 𝜌𝑛 +ℎ) 𝜆𝑛+1 +𝜌,ℎ d𝑥 + 𝜏 𝑎(𝜌𝑛+1 +ℎ , 𝜆𝑛+1 +𝜌,ℎ ) − 𝜏 𝑏(𝛽𝑛 +ℎ)(𝜌𝑛 +ℎ, 𝜆𝑛+1 +𝜌,ℎ ) +� +− +𝑁 −1 +∑︁ +𝑛=0 +𝜏 +� +𝛿1 +∫ +Ω +∇𝜙𝑛 +ℎ · ∇𝜆𝑛 +𝜙,ℎ d𝑥 + +∫ +Ω +|∇𝜙𝑛 +ℎ |2 𝜆𝑛 +𝜙,ℎ d𝑥 − +∫ +Ω +1 +𝑓 (𝜌𝑛 +ℎ)2 + 𝛿2 +𝜆𝑛 +𝜙,ℎ d𝑥 +� +− +𝑀 +∑︁ +𝑖=1 +� +(𝑥0 +𝑖 − 𝑥𝑖,0)⊤ 𝜆0 +𝒙,𝑖 + +𝑁−1 +∑︁ +𝑛=0 +� +𝑥𝑛+1 +𝑖 +− 𝑥𝑛 +𝑖 − 𝜏 𝑓 (𝜂𝑥𝑛+1 +𝑖 +∗ 𝜌𝑛+1 +ℎ ) 𝑢𝑛+1 +𝑖 +�⊤ +𝜆𝑛+1 +𝒙,𝑖 +� +. +To shorten the notation we write 𝝀𝜎 := (𝜆𝜌,𝜎, 𝜆𝜙,𝜎, 𝜆𝒙,𝜎). The adjoint equation system determining +these variables for a given control and state is +𝜆𝑛 +𝜌,ℎ ∈ 𝑉ℎ : +𝜕L𝜎 +𝜕𝜌𝑛 +ℎ +(𝒚𝜎, 𝒒𝜎, 𝝀𝜎)𝛿𝜌ℎ= 0 +∀𝛿𝜌ℎ ∈ 𝑉ℎ, 𝑛 = 0, . . . , 𝑁, +(3.14a) +𝜆𝑛 +𝜙,ℎ ∈ 𝑊ℎ,D : +𝜕L𝜎 +𝜕𝜙𝑛 +ℎ +(𝒚𝜎, 𝒒𝜎, 𝝀𝜎)𝛿𝜙ℎ= 0 +∀𝛿𝜙ℎ ∈ 𝑊ℎ, 𝑛 = 0, . . . , 𝑁 − 1, +(3.14b) +𝜆𝑛 +𝒙𝑖 ∈ R2 : +𝜕L𝜎 +𝜕𝑥𝑛 +𝑖 +(𝒚𝜎, 𝒒𝜎, 𝝀𝜎)𝛿𝑥𝑖 = 0 +∀𝛿𝑥𝑖 ∈ R2, 𝑛 = 0, . . . , 𝑁 +(3.14c) +for 𝑖 = 1, . . . , 𝑀. Note that this can be interpreted as a coupled system involving a parabolic PDE and +an ODE that run backward in time. We use the automatic diferentiation feature in FEniCS in our +implementation. As the forward system completely decouples in each time step, so does the adjoint +system and we can compute step by step: +𝜆𝑁 +𝒙,ℎ ↦→ 𝜆𝑁 +𝜌,ℎ ↦→ (𝜆𝑁 +𝜙,ℎ) ↦→ . . . ↦→ 𝜆0 +𝒙,ℎ ↦→ 𝜆0 +𝜌,ℎ ↦→ 𝜆0 +𝜙,ℎ. +With the adjoint states at hand we can assemble the derivatives of the reduced objective (3.13) and end +up with the following optimality condition for (3.12): +Theorem 3.3 (Necessary optimality condition). Let (𝒚𝜎, 𝒒𝜎) ∈ Y𝜎 × Q𝜎,ad be a local solution of (3.12). +Then, there exists 𝝀𝜎 ∈ Y𝜎 fulflling (3.14) and +𝛼1 (𝑢𝜎,𝑖, 𝑣𝜎,𝑖 − 𝑢𝜎,𝑖)𝐻 1(0,𝑇;R2),𝜏 + 𝛼2 (𝑐𝜎,𝑖,𝑑𝜎,𝑖 − 𝑐𝜎,𝑖)𝐻 1(0,𝑇),𝜏 ++ 𝜏 +𝑁 +∑︁ +𝑛=1 +� +𝑓 (𝜂𝑥𝑛 +𝑖 ∗ 𝜌𝑛 +ℎ) 𝜆𝑛 +𝒙𝑖, 𝑣𝑛 +𝑖 − 𝑢𝑖𝑛� +R2 + 𝜏 +𝑁−1 +∑︁ +𝑛=0 +𝜕 +� +𝑏(𝛽𝑛 +ℎ)(𝜌𝑛 +ℎ, 𝜆𝑛+1 +𝜌,ℎ ) +� +𝜕𝑐𝑛 +𝑖 +(𝑑𝑛 +𝑖 − 𝑐𝑛 +𝑖 ) ≥ 0 +(3.15) +for all test functions 𝒓𝜎 := (𝒗𝜎, 𝒅𝜎) ∈ Q𝜎,ad and all 𝑖 = 1, . . . , 𝑀. +Proof. It is well-known that the variational inequality 𝑗 ′ +𝜎 (𝒒𝜎)(𝒓𝜎 − 𝒒𝜎) ≥ 0 for 𝒓𝜎 ∈ Q𝜎,ad is necessary +for 𝒒𝜎 being a local minimizer of (3.13). Taking into account the the equivalence +𝑗 ′ +𝜎 (𝒒𝜎)𝛿𝒒𝜎 = 𝜕L +𝜕𝒒𝜎 +(𝒚𝜎, 𝒒𝜎, 𝝀𝜎)𝛿𝒒𝜎 +if𝝀𝜎 solves (3.14) +yields the variational inequality (3.15). +□ +2023-01-09 +cbna +page 14 of 23 + +J. Pietschmann, A. Stötzner and M. Winkler +Pedestrian dynamics +Our solution algorithm is based on a projected gradient algorithm and it remains to establish a +representation of the gradient of 𝑗𝜎. +The derivative of the objective (3.12a) towards some direction 𝛿𝒒𝜎 = (𝛿𝒖𝜎,𝛿𝒄𝜎) ∈ Q reads +𝑗 ′ +𝜎 (𝒒𝜎)𝛿𝒒𝜎 = +𝑀 +∑︁ +𝑖=1 +� +𝛼1(𝑢𝑖,𝜎,𝛿𝑢𝑖,𝜎)𝐻 1(0,𝑇),𝜏 + 𝛼2(𝑐𝑖,𝜎,𝛿𝑐𝑖,𝜎)𝐻 1(0,𝑇),𝜏 ++ 𝜏 +𝑁 +∑︁ +𝑛=1 +𝑓 (𝜂𝑥𝑛 +𝑖 ∗ 𝜌𝑛 +ℎ) 𝛿𝑢𝑛 +𝑖 +⊤𝜆𝑛 +𝒙𝑖 + 𝜏 +𝑁−1 +∑︁ +𝑛=0 +𝜕 +� +𝑏(𝛽𝑛 +ℎ)(𝜌𝑛 +ℎ, 𝜆𝑛+1 +𝜌,ℎ ) +� +𝜕𝑐𝑛 +𝑖 +𝛿𝑐𝑛 +𝑖 +� +. +To obtain a representation of the 𝐻 1(0,𝑇),𝜏-gradient of 𝑗𝜎 with respect to 𝒖𝜎 and 𝒄𝜎, we introduce the +grid functions 𝑧𝑖,𝜎 : {𝑡𝑛}𝑁 +𝑛=0 → R2 and 𝑑𝑖,𝜎 : {𝑡𝑘}𝑁 +𝑛=0 → R, 𝑖 = 1, . . . , 𝑀 solving +��������� +� +1 +𝜏2 +��������� +� +1 +−1 +−1 +2 +−1 +−1 +2 +−1 +... +... +... +−1 +2 +−1 +−1 +1 +��������� +� ++ 𝐼𝑁+1×𝑁+1 +��������� +� +��������� +� +𝑧0 +𝑖 +𝑧1 +𝑖 +𝑧2 +𝑖... +𝑧𝑁−1 +𝑖 +𝑧𝑁 +𝑖 +��������� +� += +��������� +� +0 +−𝑓 (𝜌1 +ℎ(𝒙1 +𝑖))𝜆0 +𝑥𝑖 +−𝑓 (𝜌2 +ℎ(𝒙2 +𝑖 ))𝜆1 +𝑥𝑖 +... +−𝑓 (𝜌𝑁−1 +ℎ +(𝒙𝑁−1 +𝑖 +))𝜆𝑁−2 +𝑥𝑖 +−𝑓 (𝜌𝑁 +ℎ (𝒙𝑁 +𝑖 ))𝜆𝑁−1 +𝑥𝑖 +��������� +� +(3.16a) +and +��������� +� +1 +𝜏2 +��������� +� +1 +−1 +−1 +2 +−1 +−1 +2 +−1 +... +... +... +−1 +2 +−1 +−1 +1 +��������� +� ++ 𝐼𝑁+1×𝑁+1 +��������� +� +��������� +� +𝑑0 +𝑖 +𝑑1 +𝑖 +𝑑2 +𝑖... +𝑑𝑁−1 +𝑖 +𝑑𝑁 +𝑖 +��������� +� += +������������� +� +−𝜕𝑐0 +𝑖 𝑏(𝛽0 +ℎ)(𝜌0 +ℎ, 𝜆1 +𝜌,ℎ) +−𝜕𝑐1 +𝑖𝑏(𝛽1 +ℎ)(𝜌1 +ℎ, 𝜆2 +𝜌,ℎ) +−𝜕𝑐2 +𝑖𝑏(𝛽2 +ℎ)(𝜌2 +ℎ, 𝜆3 +𝜌,ℎ) +... +−𝜕𝑐𝑁 −2 +𝑖 +𝑏(𝛽𝑁−2 +ℎ +)(𝜌𝑁 −2 +ℎ +, 𝜆𝑁−1 +𝜌,ℎ ) +−𝜕𝑐𝑁 −1 +𝑖 +𝑏(𝛽𝑁−1 +ℎ +)(𝜌𝑁−1 +ℎ +, 𝜆𝑁 +𝜌,ℎ) +0 +������������� +� +(3.16b) +for 𝑖 = 1, . . . , 𝑀. By a simple calculation we then confrm +(𝑧𝑖,𝜎,𝛿𝑢𝑖,𝜎)𝐻 1(0,𝑇;R2),𝜏 = +𝑁 +∑︁ +𝑛=1 +𝑓 (𝜂𝑥𝑛 +𝑖 ∗ 𝜌𝑛 +ℎ) 𝛿𝑢𝑛 +𝑖 +⊤𝜆𝑛 +𝒙𝑖, +(𝑑𝑖,𝜎,𝛿𝑐𝑖,𝜎)𝐻 1(0,𝑇),𝜏 = +𝑁 −1 +∑︁ +𝑛=0 +𝜕 +� +𝑏(𝛽𝑛 +ℎ)(𝜌𝑛 +ℎ, 𝜆𝑛+1 +𝜌,ℎ ) +� +𝜕𝑐𝑛 +𝑖 +𝛿𝑐𝑛 +𝑖 +We write 𝒛𝜎 = (𝑧1,𝜎, . . . ,𝑧𝑀,𝜎) ∈ U and 𝒅𝜎 = (𝑑1,𝜎, . . . ,𝑑𝑀,𝜎) ∈ C and get the following representation +of the gradient of 𝑗𝜎: +∇𝒖𝜎 𝑗𝜎 (𝒒𝜎) = 𝛼1 𝒖𝜎 + 𝒛𝜎, +(3.17a) +∇𝒄𝜎 𝑗𝜎 (𝒒𝜎) = 𝛼2 𝒄𝜎 + 𝒅𝜎. +(3.17b) +This allows an implementation of a projected gradient method which we discuss in the following +section. +2023-01-09 +cbna +page 15 of 23 + +J. Pietschmann, A. Stötzner and M. Winkler +Pedestrian dynamics +3.5 Optimization algorithms for the discretized problem +For a solution of the discretized optimal control problem 3.12 we propose a projected gradient algorithm. +In this procedure, for a given initial control 𝒒(0) = (𝒖 (0), 𝒄 (0)) ∈ Q, the new iterates are sucessively +computed by means of +𝒖 (𝑘+1) +𝜎 += 𝚷𝑢 +ad +� +𝒖 (𝑘) +𝜎 +− 𝑠(𝑘) ∇𝒖𝜎 𝑗𝜎 (𝒒(𝑘) +𝜎 ) +� +, +(3.18a) +𝒄 (𝑘+1) +𝜎 += 𝚷𝑐 +ad +� +𝒄 (𝑘) +𝜎 +− 𝑠(𝑘) ∇𝒄𝜎 𝑗𝜎 (𝒒(𝑘) +𝜎 ) +� +, +(3.18b) +with 𝚷𝑢 +ad : U𝜎 → U𝜎,ad and 𝚷𝑐 +ad : C𝜎 → C𝜎,ad the 𝐻 1(0,𝑇),𝜏 projections onto the admissible sets +U𝜎,ad and C𝜎,ad, respectively, this is, +𝚷𝑢 +ad(𝒖𝜎) := arg min +𝒗𝜎 ∈U𝜎,ad +1 +2 ∥𝒖𝜎 − 𝒗𝜎 ∥2 +𝐻 1(0,𝑇;R2),𝜏, +(3.19a) +𝚷𝑐 +ad(𝒄𝜎) := arg min +𝒅𝜎 ∈C𝜎,ad +1 +2 ∥𝒄𝜎 − 𝒅𝜎 ∥2 +𝐻 1(0,𝑇),𝜏. +(3.19b) +A formula for the gradient of 𝑗𝜎 has been derived in the previous section already, see (3.17). The step +length parameter 𝑠 (𝑘) > 0 is obtained by an Amijo line search and must fulfll the sufcient decrease +condition +𝑗𝜎 (𝒒(𝑘) +𝜎 +− 𝑠(𝑘) ∇𝑗𝜎 (𝒒(𝑘) +𝜎 )) ≤ 𝑗𝜎 (𝒒(𝑘) +𝜎 ) − 𝑑 +𝑠 (𝑘) ∥𝒒(𝑘) +𝜎 +− +� +𝒒(𝑘) +𝜎 +− 𝑠 (𝑘) ∇𝑗𝜎 (𝒒(𝑘) +𝜎 ) +� +∥2 +Q𝜎 +(3.20) +with a decrease parameter 𝑑 ∈ (0, 1) which is usually small (e.g. 10−4). A reasonable stopping criterion +for the projected gradient algorithm is +∥𝒒(𝑘) +𝜎 +− 𝚷ad +� +𝒒(𝑘) +𝜎 +− ∇𝑗𝜎 (𝒒(𝑘) +𝜎 ) +� +∥Q𝜎 ≤ 10−3. +It remains to discuss the realization of the projection operators and we propose a primal dual active set +strategy that may also be considered as semismooth Newton method. Note that the operators Πad are +semismooth, see Christof, Wachsmuth, 2021. The evaluation of the projection operator 𝚷𝑢 +ad : U𝜎 → +U𝜎,ad requires to solve the optimization problem (3.19a). The unknowns (assuming 𝑀 = 1 and omitting +the agent’s index 𝑖 for a while) are the coefcients of the functions U𝜎 ∋ 𝚷ad(𝑢𝜎) = 𝑤𝜎 ≃ �𝑤 ∈ R(𝑁+1)×2 +for some given U𝜎 ∋ 𝑢𝜎 ≃ �𝑢 ∈ R(𝑁+1)×2. We switch to a matrix-vector notation and defne +𝑤𝑛 := +�𝑤𝑛 +1 +𝑤𝑛 +2 +� +:= 𝑤𝜎 (𝑡𝑛), +�𝑤 𝑗 := (𝑤0 +𝑗, . . . ,𝑤𝑁 +𝑗 )⊤, 𝑗 = 1, 2, +as well as the matrix 𝐴 ∈ R(𝑁+1)×(𝑁+1) on the left-hand side of the linear system (3.16) inducing the +discrete 𝐻 1(0,𝑇),𝜏-norm. The Lagrangian for (3.19) reads +𝐿( �𝑤1, �𝑤2, �𝜆) = 1 +2 +2 +∑︁ +𝑖=1 +( �𝑤𝑖 − �𝑢𝑖)⊤ 𝐴( �𝑤𝑖 − �𝑢𝑖) − 1 +2 𝜆⊤� +| �𝑤|2 +∗ − �1 +� +, +2023-01-09 +cbna +page 16 of 23 + +J. Pietschmann, A. Stötzner and M. Winkler +Pedestrian dynamics +with | �𝑤|2 +∗ = (|𝑤0|2, . . . , |𝑤𝑁 |2)⊤. The Karush-Kuhn-Tucker system for (3.19) then reads +𝐴 ( �𝑤𝑖 − �𝑢𝑖) − �𝜆 · �𝑤𝑖 = 0 +𝑖 = 1, 2, +1 +2 +� +| �𝑤|2 +∗ − �1 +� +≤ 0, +�𝜆 ≥ 0, +1 +2 +�𝜆 · �| �𝑤|2 +∗ − 1� = 0, +where · is the component-wise multiplication of two vectors. We reformulate the complementarity +condition by means of a nonsmooth equation and arrive at the following equivalent form of the KKT +system +𝐹 ( �𝑤1, �𝑤2, �𝜆) := +������� +𝐴 ( �𝑤1 − �𝑢1) − �𝜆 · �𝑤1 +𝐴 ( �𝑤2 − �𝑢2) − �𝜆 · �𝑤2 +�𝜆 − max{0, − 1 +2 (| �𝑤|2 +∗ − �1) + �𝜆} +������� += 0. +(3.21) +This nonlinear system can be solved iteratively by a semismooth Newton method. Given is an initial +pair (�𝑢 (0), �𝜆(0)). Successively, one computes the active and inactive set +A (𝑘) := {𝑛 ∈ {0, . . . , 𝑁 }: − 1 +2 (|𝑤𝑛|2 +2 − 1) + 𝜆𝑛 > 0}, +I (𝑘) := {0, . . . , 𝑁 } \ A (𝑘), +solves the Newton system +������� +𝐴 − 𝐷 �𝜆(𝑘) +0 +−𝐷 �𝑤(𝑘) +1 +0 +𝐴 − 𝐷 �𝜆(𝑘) +−𝐷 �𝑤(𝑘) +2 +𝐷A (𝑘) 𝐷 �𝑤(𝑘) +1 +𝐷A (𝑘) 𝐷 �𝑤(𝑘) +2 +𝐷I (𝑘) +������� +������� +� +𝛿𝑤1 +� +𝛿𝑤2 +�𝛿𝜆 +������� += − +������� +𝐴 ( �𝑤 (𝑘) +1 +− �𝑢1) − �𝜆(𝑘) · �𝑤 (𝑘) +1 +𝐴 ( �𝑤 (𝑘) +2 +− �𝑢2) − �𝜆(𝑘) · �𝑤 (𝑘) +2 +�𝜆(𝑘) − max{0, − 1 +2 (| �𝑤 (𝑘)|2 +∗ − �1) + �𝜆(𝑘)}, +������� +with the diagonal matrices 𝐷�𝑣 = diag(�𝑣) for �𝑣 ∈ R𝑁+1 and 𝐷M = diag(𝜒M) for M ⊂ {0, . . . , 𝑁 }, and +performs the Newton update +�𝑤 (𝑘+1) = �𝑤 (𝑘) + � +𝛿𝑤, +�𝜆(𝑘+1) = �𝜆(𝑘) + �𝛿𝜆. +This procedure is repeated for 𝑘 = 0, 1, . . . until some termination criterion, e. g., ∥𝐹 ( �𝑤1, �𝑤2, �𝜆)∥ < tol, +is fulflled. +4 Numerical experiments +This section is devoted to numerical experiments. To establish the discretized system (3.12) the fnite +element library FEniCS was used, complemented by a Python implementation of the projected +gradient method from Equation (3.18) and the Armijo step size rule from (3.20). The computational +meshes were created by the mesh generator mshr integrated in FEniCS. +4.1 Example 1 +In a frst numerical test we solve the problem Equation (2.2) in the domain Ω depicted in Figure 4.1 +with the following parameters. +𝑇 = 9 +𝑛𝑇 = 300 +𝛼1 = 𝛼2 = 5 · 10−2 +𝛾 = 10 +𝜁 = 10−2 +𝜇 = 5 · 10−2 +𝜀 = 10−5 +𝛿1 = 0.2 +𝛿2 = 0.1 +𝛿3 = 10−2 +𝛿4 = 0.1. +2023-01-09 +cbna +page 17 of 23 + +J. Pietschmann, A. Stötzner and M. Winkler +Pedestrian dynamics +The initial density 𝜌0 is the sum of 6 Gaussian bells, see also Figure 4.1a. The subdomain �Ω where +densities are penalized is chosen to cover the region within the walls. Without any controlled agents, +most of the people will squeeze through the 2 smaller emergency exits in the south and north while +the large exit in the east is rarely used. To improve the evacuation 3 agents were introduced. The +initial control (𝒖, 𝒄) was chosen in such a way that the agent moves straight to the right outside of the +room having a constant the intensity. +4.2 Example 2 +In a second example we consider the domain illustrated in Figure 4.2. The initial density is concentrated +near the slit in the wall on the left-hand side. In an uncontrolled evacuation scenario the majority of +the people would leave the domain through this slit causing a massive congestion leading to a very +slow evacuation of the crowd. The model and algorithm parameters chosen in the current example are +as follows: +𝑇 = 12 +𝑛𝑇 = 300 +𝛼1 = 𝛼2 = 5 · 10−2 +𝛾 = 10 +𝜁 = 10−2 +𝜇 = 5 · 10−2 +𝜀 = 10−5 +𝛿1 = 0.2 +𝛿2 = 0.1 +𝛿3 = 10−2 +𝛿4 = 0.1. +This example shows that the evacuation can be signifcantly improved by using two agents with +optimized trajectory and intensity. Interesting is, that the intensity is non-zero only in the time interval +𝑡 ∈ (0, 3). The agents attract the people leading them sufciently far away from the slit in the west +and then they stop infuencing the crowd. When being sufciently far away from the slit the people +fnd the way to the larger exits in the north and south on their own by using the movement direction +determined by the potential 𝜙. +4.3 Example 3 +In a third example we consider a square-shaped room with exits in the south, east and north. The exits +have diferent width. The model and algorithm parameters are chosen as follows: +𝑇 = 10 +𝑛𝑇 = 300 +𝛼1 = 𝛼2 = 5 · 10−2 +𝛾 = 10 +𝜁 = 10−2 +𝜇 = 5 · 10−2 +𝜀 = 10−5 +𝛿1 = 0.2 +𝛿2 = 0.1 +𝛿3 = 10−2 +𝛿4 = 0.1. +The initial density is concentrated near the small exit and without a control of the crowd motion most +of the people are blocking each other while squeezing through this small exit. Two agents were added +in this scenario with the aim attracting the people in such a way that more of them fnd the other two +exits in the north and east. The computed agent trajectories are quite short. It is interesting to observe +that in the time interval 𝑡 ∈ (0, 2) the agents just go to an optimal position sufciently close to the +crowd and attract them in the time interval 𝑡 ∈ (2, 5), leading some of the people to the center of the +room. At this point the agents drive their intensity to zero meaning that they stop infuencing the +crowd. However, when being sufciently far away from the critical exit the people fnd the route to +the less used exits on their own due to the movement rule determined by the potential 𝜙. +2023-01-09 +cbna +page 18 of 23 + +J. Pietschmann, A. Stötzner and M. Winkler +Pedestrian dynamics +(a) Solution at time 𝑡 = 0 +(b) Solution at time 𝑡 = 1.2 +(c) Solution at time 𝑡 = 2.4 +(d) Solution at time 𝑡 = 4.2 +(e) Solution at time 𝑡 = 7.2 +100 +200 +300 +1 +2 +3 +4 +𝑘 +𝑐𝑖 +𝑐1(𝑡) +𝑐2(𝑡) +𝑐3(𝑡) +(f) Agent intensities 𝑐𝑖 (𝑡) +Figure 4.1: Solution of the problem from Section 4.1. The colored background represents the density 𝜌; +the dots are the agent positions; the black curves are the agent trajectories. +2023-01-09 +cbna +page 19 of 23 + +0 +0.1 +0.2 +0.3 +0.4 +0.5 0.6 0.7 +0.8 +0.9 +1J. Pietschmann, A. Stötzner and M. Winkler +Pedestrian dynamics +(a) Solution at time 𝑡 = 0 +(b) Solution at time 𝑡 = 2.4 +(c) Solution at time 𝑡 = 4.8 +(d) Solution at time 𝑡 = 8 +20 +40 +60 +80 +100 120 140 160 180 200 220 +1 +2 +3 +4 +5 +6 +𝑘 +𝑐𝑖 +𝑐1(𝑡) +𝑐2(𝑡) +(e) Intensities 𝑐𝑖 +Figure 4.2: Solution of the problem from Section 4.2 at various time steps 𝑡𝑘 and intensity of the agents. +2023-01-09 +cbna +page 20 of 23 + +0 +0.1 +0.2 +0.3 +0.4 +0.5 0.6 0.7 +0.8 +0.9 +1J. Pietschmann, A. Stötzner and M. Winkler +Pedestrian dynamics +(a) Solution at time 𝑡 = 0 +(b) Solution at time 𝑡 = 2 +(c) Solution at time 𝑡 = 4 +(d) Solution at time 𝑡 = 6.7 +50 +100 +150 +200 +250 +300 +1 +2 +𝑘 +𝑐𝑖 +𝑐1(𝑡) +𝑐2(𝑡) +(e) Intensities 𝑐𝑖 +Figure 4.3: Solution of the problem from Section 4.3 at various time steps 𝑡𝑘 and intensity of the agents. +2023-01-09 +cbna +page 21 of 23 + +J. Pietschmann, A. Stötzner and M. Winkler +Pedestrian dynamics +References +Albi, G.; M. Bongini; E. Cristiani; D. 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Computational Methods in Applied Mathematics 10.2, pp. 219–232. doi: +10.2478/cmam-2010-0013. +2023-01-09 +cbna +page 23 of 23 + diff --git a/b9E0T4oBgHgl3EQfnwFc/content/tmp_files/load_file.txt b/b9E0T4oBgHgl3EQfnwFc/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c064f9cacad70f4a67a86f405cdb07f9d9fa0985 --- /dev/null +++ b/b9E0T4oBgHgl3EQfnwFc/content/tmp_files/load_file.txt @@ -0,0 +1,1223 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf,len=1222 +page_content='Numerical investigation of agent controlled pedestrian dynamics using a structure preserving finite volume scheme Jan-Frederik Pietschmann∗ Ailyn Stötzner† Max Winkler† We provide a numerical realisation of an optimal control problem for pedestrian motion with agents that was analysed in Herzog, Pietschmann, Winkler, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The model consists of a regularized variant of Hughes’ model for pedestrian dynamics coupled to ordinary diferential equations that describe the motion of agents which are able to infuence the crowd via attractive forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' We devise a fnite volume scheme that preserves the box constraints that are inherent in the model and discuss some of its properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' We apply our scheme to an objective functional tailored to the case of an evacuation scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Finally, numerical simulations for several practically relevant geometries are performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' crowd motion, nonlinear transport, Eikonal equation, ODE-PDE coupling, optimal control, fnite volume, projected gradient descent AMS subject classifcations (MSC2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' 49K20, 35Q91, 35M33, 65M08 1 Introduction With more and more people living in highly populated areas, the modelling, simulation and control of (large) pedestrian crowds is an important feld of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' In this work, we study the optimal control problem for a regularized version of Hughes’ model for pedestrian motion, Hughes, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' In our approach, the (continuous) crowd can be controlled by a fxed small number of agents that can attract people in their vicinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' In terms of the model, this corresponds to an additional potential term centred at the agents positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' In a previous work, Herzog, Pietschmann, Winkler, 2020, we already studied the well-posedness and optimality conditions of this problem while here, we focus on a numerical ∗Technische Universität Chemnitz, Faculty of Mathematics, AG Inverse Probleme, 09107 Chemnitz, Ger- many (jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='pietschmann@mathematik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='tu-chemnitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='de, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='tu-chemnitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='de/mathematik/invpde/prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='php, OR- CID 0000-0003-0383-8696).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' †Technische Universität Chemnitz, Faculty of Mathematics, 09107 Chemnitz, Germany (ailyn@stoetzner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='de, http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='tu-chemnitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='de/mathematik/part dgl/people/stoetzner/, ORCID 0000-0002-8287-9738, max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='winkler@mathematik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='tu-chemnitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='de, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='tu-chemnitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='de/mathematik/part dgl/people/winkler/, ORCID 0000-0002-5292-2280).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' 2023-01-09 page 1 of 23 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='02516v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='OC] 6 Jan 2023 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Pietschmann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Stötzner and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Winkler Pedestrian dynamics implementation of the control problem and extensive numerical examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' In particular, we provide and analyse a fnite volume scheme that preserves the box constraints inherent in our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' To introduce the model, we fx Ω ⊂ R2 to be a bounded domain with 𝐶4-boundary 𝜕Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Furthermore 𝑇 > 0 is an arbitrary time horizon and 𝑄𝑇 := (0,𝑇) × Ω as well as Σ𝑇 = (0,𝑇) × 𝜕Ω denote the space-time cylinder and its lateral boundary, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The boundary is decomposed into two parts: 𝜕Ω𝐷 representing the exits and 𝜕Ω𝑊 the part where the domain is constrained by walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' For theoretical purposes (regularity of solutions) we assume 𝜕Ω𝐷 ∩ 𝜕Ω𝑊 = ∅ meaning that both boundary parts are separated from each other, see Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' In a similar way we defne Σ𝐷 = (0,𝑇) × 𝜕Ω𝐷 and Σ𝑊 = (0,𝑇) × 𝜕Ω𝑊 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The unknown variables in our system of equations are the density of the crowd 𝜌 : 𝑄𝑇 → R+, a potential specifying the current time to escape 𝜙 : 𝑄𝑇 → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' In addition, there are 𝑀 agents which may infuence the motion of the crowd via attractive forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Their positions are denoted by 𝑥𝑖 : (0,𝑇) → R2, 𝑖 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' , 𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' In addition, each agent is able to regulate the strength by which it acts on the crowd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' This is encoded in the intensities 𝑐𝑖 : (0,𝑇) → R+, 𝑖 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' , 𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Both the agent trajectories and interaction strength are summarized in a vector 𝒙 = (𝑥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' ,𝑥𝑛)⊤ and 𝒄 = (𝑐1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' ,𝑐𝑛)⊤, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The mathematical equations describing the movement of a pedestrian crowd infuenced by agents then read as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' For given agent movement directions 𝒖 = (𝑢1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' ,𝑢𝑀)⊤ with 𝑢𝑖 ∈ 𝐿∞(0,𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' R2) the unknowns 𝜌,𝜙, 𝒙 are related to each other by means of 𝜕𝑡𝜌 − ∇ · �𝜌 𝛽(𝜌,𝜙, 𝒙, 𝒄)� = 𝜀 Δ𝜌 in 𝑄𝑇, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1a) −𝛿1 Δ𝜙 + |∇𝜙|2 = 1 𝑓 (𝜌)2 + 𝛿2 in 𝑄𝑇, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1b) �𝑥𝑖(𝑡) = 𝑓 �𝜌(𝑡,𝑥𝑖(𝑡))� 𝑢𝑖(𝑡) for 𝑡 ∈ (0,𝑇), 𝑖 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' , 𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1c) Moreover, we impose the boundary conditions −�𝜀 ∇𝜌 + 𝜌 𝛽(𝜌,𝜙, 𝒙, 𝒄)� · 𝑛 = 𝛾 𝜌, 𝜙 = 0 on ΣD, �𝜀 ∇𝜌 + 𝜌 𝛽(𝜌,𝜙, 𝒙, 𝒄)� · 𝑛 = 0, ∇𝜙 · 𝑛 = 0 on ΣW, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='2) as well as the initial conditions 𝜌(0, ·) = 𝜌0 in Ω, 𝑥𝑖(0) = 𝑥𝑖,0 for 𝑖 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' , 𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='3) Here, 𝜀,𝛿1,𝛿2 > 0 are regularization parameters and the corresponding terms in the system are needed to guarantee a certain regularity for the solution, see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The domain Ω is sufciently large such that 𝑥𝑖(𝑡) ∈ Ω on [0,𝑇] for 𝑖 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' , 𝑀 and 𝑡 ∈ [0,𝑇] if |𝑢𝑖(𝑡)| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Let us briefy discuss the meaning of the respective terms: Equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1a) states that pedestrians are transported according to the velocity feld 𝛽, see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1) below, while also performing (little) random motion encoded by the Laplacian of 𝜌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The second equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1b) is a modifed and regularized Eikonal equation whose solution is the distance to the closest exit, mitigating areas of high density via the term on the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Here, the additional difusion accounts for the fact that pedestrians do not know their environment exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Then, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1c) governs the motion of the agents, whose speed is also infuenced by the surrounding pedestrian density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The function 𝑓 : [0, 1] → [0, 1] is a density-velocity rule, chosen in such a way that 𝑓 (𝜌) determines the maximum velocity an individual can move if the density in its current position is 𝜌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' We choose 𝑓 to be monotonically decreasing meaning that higher 2023-01-09 cbna page 2 of 23 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Pietschmann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Stötzner and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Winkler Pedestrian dynamics densities lead to slower movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The velocity feld 𝛽 will refect the fact that pedestrians are, on the one hand, trying the minimize their exit time which amounts to a drift term in the direction of ∇𝜙 and on the other hand, they are attracted by the agents which is realized by additional attractive potentials whose center depends on the agents’ positions 𝒙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' This results in a velocity which is the sum of two terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Furthermore, to account for the efect that the velocity will deteriorate in regions of high density, it will be modifed by an additional multiplicative factor 𝑓 (𝜌).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' As in the equations for the motion of the agents, 𝑓 is monotonically decreasing and becomes zero at a given maximal density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The boundary conditions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='2) allow for an outfow with velocity 𝛾 on parts of the boundary (Σ𝐷) while no-fux conditions on the remaining parts are to be interpreted as walls (Σ𝑊 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' A detailed description of the involved non-linearities will be given in the next section, but we also refer to Herzog, Pietschmann, Winkler, 2020 for more details on the model and the regularizing terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Analytical properties of the unregularized Hughes’ model introduced in Hughes, 2002 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' 𝜀 = 𝛿1 = 𝛿2 = 0 in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1)), without control, are difcult because of the low regularity of ∇𝜙 that appears on a set depending on the solution 𝜌 of the frst equation, but see Amadori, Goatin, Rosini, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Amadori, Di Francesco, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' El-Khatib, Goatin, Rosini, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Thus, regularized variants have been considered, see Di Francesco, Markowich, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=', 2011 for an instance where 𝜀 = 0 but 𝛿1, 𝛿2 ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' In fact, the result there is obtained as a vanishing viscosity limit 𝜀 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' There is also a number of extensions and variants of the model, aiming to understand additional properties, make it more realistic, or consider diferent settings like graphs, see Burger, Di Francesco, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Carrillo, Martin, Wolfram, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Carlini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Di Francesco, Fagioli, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Colombo, Gokieli, Rosini, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Control of systems by means of a small number of agents has received lots of interest recently, both on a discrete level (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' one considers a large system of ODEs for the motion of individuals coupled to a small number of equations for the agents), see Caponigro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Himakalasa, Wongkaew, 2021, but also for coupled PDE-ODE systems, Albi, Bongini, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Albi, Fornasier, Kalise, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Let us emphasize that whenever PDEs are coupled to ODEs in such a fashion that the solution of the PDE needs to evaluated at the solution of the ODE (as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1c)), regularity is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' While in our case, this is obtained by the additional difusion in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1a), when hyperbolic models for the transport of pedestrian are considered, an additional regularization in the ODE is needed, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Borsche, Colombo, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Borsche, Klar, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Borsche, Meurer, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Finally, let us mention that the ODE-ODE and PDE-ODE perspective are closely related by means of so-called mean feld limits when the number of agents tends to infnity, see Burger, Pinnau, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Pinnau, Totzeck, 2018 and also the recent overview on control of crowds, Banda, Herty, Trimborn, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' For the numerical discretization of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1a), we employ, as we think of the parameter 𝜀 being small, a fnite volume scheme for the spatial discretization, which may also be interpreted as a discontinuous Galerkin scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' In combination with the Lax-Friedrichs numerical fuxes the scheme is stable and preserves the bounds 0 ≤ 𝜌 ≤ 1 inherent in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Such structure-preserving discretizations of PDEs gained much attention, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=', in the context of chemotaxis problems Epshteyn, Kurganov, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Li, Shu, Yang, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Strehl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Guo, Li, Yang, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Ibrahim, Saad, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Filbet, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The previously mentioned articles difer also in the choice of the time-stepping scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' For the treatment of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1) we will use an implicit-explicit fnite diference scheme, whereas the difusion-related terms are established implicitly and the convection-related terms explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1b) is discretized with standard linear fnite elements and for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1c) we employ a backward Euler scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The paper is organized as follows: In Section 2 we give a precise defnition of our problem and recall the analytical results from Herzog, Pietschmann, Winkler, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' In Section 3 we provide a numerical 2023-01-09 cbna page 3 of 23 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Pietschmann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Stötzner and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Winkler Pedestrian dynamics discretization scheme in space and time and analyse some of its properties, in particular, we show that it preserves physical bounds of the density of pedestrians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Corresponding optimization algorithms are discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='5 and Section 4 fnally provides the results of our numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' 2 The continuous optimal control problem Let us motivate the remaining quantities arising in the system of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The function 𝑓 : [0, 𝜌max] → R is a density-velocity relation and is assumed to be 𝑊 3,∞(R) ∩ 𝐶𝑐(R) with 𝑓 (0) = 1 and 𝑓 (𝜌max) = 0 with 𝜌max denoting the maximal density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' A usual choice is 𝑓 (𝜌) = 𝜉 � 1 − 𝜌 𝜌max � with a cut-of function 𝜉 ∈ 𝐶∞ 𝑐 (−1, 2) satisfying 𝜉 ≡ 1 on (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Obviously, a higher density leads to a lower velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Throughout the article we set 𝜌max = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The movement direction of the crowd described by the function −𝛽(𝜌,𝜙, 𝒙, 𝒄) is modelled as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The primary interest of the crowd is to move either towards the closest emergency exit, this is the direction −∇𝜙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' This is mitigated by the attraction of close by agents which is the direction −∇𝜙𝐾, where is an agent potential defned by 𝜙𝐾 (𝒙, 𝒄;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='𝑡,𝑥) � 𝑀 ∑︁ 𝑖=1 𝑐𝑖(𝑡) 𝐾 �𝑥 − 𝑥𝑖(𝑡)�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Here, 𝐾(𝑥) = 𝑘(|𝑥|), 𝑘 ∈ 𝑊 3,∞(R) is a radially symmetric function and 𝑐𝑖 ∈ 𝐻 1(0,𝑇), 𝑖 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' , 𝑀, are time-dependent intensity functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Typical choices for attractive agent potentials are either the bumb function 𝑘(𝑟) = � exp � − 𝑅2 𝑅2−𝑟 2 � , if 𝑟 < 𝑅, 0, otherwise with attraction radius 𝑅 > 0, or the Morse potential 𝑘(𝑟) = e−2𝑎 (𝑟−𝑟𝑎) − 2 e−𝑎 (𝑟−𝑟𝑎) with certain parameters 𝑎,𝑟𝑎 > 0, realizing a repulsion in the near and an attraction in the far range of the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' This is useful to avoid a high density very close to the respective agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' We refer to Carillo, Huang, Martin, 2014 for a more detailed discussion on potentials in the context of focking problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' These considerations yield a velocity feld defned as follows 𝛽(𝜌,𝜙, 𝒙, 𝒄) = 𝑣0𝑓 (𝜌) ℎ(∇(𝜙 + 𝜙𝐾 (𝒙, 𝒄))) with ℎ(𝑥) = min 𝜀{1, |𝑥|} 𝑥 |𝑥|, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1) where ℎ is a smoothed projection into the unit ball in R2 and the factor 𝑓 (𝜌) again links the allowed movement speed to the current density, this is, |𝛽(𝜌,𝜙, 𝒙, 𝒄)| ≤ 𝑣0𝑓 (𝜌) in 𝑄𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' We briefy introduce the function spaces used in the sequel, see also Denk, Hieber, Prüss, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' For a domain Ω ⊂ R2 we denote by𝑊 𝑘,𝑝(Ω), 𝑘 ∈ N0, 𝑝 ∈ [1, ∞], the usual Sobolev spaces and by𝑊 𝑘−1/𝑝,𝑝(Γ) for 𝑘 ≥ 1 the corresponding trace spaces which may be equipped with the Sobolev-Slobodetskij norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' 2023-01-09 cbna page 4 of 23 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Pietschmann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Stötzner and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Winkler Pedestrian dynamics Furthermore, we write 𝐻𝑘 (Ω) = 𝑊 𝑘,2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Special spaces incorporating already boundary conditions are 𝐻 1 𝐷 (Ω) = {𝑣 ∈ 𝐻 1(Ω) : 𝑣|𝜕ΩD ≡ 0} and 𝑊 2,𝑝 DN (Ω) := {𝑣 ∈ 𝑊 2,𝑝(Ω) : 𝑣|𝜕Ω𝐷 = 0, 𝜕𝑛𝑣|𝜕Ω𝑊 = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' For time-dependent functions 𝑣 : [0,𝑇] → 𝑋 for some Banach space 𝑋 we defne 𝐿𝑝(0,𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='𝑋) := {𝑣 : (0,𝑇) → 𝑋 | ∫ 𝑇 0 ∥𝑣(𝑡)∥𝑝 𝑋 d𝑡 < ∞}, 𝑝 ∈ [1, ∞), as well as 𝑊 𝑠,𝑝(0,𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='𝑋) := {𝑣 : (0,𝑇) → 𝑋 | 𝜕ℓ 𝑡𝑣 ∈ 𝐿𝑝(0,𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='𝑋), 0 ≤ ℓ ≤ 𝑠}, 𝑠 ∈ N0, 𝑝 ∈ [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' For the application we have in mind the following spaces 𝑊 𝑟,𝑠 𝑝 (𝑄𝑇 ) � 𝐿𝑝(0,𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='𝑊 𝑟,𝑝(Ω)) ∩𝑊 𝑠,𝑝(0,𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' 𝐿𝑝(Ω)), 𝑝 ∈ [1, ∞), 𝑟,𝑠 ∈ N0, are of interest which are equipped with the natural norms ∥𝑣∥𝑊 𝑟,𝑠 𝑝 (𝑄𝑇 ) := � ∥𝑣∥𝑝 𝐿𝑝 (0,𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='𝑊 𝑟,𝑝 (Ω)) + ∥𝑣∥𝑝 𝑊 𝑠,𝑝 (0,𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='𝐿𝑝 (Ω)) �1/𝑝 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Spaces with non-integral 𝑟 and 𝑠 are defned as interpolation spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' In a previous work, Herzog, Pietschmann, Winkler, 2020, a global (in time) well-posedness and regularity result for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='3) was established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Furthermore, optimality conditions for related optimal control problems where this system occurs as a constraint were derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' For convenience of the reader we briefy summarize the most important results needed in the present article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' First, there holds the following existence and regularity result: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Assume that 𝜌0 ∈ 𝑊 3/2,4(Ω) and fx 𝑇 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Given arbitrary agent movement directions 𝒖 = (𝑢1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' ,𝑢𝑀)ᵀ ∈ 𝐿∞(0,𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' R2)𝑀 and intensities 𝒄 = (𝑐1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' ,𝑐𝑀) ∈ 𝐻 1(0,𝑇)𝑀, there exists a unique strong solution (𝜌,𝜙, 𝒙) to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='3) which satisfes, for any 2 < 𝑝 < ∞, 𝜌 ∈ 𝑊 2,1 𝑝 (𝑄𝑇 ) and 𝜙 ∈ 𝐿∞(0,𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='𝑊 2,𝑝(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The agent trajectories 𝑥𝑖, 𝑖 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' , 𝑀 are absolutely continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Moreover, the a priori estimate ∥𝜌∥𝑊 2,1 𝑝 (𝑄𝑇 ) + ∥𝜙∥𝐿∞(0,𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='𝑊 2,𝑝 (Ω)) ≤ 𝐶∥𝜌0∥𝑊 1,𝑝 (Ω), holds with 𝐶 > 0 depending only on the domain, the bounds for the coefcients and the respective kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The previous result allows to defne an operator, the so-called control–to–state operator, 𝑆 : Q → Y, 𝒒 ≔ (𝒖, 𝒄) ↦→ 𝑆(𝒒) = 𝒚 ≔ (𝜌,𝜙, 𝒙) with control and state spaces Q � U × C := 𝐿∞(0,𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' R2)𝑀 × 𝐻 1(0,𝑇)𝑀, Y � 𝑊 2,1 𝑝 (𝑄𝑇 ) × � 𝐿∞(0,𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='𝑊 2,𝑝 DN (Ω)) ∩𝑊 1,𝑝(0,𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='𝑊 1,𝑝(Ω)) � ×𝑊 1,𝑠(0,𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' R2)𝑀 for 𝑠 = � 1 2 − 1 𝑝 �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Furthermore we defne the set of admissible controls Qad := {(𝒖, 𝒄) ∈ Q : |𝑢𝑖(𝑡)| ≤ 1, 0 ≤ 𝑐𝑖(𝑡) ≤ 1 f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' 𝑡 ∈ (0,𝑇) and all 𝑖 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' , 𝑀}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' 2023-01-09 cbna page 5 of 23 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Pietschmann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Stötzner and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Winkler Pedestrian dynamics The optimal control problem we study in this article reads Minimize J (𝒚;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' 𝒒) � ∫ � 𝑄𝑇 𝑒𝜈 𝑡 𝜌(𝑡,𝑥) d𝑥 d𝑡 − 𝜇 𝑀 ∑︁ 𝑖=1 ∫ 𝑇 0 ln(𝜉(𝑥𝑖(𝑡))) d𝑡 + 𝛼1 2𝑇 𝑀 ∑︁ 𝑖=1 ∥𝑢𝑖∥2 𝐻 1(0,𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='R2) + 𝛼2 2𝑇 𝑀 ∑︁ 𝑖=1 ∥𝑐𝑖∥2 𝐻 1(0,𝑇) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='2a) subject to 𝒚 := (𝜌,𝜙, 𝒙) = 𝑆(𝒒), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='2b) 𝒒 := (𝒖, 𝒄) ∈ Qad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='2c) The objective functional J aims at a fast evacuation of the crowd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' By the factor 𝑒𝜈 𝑡 higher densities at a later time are penalized more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' We observe the density in a subregion � 𝑄𝑇 = 𝐼 × �Ω where �Ω ⊂ Ω is a subregion which the pedestrians must leave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' We use the temporal 𝐻 1-norm of the agent movement directions and the intensities as a regularization to guarantee the smoothness required in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The regularization parameters 𝛼1, 𝛼2 > 0 are arbitrary but positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The fourth term in the objective is a barrier used to avoid that the agents walk through walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The barrier function 𝜉 ∈ 𝐻 1 𝐷 (Ω) is the weak solution of the singularly perturbed problem −𝛿4Δ𝜉 + 𝜉 = 1 in Ω, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='3a) 𝜉 = 0 on 𝜕Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='3b) The barrier function − ln(𝜉(𝑥𝑖(𝑡))) tends to infnity if dist(𝑥𝑖(𝑡), 𝜕Ω) → 0 for some 𝑡 ∈ [0,𝑇].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' For 𝑥𝑖(𝑡) ∈ int Ω there holds lim𝜇→0 𝜇 ln(𝜉(𝑥𝑖(𝑡))) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Here we choose 𝜇 > 0 to be fxed but small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The control constraint (𝒖, 𝒄) ∈ Qad guarantees that the agents do not move faster than the density in their current position allows and that the intensity is bounded by reasonable values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' We have the following well-posedness result and necessary optimality condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' There exists at least one global solution (𝒚, 𝒒) ∈ Y × Qad of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Furthermore, each local minimizer (𝒚, 𝒒) ∈ Y × Qad, 𝒚 = (𝜌,𝜙, 𝒙), 𝒒 = (𝒖, 𝒄), of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='2) fulflls for all directions in the tangential cone at (𝒖, 𝒄), namely 𝛿𝒒 = (𝛿𝒖,𝛿𝒄) ∈ TQad(𝒖, 𝒄), ∫ � 𝑄𝑇 𝑒𝜈 𝑡 𝛿𝜌(𝑡,𝑥) d𝑥 d𝑡 + 𝛼1 𝑇 (𝒖 , 𝛿𝒖)𝐻 1(0,𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='R2)𝑀 + 𝛼2 𝑇 (𝒄 , 𝛿𝒄)𝐻 1(0,𝑇)𝑀 − 𝜇 𝑀 ∑︁ 𝑖=1 ∫ 𝑇 0 ∇𝜉(𝑥𝑖(𝑡))⊤ 𝛿𝑥𝑖(𝑡) 𝜉(𝑥𝑖(𝑡)) d𝑡 ≥ 0, with 𝒚 = 𝑆(𝒖, 𝒄) and 𝛿𝒚 = (𝛿𝜌,𝛿𝜙,𝛿𝒙) = 𝑆 ′(𝒖, 𝒄) (𝛿𝒖,𝛿𝒄) characterized by the system 𝜕𝑡𝛿𝜌 − 𝜀Δ𝛿𝜌 − ∇ · � 𝛿𝜌 𝛽(𝜌,𝜙, 𝒙, 𝒄) + 𝜌 � 𝜕𝛽(𝜌,𝜙, 𝒙, 𝒄) 𝜕(𝜌,𝜙, 𝒙, 𝒄) (𝛿𝜌,𝛿𝜙,𝛿𝒙,𝛿𝒄) �� = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='4a) 2023-01-09 cbna page 6 of 23 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Pietschmann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Stötzner and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Winkler Pedestrian dynamics −𝛿1 Δ𝛿𝜙 + 2∇𝜙 · ∇𝛿𝜙 + 2𝑓 (𝜌) 𝑓 ′(𝜌) (𝑓 2(𝜌) + 𝛿2)2𝛿𝜌 = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='4b) �𝛿𝑥𝑖 − 𝑣0 𝑓 ′(𝜌(·,𝑥𝑖)) �∇𝜌(·,𝑥𝑖)ᵀ𝛿𝑥𝑖 + 𝛿𝜌(·,𝑥𝑖)� 𝑢𝑖 = 𝑣0 𝑓 (𝜌(·,𝑥𝑖)) 𝛿𝑢𝑖, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='4c) for 𝑖 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' , 𝑀, together with the boundary conditions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='2) and homogeneous initial conditions 𝛿𝜌(0, ·) = 0 and 𝛿𝑥𝑖(0) = 0, 𝑖 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' , 𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='5) The proof of the theorem above is very close to those of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='8 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='4 Herzog, Pietschmann, Winkler, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The main diference is that the model in Herzog, Pietschmann, Winkler, 2020 only allows to control the velocity 𝒖 of the agents, yet not their strength 𝒄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' As for the existence proof, this does not impose any additional difculty due to the uniform 𝐿∞-boundedness of 𝒄 as a consequence of the embedding 𝐻 1 ↩→ 𝐿∞ in one spatial dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' For the diferentiability result, one has to add the derivatives with respect to 𝒄, yielding an additional term in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='4a) that, however, can be estimated similarly to the remaining terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' 3 Discretization of the state equation In this section we introduce the numerical scheme used to compute approximate solutions of the forward system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' To this end, we introduce a semi-implicit time-stepping scheme and use a fnite volume discretization for the density function 𝜌 and continuous Lagrange fnite elements for the potential function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1 Space discretization For the spatial discretization of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='3) we defne a family of geometrically conforming triangular meshes {Tℎ}ℎ>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' For each 𝑇 ∈ Tℎ we denote by ℎ𝑇 = diam(𝑇) the element diameter and by 𝜌𝑇 the diameter of the largest inscribed ball in 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The mesh parameter is then ℎ = max𝑇 ∈Tℎ ℎ𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The mesh family is assumed to be shape regular meaning that there is a constant 𝜅 > 0 such that ℎ𝑇 /𝜌𝑇 ≤ 𝜅 for all 𝑇 ∈ Tℎ and all ℎ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' By F i ℎ we denote the set of interior element edges, by F bd ℎ the boundary edges and write Fℎ := F i ℎ ∪ F bd ℎ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Furthermore, to each edge 𝐹 ∈ Fℎ we associate the normal vector 𝑛𝐹 which is pointing outward in case of a boundary edge and has a fxed orientation in case of an interior edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' We propose a fnite volume scheme for the transport equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' As we use the fnite element package FEniCS for our implementation, we use a notation which is rather usual for discontinuous Galerkin discretizations, see Di Pietro, Ern, 2012 for an overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The fnite-dimensional function spaces are defned by 𝑉ℎ = {𝑣 ∈ 𝐿∞(Ω) : 𝑣|𝑇 ∈ P0(𝑇) for all 𝑇 ∈ Tℎ, }, 𝑊ℎ = {𝑣 ∈ 𝐶(Ω) : 𝑣|𝑇 ∈ P1(𝑇) for all 𝑇 ∈ Tℎ}, 𝑊ℎ,D := 𝑊ℎ ∩ 𝐻 1 𝐷 (Ω), 2023-01-09 cbna page 7 of 23 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Pietschmann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Stötzner and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Winkler Pedestrian dynamics where P𝑘 (𝑇) denotes the space of polynomials on 𝑇 of degree not larger than 𝑘 ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' For a function 𝑣 : Ω → R, we defne interface averages and jumps in the following way {𝑣}𝐹 := 1 2 (𝑣|𝑇1 + 𝑣|𝑇2), ⟦𝑣⟧𝐹 := 𝑣|𝑇1 − 𝑣|𝑇2 , ∀𝐹 ∈ F i ℎ, where 𝑇1,𝑇2 ∈ Tℎ are chosen in such a way that 𝑛𝐹 = 𝑛𝜕𝑇1 |𝐹 = −𝑛𝜕𝑇2 |𝐹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' In order to discretize the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1) we use discontinuous approximations for 𝜌 and continuous ones for 𝜙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The unknowns in our semi-discrete scheme are 𝜌ℎ(𝑡) ∈ 𝑉ℎ, 𝜙ℎ(𝑡) ∈ 𝑊ℎ,D, 𝑥1(𝑡), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' ,𝑥𝑀 (𝑡) ∈ R2, 𝑐1(𝑡), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' ,𝑐𝑀 (𝑡) ∈ R for all 𝑡 ∈ [0,𝑇].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The approximate transport direction is then given by 𝛽ℎ(𝜌ℎ,𝜙ℎ, 𝒙, 𝒄) ≔ 𝑓 (𝜌ℎ) ℎ(∇𝜙ℎ + 𝜙𝐾 (𝒙, 𝒄)) with 𝜙𝐾 (𝒙, 𝒄;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='𝑡,𝑥) := 𝑀 ∑︁ 𝑗=1 𝑐 𝑗 (𝑡) 𝐾(𝑥 − 𝑥𝑖(𝑡)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The semi-discretization of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1a) then reads Find 𝜌ℎ : [0,𝑇] → 𝑉ℎ with 𝜌ℎ(0) = proj𝑉ℎ (𝜌0) and (𝜕𝑡𝜌ℎ(𝑡), 𝑣ℎ)Ω + 𝑎(𝜌ℎ(𝑡), 𝑣ℎ) + 𝑏(𝛽ℎ)(𝜌ℎ(𝑡), 𝑣ℎ) = 0 ∀𝑣ℎ ∈ 𝑉ℎ,𝑡 ∈ (0,𝑇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1) Here, proj𝑉ℎ : 𝐿2(Ω) → 𝑉ℎ is some projection operator, (·, ·)Ω is the standard 𝐿2(Ω)-inner product and the bilinear forms are defned by 𝑎(𝜌ℎ, 𝑣ℎ) = 𝜀 ∑︁ 𝐹 ∈Fi ℎ ∫ 𝐹 𝜏𝐹 ⟦𝜌ℎ⟧⟦𝑣ℎ⟧ d𝑠 + ∑︁ 𝐹 ∈Fbd ℎ 𝜒𝜕Ω𝐷𝛾 ∫ 𝐹 𝜌ℎ 𝑣ℎ d𝑠 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='2a) 𝑏(𝛽ℎ)(𝜌ℎ, 𝑣ℎ) = − ∑︁ 𝐹 ∈Fi ℎ ∫ 𝐹 (𝜌ℎ 𝛽ℎ)∗ 𝐹 ⟦𝑣ℎ⟧ d𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='2b) The parameter 𝜏𝐹 is defned by 𝜏𝐹 ≔ 1 |𝑥𝑇1 − 𝑥𝑇2|, where 𝑥𝑇 is the intersection of the orthogonal edge bisectors of 𝑇 ∈ Tℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The term 𝜏𝐹 ⟦𝜌ℎ⟧⟦𝑣ℎ⟧ with 𝑣ℎ = 𝜒𝑇 for some 𝑇 ∈ Tℎ approximates the difusive fux ∇𝜌ℎ · 𝑛𝜕𝑇 over the edge 𝐹 ⊂ 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The bilinear form 𝑏 establishes the convective fux 𝛽 𝜌 · 𝑛𝜕𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' As numerical fux function (·)∗, we choose the Lax-Friedrichs fux, see Rider, Lowrie, 2002, defned by (𝜌ℎ 𝛽)∗ 𝐹 = {𝜌ℎ 𝛽}𝐹 · 𝑛𝐹 − 𝜂 2 ⟦𝜌ℎ⟧𝐹 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='3) The stabilization parameter 𝜂 ∈ R is specifed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' For the closely related chemotaxis model such an approach has been sucessfully applied in Li, Shu, Yang, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Guo, Li, Yang, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Of course, also other fux functions are possible, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=', the central upwind fux, Epshteyn, Kurganov, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' 2023-01-09 cbna page 8 of 23 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Pietschmann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Stötzner and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Winkler Pedestrian dynamics The Eikonal equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1b) is discretized in space using standard linear Lagrange elements which yields 𝛿1 (∇𝜙ℎ(𝑡), ∇𝑤ℎ)Ω + �|∇𝜙ℎ(𝑡)|2,𝑤ℎ � Ω = � 1 𝑓 (𝜌ℎ(𝑡))2 + 𝛿2 ,𝑤ℎ � Ω ∀𝑤ℎ ∈ 𝑊ℎ,D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='4) In our numerical experiments we used the Newton solver integrated in FEniCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The Jacobian is established by automatic diferentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The ordinary diferential equations for the agent trajectories (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1c) depend on a point evaluation 𝜌(𝑡,𝑥𝑖(𝑡)) of a function which is discontinuous in the discrete setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' In particular, this term would not be diferentiable with respect to 𝑥𝑖(𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' As a remedy, we use instead of a point evaluation, see also Borsche, Colombo, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=', 2015, the following regularization 𝜌ℎ(𝑡,𝑥𝑖(𝑡)) ≈ 𝜂𝑥𝑖 (𝑡) ∗ 𝜌ℎ(𝑡), with 𝜂𝑥𝑖 (𝑡) := 𝛿𝑥𝑖 (𝑡) 𝛿𝑥𝑖 (𝑡) ∗ 1, where ∗ stands for the convolution integral 𝛿𝑥0 ∗ 𝑣 = ∫ Ω 𝛿𝑥0 𝑣 d𝑥 of the functions 𝑣 ∈ 𝐿1(Ω) and some kernel function 𝛿𝑥0 ∈ 𝐶∞(R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' An obvious choice is the regularized Dirac delta function 𝛿𝑥0(𝑥) := 1 2 𝜋 𝜁 e− ∥𝑥−𝑥0∥2 2𝜁 with small locality parameter 𝜁 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Note that for 𝜁 → 0 there holds 𝛿𝑥0 ∗ 𝑣 → 𝑣(𝑥0) for any 𝑣 ∈ 𝐶(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Furthermore, the regularized Dirac delta fulflls ∫ R2 𝛿𝑥0 d𝑥 = 1 for arbitrary 𝑥0 ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The discretized ordinary diferential equation then reads �𝑥𝑖(𝑡) = 𝑣0 𝑓 �𝜂𝑥𝑖 (𝑡) ∗ 𝜌ℎ(𝑡)� 𝑢𝑖(𝑡), 𝑛 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' , 𝑁 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='5) and initial conditions 𝑥𝑖(0) = 𝑥𝑖,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='2 Time discretization For the temproal discretization we cover the time interval [0,𝑇] by an equidistant grid 𝐼𝜏 � {𝑡𝑛}𝑁 𝑛=0 with grid points 𝑡𝑛 := 𝑛 𝜏 and grid size 𝜏 := 𝑇/𝑁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The spatial and temporal discretization parameters are summarized in 𝜎 = (ℎ,𝜏).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Moreover, we defne the space of grid functions 𝐻𝜏 (𝑉 ) = {𝑣 : 𝐼𝜏 → 𝑉 }, with 𝑉 an arbitrary linear space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' If 𝑉 is again a function space containing functions 𝑣 : Ω → R we write 𝑣(𝑡𝑛) = 𝑣(𝑡𝑛, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The functions 𝜌ℎ, 𝜙ℎ, 𝑥𝑖, 𝑢𝑖 and 𝑐𝑖 arising in the semidiscrete equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='4) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='5) are approximated by grid functions 𝜌𝜎 ∈ 𝐻𝜏 (𝑉ℎ), 𝜙𝜎 ∈ 𝐻𝜏 (𝑊ℎ,D), 𝑢𝑖,𝜎,𝑥𝑖,𝜎 ∈ 𝐻𝜏 (R2), 𝑐𝑖,𝜎 ∈ 𝐻𝜏 (R) for 𝑖 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' , 𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' For brevity we write for 𝑛 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' , 𝑁 𝜌𝑛 ℎ ≔ 𝜌𝜎 (𝑡𝑛, ·), 𝜙𝑛 ℎ ≔ 𝜙𝜎 (𝑡𝑛, ·), 𝑥𝑛 𝑖 ≔ 𝑥𝑖,𝜎 (𝑡𝑛), 𝑢𝑛 𝑖 ≔ 𝑢𝑖,𝜎 (𝑡𝑛) 2023-01-09 cbna page 9 of 23 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Pietschmann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Stötzner and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Winkler Pedestrian dynamics and for the transport vector we use 𝛽𝑛 ℎ = 𝛽ℎ(𝜌𝑛 ℎ,𝜙𝑛 ℎ, 𝒙𝑛, 𝒄𝑛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' We replace the temporal derivative by a diference quotient and use a semi-implicit time-stepping scheme, more precisely, the difusion-related terms are evaluated implicitly and the convection-related terms explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' This yields the fully-discrete system (𝜌𝑛+1 ℎ , 𝑣ℎ)Ω + 𝜏 𝑎(𝜌𝑛+1 ℎ , 𝑣ℎ) = (𝜌𝑛 ℎ, 𝑣ℎ)Ω − 𝜏 𝑏(𝛽𝑛 ℎ)(𝜌𝑛 ℎ, 𝑣ℎ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='6a) 𝛿1 �∇𝜙𝑛 ℎ, ∇𝑤ℎ � Ω + �|∇𝜙𝑛 ℎ |2,𝑤ℎ � Ω = � 1 𝑓 (𝜌𝑛 ℎ)2 + 𝛿2 ,𝑤ℎ � Ω , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='6b) 𝑥𝑛+1 𝑖 − 𝑥𝑛 𝑖 = 𝜏 𝑣0 𝑓 � 𝜂𝑥𝑛+1 𝑖 ∗ 𝜌𝑛+1 ℎ � 𝑢𝑛+1 𝑖 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='6c) for all test functions 𝑣ℎ ∈ 𝑉ℎ, 𝑤ℎ ∈ 𝑊ℎ,D and indices 𝑖 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' , 𝑀, 𝑛 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' , 𝑁 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Furthermore, the initial conditions are established by means of: 𝜌0 ℎ = proj𝑉ℎ (𝜌0), 𝑥0 𝑖 = 𝑥𝑖,0, 𝑖 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' , 𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Note that the system of equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='6) completely decouples and we can compute each variable after the other, in the following order 𝜌0 ℎ, 𝒙0 ↦→ 𝜙0 ℎ ↦→ 𝜌1 ℎ ↦→ 𝒙1 ↦→ 𝜙1 ℎ ↦→ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' ↦→ 𝜌𝑁−1 ℎ ↦→ 𝒙𝑁−1 ↦→ 𝜙𝑁−1 ℎ ↦→ 𝜌𝑁 ℎ ↦→ 𝒙𝑁 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='7) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='3 Quality of discrete solutions In this section we study some basic properties for the solutions of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' In particular, it is of inter- est whether the physical bounds observed for the solution of the continuous system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='3) are transferred to the discrete setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The basis functions of the fnite element space 𝑉ℎ are denoted by {𝜒𝑇 }𝑇 ∈Tℎ defned by 𝜒𝑇 |𝑇 ′ ≡ 𝛿𝑇,𝑇 ′ for all 𝑇,𝑇 ′ ∈ Tℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Note that by a slight abuse of notation we use the elements of Tℎ as indices here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Introducing the matrices 𝑀 = (𝑚𝑇,𝑇 ′)𝑇,𝑇 ′∈Tℎ, 𝐴 = (𝑎𝑇,𝑇 ′)𝑇,𝑇 ′∈Tℎ and 𝐵𝑛 = (𝑏𝑛 𝑇,𝑇 ′)𝑇,𝑇 ′∈Tℎ with entries 𝑚𝑇,𝑇 ′ = � |𝑇 |, if 𝑇 = 𝑇 ′, 0, otherwise, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='8a) 𝑎𝑇,𝑇 ′ = ������ ������ 𝜀 ∑︁ 𝐹 ∈F𝑇 ∩Fi ℎ 𝜏𝐹 |𝐹 | + 𝛾 ∑︁ 𝐹 ∈F𝑇 ∩Fbd ℎ |𝐹 |, if 𝑇 = 𝑇 ′, −𝜀 𝜏𝐹 |𝐹 |, if 𝑇 ≠ 𝑇 ′ and 𝐹 := 𝜕𝑇 ∩ 𝜕𝑇 ′ ≠ ∅, 0, otherwise, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='8b) 𝑏𝑛 𝑇,𝑇 ′ = �������� �������� − 1 2 ∑︁ 𝐹 ∈F𝑇 ∩Fi ℎ �∫ 𝐹 𝛽𝑛 ℎ |𝑇 · 𝑛𝜕𝑇 d𝑠 − 𝜂 |𝐹 | � , if 𝑇 = 𝑇 ′, − 1 2 �∫ 𝐹 𝛽𝑛 ℎ |𝑇 ′ · 𝑛𝜕𝑇 d𝑠 + 𝜂 |𝐹 | � , if 𝑇 ≠ 𝑇 ′ and 𝐹 = 𝜕𝑇 ∩ 𝜕𝑇 ′ ≠ ∅, 0, otherwise, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='8c) 2023-01-09 cbna page 10 of 23 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Pietschmann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Stötzner and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Winkler Pedestrian dynamics allows to rewrite the system of equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='6a) as (𝑀 + 𝜏 𝐴) �𝜌𝑛+1 = (𝑀 − 𝜏 𝐵𝑛) �𝜌𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='9) Here, �𝜌𝑛, 𝑛 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' , 𝑁, are the vector representations of 𝜌𝑛 ℎ with respect to the basis {𝜒𝑇 }𝑇 ∈Tℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Note that the matrix 𝐵𝑛 depends also on �𝜌𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The numerical scheme (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='6a) is mass conserving in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Assuming that 𝛾 = 0 holds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=', there are no-fux boundary conditions present at all boundary parts 𝜕ΩD and 𝜕ΩW, the solution 𝜌𝜎 fulflls ∫ Ω 𝜌𝑛 ℎ d𝑥 = ∫ Ω proj𝑉ℎ (𝜌0) d𝑥 ∀𝑛 = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' , 𝑁 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The assertion is trivially fulflled for 𝑛 = 0 as the initial condition is established by 𝜌0 ℎ = proj𝑉ℎ (𝜌0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' In matrix-vector notation the assertion is equivalent to �1⊤𝑀 �𝜌𝑛+1 = �1⊤𝑀 �𝜌𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' This follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='9) after using �1⊤𝐴 �𝜌𝑛+1 = ∑︁ 𝑇 ∈Tℎ ∑︁ 𝑇 ′∈Tℎ 𝑎𝑇,𝑇 ′𝜌𝑛+1 𝑇 ′ = 0 and �1⊤𝐵𝑛 �𝜌𝑛 = ∑︁ 𝑇 ∈Tℎ ∑︁ 𝑇 ′∈Tℎ 𝑏𝑇,𝑇 ′𝜌𝑛 𝑇 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' In this expression, the stabilization terms (the ones multiplied with 𝜂) cancel out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Furthermore, after sorting terms in the sum by the edges 𝐹 ∈ F i ℎ and denoting by 𝑇𝐹,1,𝑇𝐹,2 the two triangles meeting in 𝐹, we obtain the terms �1⊤𝐵𝑛 �𝜌𝑛 = − 1 2 ∑︁ 𝐹 ∈Fi ℎ ∫ 𝐹 � 𝛽ℎ|𝑇𝐹,1 𝑛𝜕𝑇𝐹,1 𝜌𝑇𝐹,1 + 𝛽ℎ|𝑇𝐹,2 𝑛𝜕𝑇𝐹,1 𝜌𝑇𝐹,2 +𝛽ℎ|𝑇𝐹,2 𝑛𝜕𝑇𝐹,2 𝜌𝑇𝐹,2 + 𝛽ℎ|𝑇𝐹,1 𝑛𝜕𝑇𝐹,2 𝜌𝑇𝐹,1 � d𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The last step follows due to 𝑛𝜕𝑇𝐹,2 = −𝑛𝜕𝑇𝐹,2 which implies �1⊤𝐵𝑛 �𝜌𝑛 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' □ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Choose 𝜂 = 1 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='3) and denote by 𝜅 > 0 the maximal aspect ratio of the mesh family Tℎ, see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Let 𝜏 be chosen to satisfy the CFL condition 𝜏 ≤ 𝜋 3𝜅2 min 𝑇 ∈Tℎ ℎ𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='10) If furthermore, there holds proj𝑉ℎ (𝜌0) ∈ [0, 1] a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' in Ω and 𝛽𝑛 ℎ = (1 − 𝜌𝑛 ℎ) Φ𝑛 with |Φ𝑛| ≤ 1, 𝑛 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' , 𝑁, the solutions of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='6) fulfll for all 𝑛 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' , 𝑁 𝜌𝑛 ℎ (𝑥) ∈ [0, 1] f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' 𝑥 ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The diagonal entries of (𝑀 + 𝜏 𝐴) are all positive and the of-diagonal entries are negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Moreover, one easily concludes the strict diagonal dominance, this is, ∑︁ 𝑇′∈Tℎ 𝑇′≠𝑇 |𝑚𝑇,𝑇 ′ + 𝜏 𝑎𝑇,𝑇 ′| < 𝑚𝑇,𝑇 + 𝜏 𝑎𝑇,𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' 2023-01-09 cbna page 11 of 23 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Pietschmann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Stötzner and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Winkler Pedestrian dynamics This implies that (𝑀 +𝜏 𝐴) is an M-matrix and consequently, the inverse (𝑀 +𝜏 𝐴)−1 exists and fulflls (𝑀 + 𝜏 𝐴)−1 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Let 𝑛 ∈ N0 be fxed and assume that 𝜌𝑛 ℎ (𝑥) ∈ [0, 1] for almost all 𝑥 ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' We show 𝜌𝑛+1 ℎ ≥ 0 by confrming that the right-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='9) has non-negative entries only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Assuming that 𝜌𝑛 𝑇 ≥ 0, 𝑇 ∈ Tℎ, we show that the entries of (𝑀 − 𝜏 𝐵𝑛) are non-negative as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The entries on the diagnoal have the form |𝑇 | + 𝜏 2 ∑︁ 𝐹 ∈F𝑇 ∩Fi ℎ ∫ 𝐹 (𝛽𝑛 ℎ |𝑇 · 𝑛𝜕𝑇 − 1) d𝑠 ≥ |𝑇 | − 𝜏 |𝜕𝑇 |, where the frst step follows from the assumption (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1) implying |𝛽𝑛 ℎ | ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' To estimate further we take the geometric mesh quantities and the shape regularity ℎ𝑇 ≤ 𝜅 𝜌𝑇 into account and arrive at |𝑇 | |𝜕𝑇 | ≥ 𝜋 𝜌2 𝑇 3ℎ𝑇 ≥ 𝜋 ℎ𝑇 3𝜅2 ≥ 𝜏, where the last step is the CFL condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The previous two estimates confrm (𝑚𝑇,𝑇 −𝜏 𝑏𝑛 𝑇,𝑇 ) ≥ 0 for all 𝑇 ∈ Tℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The remaining entries in the matrix, namely (𝑚𝑇,𝑇 ′ − 𝜏 𝑏𝑛 𝑇,𝑇 ′) with 𝐹 = 𝜕𝑇 ∩ 𝜕𝑇 ′ ≠ ∅, have the form 𝜏 2 ∫ 𝐹 (𝛽𝑛 ℎ |𝑇 ′ · 𝑛𝜕𝑇 + 1) d𝑠 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The non-negativity follows again from |𝛽𝑛 ℎ | ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Combining the previous arguments provides the lower bound 𝜌𝑛+1 ℎ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' To show the upper bound we rearrange the equation system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='9) in the form (𝑀 + 𝜏 𝐴) (�1 − �𝜌𝑛+1) = 𝑀 (�1 − �𝜌𝑛) + 𝜏 𝐵𝑛 �𝜌𝑛 + 𝜏 𝐴�1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='11) We may rewrite the transport term using 𝜌𝑛 ℎ 𝛽𝑛 ℎ = (1 − 𝜌𝑛 ℎ) �𝛽𝑛 ℎ with �𝛽𝑛 ℎ = 𝜌𝑛 ℎ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' In (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='11) we reformulate the expression involving 𝐵𝑛 by means of [𝐵𝑛 �𝜌𝑛]𝑇 = − ∑︁ 𝐹 ∈F𝑇 ∩Fi ℎ ∫ 𝐹 � {(1 − 𝜌𝑛 𝑇 ) �𝛽𝑛 ℎ}𝐹 · 𝑛𝐹 + 1 2 ⟦1 − 𝜌𝑛 ℎ⟧ � 𝐹 ⟦𝜒𝑇 ⟧𝐹 d𝑠 = − 1 2 ∑︁ 𝐹 ∈F𝑇 ∩Fi ℎ ∫ 𝐹 (�𝛽𝑛 ℎ |𝑇 · 𝑛𝜕𝑇 + 1) d𝑠 · (1 − 𝜌𝑛 𝑇 ) − 1 2 ∑︁ 𝐹 ∈F𝑇 ∩Fi ℎ ∫ 𝐹 (�𝛽𝑛 ℎ |𝑇𝐹 · 𝑛𝜕𝑇 − 1) d𝑠 · (1 − 𝜌𝑛 𝑇𝐹 ) =: [�𝐵𝑛(1 − �𝜌𝑛)]𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' with 𝑇𝐹 ∈ Tℎ \\ {𝑇 }, 𝑇𝐹 ∩𝑇 = 𝐹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' With the same arguments like above one can show that the entries of 𝑀 +𝜏 �𝐵𝑛 are non-negative and together with 1 − �𝜌𝑛 ≥ 0, 𝐴�1 ≥ 0 and the M-matrix property of 𝑀 +𝜏 𝐴 we arrive at the desired bound 1 − �𝜌𝑛+1 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='4 The discrete optimal control problem Next, we study a discrete version of the optimal control problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The control and state variables are grid functions in time and thus, we introduce the discrete 𝐻 1(0,𝑇)-inner product for functions 2023-01-09 cbna page 12 of 23 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Pietschmann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Stötzner and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Winkler Pedestrian dynamics 𝑢𝜏, 𝑣𝜏 : {𝑡𝑛}𝑁 𝑛=0 → 𝑉 , with 𝑉 some Hilbert space, (𝑢𝜏, 𝑣𝜏)𝐻 1(0,𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='𝑉 ),𝜏 := 𝜏 𝑁 ∑︁ 𝑛=0 (𝑢𝑛, 𝑣𝑛)𝑉 ×𝑉 + 𝜏−1 𝑁−1 ∑︁ 𝑛=0 �𝑢𝑛+1 − 𝑢𝑛, 𝑣𝑛+1 − 𝑣𝑛� 𝑉 ×𝑉 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' This induces the norm ∥𝑢𝜏 ∥2 𝐻 1(0,𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='𝑉 ),ℎ := (𝑢𝜏,𝑢𝜏)𝐻 1(0,𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='𝑉 ),ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' For the discrete control space we obtain U𝜎 := {𝒖𝜎 = (𝑢1,𝜎, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' ,𝑢𝑀,𝜎) : 𝑢𝑖,𝜎 ∈ 𝐻𝜏 (R2) for 𝑖 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' , 𝑀}} C𝜎 := {𝒄𝜎 = (𝑐1,𝜎, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' ,𝑐𝑀,𝜎) : 𝑐𝑖,𝜎 ∈ 𝐻𝜏 (R) for 𝑖 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' , 𝑀}, Q𝜎 := U𝜎 × C𝜎, and the admissible set by U𝜎,ad := {𝒖𝜎 ∈ U𝜎 : |𝑢𝑛 𝑖 | ≤ 1, 𝑖 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' , 𝑀, 𝑛 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' , 𝑁 }, C𝜎,ad := {𝒄𝜎 ∈ C𝜎 : 0 ≤ 𝑐𝑛 𝑖 ≤ 1,𝑖 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' , 𝑀,𝑛 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' , 𝑁 }, Q𝜎,ad := U𝜎,ad × C𝜎,ad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The discrete state space is defned by Y𝜎 := 𝐻𝜏 (𝑉ℎ) × 𝐻𝜏 (𝑊ℎ,D) × 𝐻𝜏 (R2)𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' With these defnitions, the discrete optimal control problem related to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='2) reads as Minimize J𝜎 (𝒚𝜎, 𝒒𝜎) = 𝜏 𝑁 ∑︁ 𝑛=1 𝑒𝜈 𝑡𝑛 ∫ Ω 𝜌𝑛 ℎ (𝑥) d𝑥 − 𝜇 𝜏 𝑀 ∑︁ 𝑖=1 𝑁 ∑︁ 𝑛=1 ln(𝜂𝑥𝑛 𝑖 ∗ 𝜉ℎ) + 𝛼1 2𝑇 𝑀 ∑︁ 𝑖=1 ∥𝑢𝑖,𝜎 ∥2 𝐻 1(0,𝑇),𝜏 + 𝛼2 2𝑇 𝑀 ∑︁ 𝑖=1 ∥𝑐𝑖,𝜎 ∥2 𝐻 1(0,𝑇),𝜏 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='12a) subject to 𝒚𝜎 := (𝜌𝜎,𝜙𝜎, 𝒙𝜎) = 𝑆𝜎 (𝒒𝜎), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='12b) 𝒒𝜎 := (𝒖𝜎, 𝒄𝜎) ∈ Q𝜎,ad, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='12c) where is 𝜉ℎ ∈ 𝑊ℎ the fnite element approximation of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='3) with frst-order Lagrange elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Fur- thermore, 𝑆𝜎 is the solution operator of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Note that, in order to maintain the diferentiability of the barrier term with respect to 𝑥𝑛 𝑖 , we use a regularization of the point evaluation of the nonsmooth function 𝜉ℎ, compare also (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='6c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' We may write the control problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='12) in the more compact reduced form 𝑗𝜎 (𝒒𝜎) := J𝜎 (𝑆𝜎 (𝒒𝜎), 𝒒𝜎) → min!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' subject to 𝒒𝜎 ∈ Q𝜎,ad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='13) To deduce a necessary optimality condition we apply the Lagrange formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The Lagrange function L𝜎 : Y𝜎 × Q𝜎 × Y𝜎 → R 2023-01-09 cbna page 13 of 23 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Pietschmann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Stötzner and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Winkler Pedestrian dynamics coupling the discrete state equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='6) reads L𝜎 (𝜌𝜎,𝜙𝜎, 𝒙𝜎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' 𝒖𝜎, 𝒄𝜎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' 𝜆𝜌,𝜎, 𝜆𝜙,𝜎, 𝜆𝒙,𝜎) = J𝜎 (𝜌𝜎,𝜙𝜎, 𝒙𝜎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' 𝒖𝜎, 𝒄𝜎) − ∫ Ω (𝜌0 ℎ − proj𝑉ℎ (𝜌0)) 𝜆0 𝜌,ℎ d𝑥 − 𝑁−1 ∑︁ 𝑛=0 �∫ Ω (𝜌𝑛+1 ℎ − 𝜌𝑛 ℎ) 𝜆𝑛+1 𝜌,ℎ d𝑥 + 𝜏 𝑎(𝜌𝑛+1 ℎ , 𝜆𝑛+1 𝜌,ℎ ) − 𝜏 𝑏(𝛽𝑛 ℎ)(𝜌𝑛 ℎ, 𝜆𝑛+1 𝜌,ℎ ) � − 𝑁 −1 ∑︁ 𝑛=0 𝜏 � 𝛿1 ∫ Ω ∇𝜙𝑛 ℎ · ∇𝜆𝑛 𝜙,ℎ d𝑥 + ∫ Ω |∇𝜙𝑛 ℎ |2 𝜆𝑛 𝜙,ℎ d𝑥 − ∫ Ω 1 𝑓 (𝜌𝑛 ℎ)2 + 𝛿2 𝜆𝑛 𝜙,ℎ d𝑥 � − 𝑀 ∑︁ 𝑖=1 � (𝑥0 𝑖 − 𝑥𝑖,0)⊤ 𝜆0 𝒙,𝑖 + 𝑁−1 ∑︁ 𝑛=0 � 𝑥𝑛+1 𝑖 − 𝑥𝑛 𝑖 − 𝜏 𝑓 (𝜂𝑥𝑛+1 𝑖 ∗ 𝜌𝑛+1 ℎ ) 𝑢𝑛+1 𝑖 �⊤ 𝜆𝑛+1 𝒙,𝑖 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' To shorten the notation we write 𝝀𝜎 := (𝜆𝜌,𝜎, 𝜆𝜙,𝜎, 𝜆𝒙,𝜎).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The adjoint equation system determining these variables for a given control and state is 𝜆𝑛 𝜌,ℎ ∈ 𝑉ℎ : 𝜕L𝜎 𝜕𝜌𝑛 ℎ (𝒚𝜎, 𝒒𝜎, 𝝀𝜎)𝛿𝜌ℎ= 0 ∀𝛿𝜌ℎ ∈ 𝑉ℎ, 𝑛 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' , 𝑁, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='14a) 𝜆𝑛 𝜙,ℎ ∈ 𝑊ℎ,D : 𝜕L𝜎 𝜕𝜙𝑛 ℎ (𝒚𝜎, 𝒒𝜎, 𝝀𝜎)𝛿𝜙ℎ= 0 ∀𝛿𝜙ℎ ∈ 𝑊ℎ, 𝑛 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' , 𝑁 − 1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='14b) 𝜆𝑛 𝒙𝑖 ∈ R2 : 𝜕L𝜎 𝜕𝑥𝑛 𝑖 (𝒚𝜎, 𝒒𝜎, 𝝀𝜎)𝛿𝑥𝑖 = 0 ∀𝛿𝑥𝑖 ∈ R2, 𝑛 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' , 𝑁 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='14c) for 𝑖 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' , 𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Note that this can be interpreted as a coupled system involving a parabolic PDE and an ODE that run backward in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' We use the automatic diferentiation feature in FEniCS in our implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' As the forward system completely decouples in each time step, so does the adjoint system and we can compute step by step: 𝜆𝑁 𝒙,ℎ ↦→ 𝜆𝑁 𝜌,ℎ ↦→ (𝜆𝑁 𝜙,ℎ) ↦→ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' ↦→ 𝜆0 𝒙,ℎ ↦→ 𝜆0 𝜌,ℎ ↦→ 𝜆0 𝜙,ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' With the adjoint states at hand we can assemble the derivatives of the reduced objective (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='13) and end up with the following optimality condition for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='12): Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='3 (Necessary optimality condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Let (𝒚𝜎, 𝒒𝜎) ∈ Y𝜎 × Q𝜎,ad be a local solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Then, there exists 𝝀𝜎 ∈ Y𝜎 fulflling (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='14) and 𝛼1 (𝑢𝜎,𝑖, 𝑣𝜎,𝑖 − 𝑢𝜎,𝑖)𝐻 1(0,𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='R2),𝜏 + 𝛼2 (𝑐𝜎,𝑖,𝑑𝜎,𝑖 − 𝑐𝜎,𝑖)𝐻 1(0,𝑇),𝜏 + 𝜏 𝑁 ∑︁ 𝑛=1 � 𝑓 (𝜂𝑥𝑛 𝑖 ∗ 𝜌𝑛 ℎ) 𝜆𝑛 𝒙𝑖, 𝑣𝑛 𝑖 − 𝑢𝑖𝑛� R2 + 𝜏 𝑁−1 ∑︁ 𝑛=0 𝜕 � 𝑏(𝛽𝑛 ℎ)(𝜌𝑛 ℎ, 𝜆𝑛+1 𝜌,ℎ ) � 𝜕𝑐𝑛 𝑖 (𝑑𝑛 𝑖 − 𝑐𝑛 𝑖 ) ≥ 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='15) for all test functions 𝒓𝜎 := (𝒗𝜎, 𝒅𝜎) ∈ Q𝜎,ad and all 𝑖 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' , 𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' It is well-known that the variational inequality 𝑗 ′ 𝜎 (𝒒𝜎)(𝒓𝜎 − 𝒒𝜎) ≥ 0 for 𝒓𝜎 ∈ Q𝜎,ad is necessary for 𝒒𝜎 being a local minimizer of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Taking into account the the equivalence 𝑗 ′ 𝜎 (𝒒𝜎)𝛿𝒒𝜎 = 𝜕L 𝜕𝒒𝜎 (𝒚𝜎, 𝒒𝜎, 𝝀𝜎)𝛿𝒒𝜎 if𝝀𝜎 solves (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='14) yields the variational inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' □ 2023-01-09 cbna page 14 of 23 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Pietschmann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Stötzner and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Winkler Pedestrian dynamics Our solution algorithm is based on a projected gradient algorithm and it remains to establish a representation of the gradient of 𝑗𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The derivative of the objective (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='12a) towards some direction 𝛿𝒒𝜎 = (𝛿𝒖𝜎,𝛿𝒄𝜎) ∈ Q reads 𝑗 ′ 𝜎 (𝒒𝜎)𝛿𝒒𝜎 = 𝑀 ∑︁ 𝑖=1 � 𝛼1(𝑢𝑖,𝜎,𝛿𝑢𝑖,𝜎)𝐻 1(0,𝑇),𝜏 + 𝛼2(𝑐𝑖,𝜎,𝛿𝑐𝑖,𝜎)𝐻 1(0,𝑇),𝜏 + 𝜏 𝑁 ∑︁ 𝑛=1 𝑓 (𝜂𝑥𝑛 𝑖 ∗ 𝜌𝑛 ℎ) 𝛿𝑢𝑛 𝑖 ⊤𝜆𝑛 𝒙𝑖 + 𝜏 𝑁−1 ∑︁ 𝑛=0 𝜕 � 𝑏(𝛽𝑛 ℎ)(𝜌𝑛 ℎ, 𝜆𝑛+1 𝜌,ℎ ) � 𝜕𝑐𝑛 𝑖 𝛿𝑐𝑛 𝑖 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' To obtain a representation of the 𝐻 1(0,𝑇),𝜏-gradient of 𝑗𝜎 with respect to 𝒖𝜎 and 𝒄𝜎, we introduce the grid functions 𝑧𝑖,𝜎 : {𝑡𝑛}𝑁 𝑛=0 → R2 and 𝑑𝑖,𝜎 : {𝑡𝑘}𝑁 𝑛=0 → R, 𝑖 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' , 𝑀 solving ��������� � 1 𝜏2 ��������� � 1 −1 −1 2 −1 −1 2 −1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' −1 2 −1 −1 1 ��������� � + 𝐼𝑁+1×𝑁+1 ��������� � ��������� � 𝑧0 𝑖 𝑧1 𝑖 𝑧2 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' 𝑧𝑁−1 𝑖 𝑧𝑁 𝑖 ��������� � = ��������� � 0 −𝑓 (𝜌1 ℎ(𝒙1 𝑖))𝜆0 𝑥𝑖 −𝑓 (𝜌2 ℎ(𝒙2 𝑖 ))𝜆1 𝑥𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' −𝑓 (𝜌𝑁−1 ℎ (𝒙𝑁−1 𝑖 ))𝜆𝑁−2 𝑥𝑖 −𝑓 (𝜌𝑁 ℎ (𝒙𝑁 𝑖 ))𝜆𝑁−1 𝑥𝑖 ��������� � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='16a) and ��������� � 1 𝜏2 ��������� � 1 −1 −1 2 −1 −1 2 −1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' −1 2 −1 −1 1 ��������� � + 𝐼𝑁+1×𝑁+1 ��������� � ��������� � 𝑑0 𝑖 𝑑1 𝑖 𝑑2 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' 𝑑𝑁−1 𝑖 𝑑𝑁 𝑖 ��������� � = ������������� � −𝜕𝑐0 𝑖 𝑏(𝛽0 ℎ)(𝜌0 ℎ, 𝜆1 𝜌,ℎ) −𝜕𝑐1 𝑖𝑏(𝛽1 ℎ)(𝜌1 ℎ, 𝜆2 𝜌,ℎ) −𝜕𝑐2 𝑖𝑏(𝛽2 ℎ)(𝜌2 ℎ, 𝜆3 𝜌,ℎ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' −𝜕𝑐𝑁 −2 𝑖 𝑏(𝛽𝑁−2 ℎ )(𝜌𝑁 −2 ℎ , 𝜆𝑁−1 𝜌,ℎ ) −𝜕𝑐𝑁 −1 𝑖 𝑏(𝛽𝑁−1 ℎ )(𝜌𝑁−1 ℎ , 𝜆𝑁 𝜌,ℎ) 0 ������������� � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='16b) for 𝑖 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' , 𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' By a simple calculation we then confrm (𝑧𝑖,𝜎,𝛿𝑢𝑖,𝜎)𝐻 1(0,𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='R2),𝜏 = 𝑁 ∑︁ 𝑛=1 𝑓 (𝜂𝑥𝑛 𝑖 ∗ 𝜌𝑛 ℎ) 𝛿𝑢𝑛 𝑖 ⊤𝜆𝑛 𝒙𝑖, (𝑑𝑖,𝜎,𝛿𝑐𝑖,𝜎)𝐻 1(0,𝑇),𝜏 = 𝑁 −1 ∑︁ 𝑛=0 𝜕 � 𝑏(𝛽𝑛 ℎ)(𝜌𝑛 ℎ, 𝜆𝑛+1 𝜌,ℎ ) � 𝜕𝑐𝑛 𝑖 𝛿𝑐𝑛 𝑖 We write 𝒛𝜎 = (𝑧1,𝜎, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' ,𝑧𝑀,𝜎) ∈ U and 𝒅𝜎 = (𝑑1,𝜎, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' ,𝑑𝑀,𝜎) ∈ C and get the following representation of the gradient of 𝑗𝜎: ∇𝒖𝜎 𝑗𝜎 (𝒒𝜎) = 𝛼1 𝒖𝜎 + 𝒛𝜎, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='17a) ∇𝒄𝜎 𝑗𝜎 (𝒒𝜎) = 𝛼2 𝒄𝜎 + 𝒅𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='17b) This allows an implementation of a projected gradient method which we discuss in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' 2023-01-09 cbna page 15 of 23 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Pietschmann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Stötzner and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Winkler Pedestrian dynamics 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='5 Optimization algorithms for the discretized problem For a solution of the discretized optimal control problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='12 we propose a projected gradient algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' In this procedure, for a given initial control 𝒒(0) = (𝒖 (0), 𝒄 (0)) ∈ Q, the new iterates are sucessively computed by means of 𝒖 (𝑘+1) 𝜎 = 𝚷𝑢 ad � 𝒖 (𝑘) 𝜎 − 𝑠(𝑘) ∇𝒖𝜎 𝑗𝜎 (𝒒(𝑘) 𝜎 ) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='18a) 𝒄 (𝑘+1) 𝜎 = 𝚷𝑐 ad � 𝒄 (𝑘) 𝜎 − 𝑠(𝑘) ∇𝒄𝜎 𝑗𝜎 (𝒒(𝑘) 𝜎 ) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='18b) with 𝚷𝑢 ad : U𝜎 → U𝜎,ad and 𝚷𝑐 ad : C𝜎 → C𝜎,ad the 𝐻 1(0,𝑇),𝜏 projections onto the admissible sets U𝜎,ad and C𝜎,ad, respectively, this is, 𝚷𝑢 ad(𝒖𝜎) := arg min 𝒗𝜎 ∈U𝜎,ad 1 2 ∥𝒖𝜎 − 𝒗𝜎 ∥2 𝐻 1(0,𝑇;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='R2),𝜏, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='19a) 𝚷𝑐 ad(𝒄𝜎) := arg min 𝒅𝜎 ∈C𝜎,ad 1 2 ∥𝒄𝜎 − 𝒅𝜎 ∥2 𝐻 1(0,𝑇),𝜏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='19b) A formula for the gradient of 𝑗𝜎 has been derived in the previous section already, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The step length parameter 𝑠 (𝑘) > 0 is obtained by an Amijo line search and must fulfll the sufcient decrease condition 𝑗𝜎 (𝒒(𝑘) 𝜎 − 𝑠(𝑘) ∇𝑗𝜎 (𝒒(𝑘) 𝜎 )) ≤ 𝑗𝜎 (𝒒(𝑘) 𝜎 ) − 𝑑 𝑠 (𝑘) ∥𝒒(𝑘) 𝜎 − � 𝒒(𝑘) 𝜎 − 𝑠 (𝑘) ∇𝑗𝜎 (𝒒(𝑘) 𝜎 ) � ∥2 Q𝜎 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='20) with a decrease parameter 𝑑 ∈ (0, 1) which is usually small (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' 10−4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' A reasonable stopping criterion for the projected gradient algorithm is ∥𝒒(𝑘) 𝜎 − 𝚷ad � 𝒒(𝑘) 𝜎 − ∇𝑗𝜎 (𝒒(𝑘) 𝜎 ) � ∥Q𝜎 ≤ 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' It remains to discuss the realization of the projection operators and we propose a primal dual active set strategy that may also be considered as semismooth Newton method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Note that the operators Πad are semismooth, see Christof, Wachsmuth, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The evaluation of the projection operator 𝚷𝑢 ad : U𝜎 → U𝜎,ad requires to solve the optimization problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='19a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The unknowns (assuming 𝑀 = 1 and omitting the agent’s index 𝑖 for a while) are the coefcients of the functions U𝜎 ∋ 𝚷ad(𝑢𝜎) = 𝑤𝜎 ≃ �𝑤 ∈ R(𝑁+1)×2 for some given U𝜎 ∋ 𝑢𝜎 ≃ �𝑢 ∈ R(𝑁+1)×2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' We switch to a matrix-vector notation and defne 𝑤𝑛 := �𝑤𝑛 1 𝑤𝑛 2 � := 𝑤𝜎 (𝑡𝑛), �𝑤 𝑗 := (𝑤0 𝑗, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' ,𝑤𝑁 𝑗 )⊤, 𝑗 = 1, 2, as well as the matrix 𝐴 ∈ R(𝑁+1)×(𝑁+1) on the left-hand side of the linear system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='16) inducing the discrete 𝐻 1(0,𝑇),𝜏-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The Lagrangian for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='19) reads 𝐿( �𝑤1, �𝑤2, �𝜆) = 1 2 2 ∑︁ 𝑖=1 ( �𝑤𝑖 − �𝑢𝑖)⊤ 𝐴( �𝑤𝑖 − �𝑢𝑖) − 1 2 𝜆⊤� | �𝑤|2 ∗ − �1 � , 2023-01-09 cbna page 16 of 23 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Pietschmann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Stötzner and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Winkler Pedestrian dynamics with | �𝑤|2 ∗ = (|𝑤0|2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' , |𝑤𝑁 |2)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The Karush-Kuhn-Tucker system for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='19) then reads 𝐴 ( �𝑤𝑖 − �𝑢𝑖) − �𝜆 · �𝑤𝑖 = 0 𝑖 = 1, 2, 1 2 � | �𝑤|2 ∗ − �1 � ≤ 0, �𝜆 ≥ 0, 1 2 �𝜆 · �| �𝑤|2 ∗ − 1� = 0, where · is the component-wise multiplication of two vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' We reformulate the complementarity condition by means of a nonsmooth equation and arrive at the following equivalent form of the KKT system 𝐹 ( �𝑤1, �𝑤2, �𝜆) := ������� 𝐴 ( �𝑤1 − �𝑢1) − �𝜆 · �𝑤1 𝐴 ( �𝑤2 − �𝑢2) − �𝜆 · �𝑤2 �𝜆 − max{0, − 1 2 (| �𝑤|2 ∗ − �1) + �𝜆} ������� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='21) This nonlinear system can be solved iteratively by a semismooth Newton method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Given is an initial pair (�𝑢 (0), �𝜆(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Successively, one computes the active and inactive set A (𝑘) := {𝑛 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' , 𝑁 }: − 1 2 (|𝑤𝑛|2 2 − 1) + 𝜆𝑛 > 0}, I (𝑘) := {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' , 𝑁 } \\ A (𝑘), solves the Newton system ������� 𝐴 − 𝐷 �𝜆(𝑘) 0 −𝐷 �𝑤(𝑘) 1 0 𝐴 − 𝐷 �𝜆(𝑘) −𝐷 �𝑤(𝑘) 2 𝐷A (𝑘) 𝐷 �𝑤(𝑘) 1 𝐷A (𝑘) 𝐷 �𝑤(𝑘) 2 𝐷I (𝑘) ������� ������� � 𝛿𝑤1 � 𝛿𝑤2 �𝛿𝜆 ������� = − ������� 𝐴 ( �𝑤 (𝑘) 1 − �𝑢1) − �𝜆(𝑘) · �𝑤 (𝑘) 1 𝐴 ( �𝑤 (𝑘) 2 − �𝑢2) − �𝜆(𝑘) · �𝑤 (𝑘) 2 �𝜆(𝑘) − max{0, − 1 2 (| �𝑤 (𝑘)|2 ∗ − �1) + �𝜆(𝑘)}, ������� with the diagonal matrices 𝐷�𝑣 = diag(�𝑣) for �𝑣 ∈ R𝑁+1 and 𝐷M = diag(𝜒M) for M ⊂ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' , 𝑁 }, and performs the Newton update �𝑤 (𝑘+1) = �𝑤 (𝑘) + � 𝛿𝑤, �𝜆(𝑘+1) = �𝜆(𝑘) + �𝛿𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' This procedure is repeated for 𝑘 = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' until some termination criterion, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=', ∥𝐹 ( �𝑤1, �𝑤2, �𝜆)∥ < tol, is fulflled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' 4 Numerical experiments This section is devoted to numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' To establish the discretized system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='12) the fnite element library FEniCS was used, complemented by a Python implementation of the projected gradient method from Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='18) and the Armijo step size rule from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The computational meshes were created by the mesh generator mshr integrated in FEniCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1 Example 1 In a frst numerical test we solve the problem Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='2) in the domain Ω depicted in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1 with the following parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' 𝑇 = 9 𝑛𝑇 = 300 𝛼1 = 𝛼2 = 5 · 10−2 𝛾 = 10 𝜁 = 10−2 𝜇 = 5 · 10−2 𝜀 = 10−5 𝛿1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='2 𝛿2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1 𝛿3 = 10−2 𝛿4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' 2023-01-09 cbna page 17 of 23 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Pietschmann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Stötzner and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Winkler Pedestrian dynamics The initial density 𝜌0 is the sum of 6 Gaussian bells, see also Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The subdomain �Ω where densities are penalized is chosen to cover the region within the walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Without any controlled agents, most of the people will squeeze through the 2 smaller emergency exits in the south and north while the large exit in the east is rarely used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' To improve the evacuation 3 agents were introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The initial control (𝒖, 𝒄) was chosen in such a way that the agent moves straight to the right outside of the room having a constant the intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='2 Example 2 In a second example we consider the domain illustrated in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The initial density is concentrated near the slit in the wall on the left-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' In an uncontrolled evacuation scenario the majority of the people would leave the domain through this slit causing a massive congestion leading to a very slow evacuation of the crowd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The model and algorithm parameters chosen in the current example are as follows: 𝑇 = 12 𝑛𝑇 = 300 𝛼1 = 𝛼2 = 5 · 10−2 𝛾 = 10 𝜁 = 10−2 𝜇 = 5 · 10−2 𝜀 = 10−5 𝛿1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='2 𝛿2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1 𝛿3 = 10−2 𝛿4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' This example shows that the evacuation can be signifcantly improved by using two agents with optimized trajectory and intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Interesting is, that the intensity is non-zero only in the time interval 𝑡 ∈ (0, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The agents attract the people leading them sufciently far away from the slit in the west and then they stop infuencing the crowd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' When being sufciently far away from the slit the people fnd the way to the larger exits in the north and south on their own by using the movement direction determined by the potential 𝜙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='3 Example 3 In a third example we consider a square-shaped room with exits in the south, east and north.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The exits have diferent width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The model and algorithm parameters are chosen as follows: 𝑇 = 10 𝑛𝑇 = 300 𝛼1 = 𝛼2 = 5 · 10−2 𝛾 = 10 𝜁 = 10−2 𝜇 = 5 · 10−2 𝜀 = 10−5 𝛿1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='2 𝛿2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1 𝛿3 = 10−2 𝛿4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The initial density is concentrated near the small exit and without a control of the crowd motion most of the people are blocking each other while squeezing through this small exit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Two agents were added in this scenario with the aim attracting the people in such a way that more of them fnd the other two exits in the north and east.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The computed agent trajectories are quite short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' It is interesting to observe that in the time interval 𝑡 ∈ (0, 2) the agents just go to an optimal position sufciently close to the crowd and attract them in the time interval 𝑡 ∈ (2, 5), leading some of the people to the center of the room.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' At this point the agents drive their intensity to zero meaning that they stop infuencing the crowd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' However, when being sufciently far away from the critical exit the people fnd the route to the less used exits on their own due to the movement rule determined by the potential 𝜙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' 2023-01-09 cbna page 18 of 23 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Pietschmann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Stötzner and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Winkler Pedestrian dynamics (a) Solution at time 𝑡 = 0 (b) Solution at time 𝑡 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='2 (c) Solution at time 𝑡 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='4 (d) Solution at time 𝑡 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='2 (e) Solution at time 𝑡 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='2 100 200 300 1 2 3 4 𝑘 𝑐𝑖 𝑐1(𝑡) 𝑐2(𝑡) 𝑐3(𝑡) (f) Agent intensities 𝑐𝑖 (𝑡) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1: Solution of the problem from Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' The colored background represents the density 𝜌;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' the dots are the agent positions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' the black curves are the agent trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' 2023-01-09 cbna page 19 of 23 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='9 1J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Pietschmann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Stötzner and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Winkler Pedestrian dynamics (a) Solution at time 𝑡 = 0 (b) Solution at time 𝑡 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='4 (c) Solution at time 𝑡 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='8 (d) Solution at time 𝑡 = 8 20 40 60 80 100 120 140 160 180 200 220 1 2 3 4 5 6 𝑘 𝑐𝑖 𝑐1(𝑡) 𝑐2(𝑡) (e) Intensities 𝑐𝑖 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='2: Solution of the problem from Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='2 at various time steps 𝑡𝑘 and intensity of the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' 2023-01-09 cbna page 20 of 23 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='9 1J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Pietschmann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Stötzner and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Winkler Pedestrian dynamics (a) Solution at time 𝑡 = 0 (b) Solution at time 𝑡 = 2 (c) Solution at time 𝑡 = 4 (d) Solution at time 𝑡 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='7 50 100 150 200 250 300 1 2 𝑘 𝑐𝑖 𝑐1(𝑡) 𝑐2(𝑡) (e) Intensities 𝑐𝑖 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='3: Solution of the problem from Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='3 at various time steps 𝑡𝑘 and intensity of the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' 2023-01-09 cbna page 21 of 23 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Pietschmann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Stötzner and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Winkler Pedestrian dynamics References Albi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Bongini;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Cristiani;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Kalise (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' “Invisible Control of Self-Organizing Agents Leav- ing Unknown Environments”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' SIAM Journal on Applied Mathematics 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' 1683–1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' 1137/15M1017016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' eprint: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1137/15M1017016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1137/ 15M1017016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Albi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Fornasier;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Kalise (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' “A Boltzmann approach to mean-feld sparse feedback control”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' IFAC-PapersOnLine 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' 2898–2903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Amadori, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Di Francesco (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' “The one-dimensional Hughes model for pedestrian fow: Riemann- type solutions”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Acta Mathematica Scientia 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' 259–280.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1016/s0252-9602(12)60016-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Amadori, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Goatin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Rosini (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' “Existence results for Hughes’ model for pedestrian fows”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Journal of Mathematical Analysis and Applications 420.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' 387–406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='jmaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' 05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='072.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Banda, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' M.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' “Deterministic particle approximation of the Hughes model in one space dimension”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' Kinetic & Related Models 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' 215–237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E0T4oBgHgl3EQfnwFc/content/2301.02516v1.pdf'} +page_content='3934/krm.' metadata={'source': 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b/c9E1T4oBgHgl3EQfdwSQ/content/tmp_files/2301.03199v1.pdf.txt @@ -0,0 +1,1712 @@ +Grid-Adaptation for Wall-Modeled Large Eddy +Simulation Using Unstructured High-Order Methods +Marcel Blinda,∗, Ali Berk Kahramanb, Johan Larssonb, Andrea Becka,c +aInstitute for Aerodynamics and Gas Dynamics, University of Stuttgart, Pfaffenwaldring +21, Stuttgart, 70569, Germany +bDepartment of Mechanical Engineering, University of Maryland, College +Park, 20742, MD, USA +cInstitute of Fluid Dynamics and Thermodynamics, Otto-von-Guericke-University +Magdeburg, Universitätsplatz 2, Magdeburg, 39106, Germany +Abstract +The accuracy and computational cost of a large eddy simulation are highly +dependent on the computational grid. +Building optimal grids manually +from a priori knowledge is not feasible in most practical use cases; instead, +solution-adaptive strategies can provide a robust and cost-efficient method +to generate a grid with the desired accuracy. We adapt the grid-adaptation +algorithm developed by Toosi and Larsson [1] to a Discontinuous Galerkin +Spectral Elements Method (DGSEM) and show its potential on fully un- +structured grids. The core of the method is the computation of the estimated +modeling residual using the polynomial basis functions used in DGSEM, and +the averaging of the estimated residual over each element. The final method +is assessed in multiple channel flow test cases and for the transonic flow over +an airfoil, in both cases making use of mortar interfaces between elements +with hanging nodes. The method is found to be robust and reliable, and to +provide solutions at up to 50% lower cost at comparable accuracy compared +to when using human-generated grids. +Keywords: +high-order methods, WMLES, non-conforming grids, grid error +indicator +∗corresponding author +Email address: blind@iag.uni-stuttgart.de (Marcel Blind) +Preprint submitted to arXiv +January 10, 2023 +arXiv:2301.03199v1 [physics.flu-dyn] 9 Jan 2023 + +1. Introduction +As the large eddy simulation (LES) technique gathers increased attention +from the engineering community, the open research questions for the method +are also changing. While much of the focus in the past was on the devel- +opment of subgrid scale stress models and numerical methods with limited +dissipation, today other issues in the LES method demand more attention. +First and foremost among those is the question of how to design a good grid, +especially for more complicated geometries. The grid-generation process is +still mainly an art that relies on the experience of the user, who needs to +incorporate knowledge of numerics and the resolution requirements of flow +physics into his/her judgment. This can even lead to a point where differ- +ent users can create different grids for the same geometry and get different +results. +Creating an objectively good mesh is crucial for an efficient and +resource saving simulation, as well as increasing the dependability of the +solutions. The ultimate objective of this work is to provide a workflow to +generate a problem-tailored mesh that is compatible with unstructured high- +order methods starting from a very coarse initial mesh generated without +any problem-specific insight. +There have been several attempts at devising ways to quantify how well +resolved an LES is. Early arguments were based on the ratio of subgrid to +viscous dissipation or viscosity (e.g. [2, 3]), but these concepts are meaningful +only in the buffer layer of wall-bounded turbulence since LES should be ap- +plicable to free shear flows at any Reynolds number. Others have suggested +that the sufficiency of a grid in LES should be measured by the ratio of +modeled to resolved (or total) turbulence kinetic energy (cf. [4, 5]), but this +has been found to correlate poorly to the known behavior of length scales +in wall-bounded flows [1]. Methods that approximate a local turbulent spec- +trum have some basis for isotropic flows, but fail for more relevant cases [6]. +The most well-grounded approach to date is that by Toosi and Larsson [1] +which can be viewed as an estimate of the LES modeling residual, i.e., the +source term in an error transport equation [7]. +The objective of the present work is to extend the residual estimate of [1, +7] to high-order codes and to test the ability of the resulting grid-adaptation +method to produce efficient (high accuracy at low cost) grids in a high-order +type solver. In this paper we use the Discontinuous Galerkin (DG) method. +Implementation in a DG-type code differs from that in finite-difference or +2 + +finite-volume codes (where the residual estimate has been tested before) in +several ways, including how one performs the low-pass filtering, the sensitivity +to aliasing errors, and how one locally averages the results to be meaningful +for grid-adaptation. The method is mainly assessed in several channel flows +where the behavior of LES is relatively well known and where we therefore +can judge the optimality of the resulting grids. The method is finally applied +to the transonic flow around an airfoil to verify that it works for more realistic +and complex problems. +2. Methodology +Any grid-adaptation method necessarily starts by estimating the spatial +distribution of the error generation process (i.e., the residual, or source term +for the solution error). It then proceeds by generating a new mesh that would +minimize the residual field. The main focus of this paper is on the first step, +specifically on how to compute the estimated residual field in the context of +a Discontinuous Galerkin Spectral Element Method (DGSEM) code for grids +with primarily hexahedral elements. +2.1. The DGSEM code FLEXI +The simulations are run using the Discontinuous Galerkin Spectral Ele- +ment Method (DGSEM) framework FLEXI. The code is developed in the +Numerics Research Group at the University of Stuttgart [8]. +It is open +source and can be downloaded from http://www.flexi-project.org/. The +DGSEM [8, 9, 10] uses piecewise smooth functions in every element, but al- +lows for discontinuities at element interfaces. However, the residual estima- +tion procedure (Sec. 2.2) and subsequent grid-adaptation can be transferred +to any other DG or related scheme with minimal changes, as long as an +element-local filter operation can be specified. Thus, the details regarding +implementation given below serve to elucidate the procedure for this specific +DG variant. +The computational domain Ω is partitioned in non-overlapping elements +C ∈ Ω. The solution in each element is described using a polynomial basis of +degree P. We discretize the compressible Navier-Stokes equation using the +same polynomial basis function as ansatz function in each three-dimensional +element. The polynomials in each direction are represented using nodal 1D +3 + +Lagrange basis functions on P + 1 Legendre-Gauss-Lobatto points. Other +node choices are possible, here we stick with these to be able to formulate the +so-called split forms for the advective operators. To apply common integra- +tion rules and to build an efficient scheme, we define the three-dimensional +hexahedral reference element [−1, 1]3. We create the three-dimensional basis +as the tensor product of three one-dimensional polynomials. We integrate +the Navier-Stokes equation in space using the associated P + 1 Legendre- +Gauss-Lobatto quadrature to compute the projection integral. This results +in the very efficient DGSEM. At the element boundaries we use a Riemann +solver e.g. [11]. To take the parabolic terms into account, the BR1 lifting +procedure [12] is applied. According to the method of lines we advance the +resulting ODE in time by applying an explicit-in-time low storage Runge- +Kutta method with optimized stability region [13]. +We rely on skew-symmetric splitting of the advective terms of the com- +pressible Navier-Stokes equations for stability [14], since in implicitly filtered +LES the simulations are typically under-resolved. To obtain a dissipation- +free and kinetic-energy preserving semi-discrete system we use the skew- +symmetric split fluxes by Pirozzoli [15]. The Roe numerical flux is used to +solve the Riemann problem at the cell interfaces [16]. We use the Vreman +model for explicit sub grid scale modeling in the following simulation [17]. +2.2. Estimating the LES Residual +Consider a general evolution equation ∂q/∂t = R(q) solved numerically +on a grid with spacing ∆, the solution of which can be denoted by +N∆(q∆) ≡ R∆(q∆) − ∂q∆ +∂t = 0 , +(1) +where R∆ implies that R is approximated at grid-spacing ∆ and q∆ means +the solution to the discrete problem at this grid-spacing. The error in this +equation is then q∆ − q, which satisfies the error equation +∂N∆ +∂q (q∆ − q) ≈ N∆(q∆) +� �� � +=0 +− N∆(q) +� �� � +F +, +where F is the residual, i.e., the source term for error. +The residual is +defined based on the exact solution q which is, of course, not known. All grid- +adaptation methods therefore involve some type of process for estimating the +4 + +residual from the numerical solution q∆, for example using the leading terms +in Taylor expansions of the numerical operators or by directly approximating +q by interpolating q∆ onto a refined mesh. Neither approach works in large +eddy simulation (LES) which by definition seeks a solution to the coarse- +grained Navier-Stokes equation. While one could interpolate the solution +q∆ onto a finer grid, in reality the solution on this finer grid should have +developed smaller scales due to the broadband nature of turbulence; the +resulting residual estimate would therefore be incorrect. For the same reason, +the fact that an LES solution is, by definition, a rough solution far from +the asymptotic range of numerical convergence means that many terms in +a Taylor expansion should be expected to be large, and thus one could not +approximate the error behavior from the leading term only as is traditionally +done in numerical analysis. +Toosi and Larsson [1, 7] proposed that the modeling residual (i.e., the +residual due to imperfect resolution/modeling of the small scale turbulence, +excluding the residual due to numerical errors) in LES can be estimated in +a post-processing step using low-pass test-filtering. Specifically, they argued +that the residual must be estimated at an imagined coarser resolution �∆ than +the resolution ∆ used in the actual LES. Their argument went as follows: +Assume that the LES equations at resolution ∆ (i.e., the LES solved in +the code) can be written as in Eqn. (1). The residual at the test-filtered +(or additionally coarse-grained) level is then N�∆(�q), where �q is the exact +solution restricted (or test-filtered) to the resolution �∆. Note that one could +not define the residual with q directly, since this would contain smaller scales +that would then be double-counted due to the possible subgrid-model in the +LES. Toosi and Larsson [1, 7] then suggested that �q can be approximated by +� +q∆, and thus defined the approximate modeling residual as +F�∆ = N�∆ (� +q∆) = R�∆ (� +q∆) − ∂� +q∆ +∂t . +Assuming that the test-filter commutes with the time-derivative and using +Eqn. (1) then yields +F�∆ = R�∆ (� +q∆) − R∆(q∆) +� . +(2) +In the context of subsonic flows without shocks or other multi-physics +effects (heat release, multi-phase, etc), the modeling residual for the momen- +5 + +tum equation then becomes +Fi,�∆ = +∂ +∂xj +� +ρuiuj + pδij + τ mod +ij +(ρ, ui, ∆) − σij +� +� +(3) +− ∂ +∂xj +� +�ρ�ui�uj + �pδij + τ mod +ij +(�ρ, �ui, �∆) − � +σij +� +, +where the test-filtered velocity should be understood as a Favre-weighted +filter. +All operations above were defined based on the instantaneous flow fields, +and thus the estimated modeling residual Fi,�∆ is necessarily a chaotic field +in space and time. +Assuming that one seeks a stationary grid (i.e., that +one runs a full LES and then adapts the grid before running another full +LES), one must then reduce the residual field in time (and possibly in any +spatially homogeneous directions). While there is no theoretical reason for +any specific choice of reduction, Toosi and Larsson [1, 7] suggested using +the L2 norm. In addition, they argued that the test-filter should be chosen +as a uni-directional one, providing filtering in one direction at a time. For +hexahedral elements, this implies that one should compute 3 different residual +fields (using test-filtering in each of the 3 directions of a hexahedral element) +which then provides information about the lack of resolution in each direction +separately; this then allows for anisotropic grid-adaptation. +Putting these things together, the final time-averaged modeling residual +is written as +G(⃗x,⃗n) = +� +⟨Fi(⃗x,⃗n)Fi(⃗x,⃗n)⟩ , +(4) +where ⃗x denotes the spatial location, ⃗n denotes the direction of the uni- +directional test-filter, and the angular brackets denote averaging in time, +spatially homogeneous directions, and possibly locally in space as well (to be +discussed below). +We emphasize that the subgrid model contribution in the residual (3) +must be computed with the correct length scale: when evaluated for the +test-filtered field, it must use the length scale of the imagined �∆ resolution. +This length scale must be estimated from the properties of the test filter. +For implicit LES run without an explicit subgrid model, the subgrid model +terms are simply zero in the residual estimate. +6 + +2.3. Computation of the LES modeling residual in a DGSEM code +The implementation of the LES residual estimator described above in a +Discontinuous Galerkin Spectral Element Method (DGSEM) requires some +care and specific implementation details, which are described in this section. +The low-pass test-filtering operator �· is realized as a uni-directional +modal cut-off filter, which removes the highest modes of the polynomial rep- +resentation in one direction only. Throughout this work we used 5th order +polynomials, i.e. P = 5 where P represents the polynomial order, to repre- +sent the solution in each direction resulting in a 6th order accurate scheme, +and chose to low-pass filter by setting the coefficients of the 4th and 5th order +polynomials to zero. Importantly, this was done in only one direction, with +the high-order polynomial coefficients in the other directions remaining un- +changed. A nice feature of the DGSEM method is that the filtering in one +element is independent from all other elements, and thus it works equally +well in an unstructured grid (of hexahedral elements). +The LES residual F�∆ could be computed using either the generic Eqn. (2) +or the more flow-specific Eqn. (3). The most convenient and easy-to-implement +option is the former one, as most codes (especially ones with explicit time- +stepping) have functions to compute R∆ for a given solution field. Given an +instantaneous LES field q∆, it is then trivial to compute R∆(q∆) using a sin- +gle function-call in the code. The resulting field is then low-pass test-filtered +three times, one for each natural direction of each hexahedral element, to +form three different instantiations of the second part of Eqn. (2). +In a separate process, the LES solution itself is test-filtered to form � +q∆. +In the present work, we apply the test-filter to the conserved variables which +then implies that the filtered velocity is Favre-weighted. This test-filtered +LES solution is then fed into the computation of R�∆(� +q∆) to complete Eqn. (2). +This last step requires the most care, as the function in the LES code will +(for the given grid and polynomial order) compute R∆(� +q∆) rather than the +correct R�∆(� +q∆). If we are interested only in the LES modeling residual rather +than the numerical errors (as was the case in Toosi and Larsson [1, 7]), then +the only modification required is to ensure that the length scale in the subgrid +model is reflective of the coarse-grained state. This is the option chosen in +this work. In principle one could also use a lower polynomial order when +evaluating R�∆(� +q∆) as a way to also approximately account for the numerical +error, but this is left for future work. +7 + +The present simulations use the Vreman subgrid model [17] but in a +modified form in which the length scale is taken as +∆ = V1/3 +P + 1 , +where V is the element volume and P is the polynomial order, taken as P = 5 +in this work. Since we use a uni-directional filter that keeps modes up to the +Mth order (taken as M = 3 in this work), the consistent length scale after +test-filtering is +�∆ = +� P + 1 +M + 1 +�1/3 +∆ . +Since the model implemented in the FLEXI code uses an isotropic length +scale, this implies that the eddy viscosity at the test-filtered level is simply a +factor of [(P +1)/(M +1)]2/3 ≈ 1.59 larger than the eddy viscosity computed +by the code. +The norm operation in equation Eqn. (4) can be tricky because the solu- +tion is represented as a polynomial in an appropriate number of collocation +points, but the included squaring operator in the norm operation needs a +higher order of polynomial, thus more collocation points. The approach we +took is to map the F�∆ from the original P degree of polynomial to 2P degree +of polynomial and its associated collocation points, and then take the square +of the polynomial at that order. This way we got an exact representation of +the term Fi(⃗x,⃗n)Fi(⃗x,⃗n). +Another issue has risen in the computation, which is the cell edges. Since +the DG solution is discontinuous at the element boundaries and local projec- +tions tend to produce large gradients at these boundaries, the local gradients +at the cell boundaries in an underresolved setting can overshoot. This ba- +sically results in the residual estimator showing the element boundaries as +high error regions, when this is in fact a numerical effect that is excluded +from its goal, to find high error regions in the modeling of turbulence. To +mitigate this issue, and to stay true to the cell based nature of the discontin- +uous Galerkin method, we average the Fi(⃗x,⃗n)Fi(⃗x,⃗n) term over the whole +cell before the periodic direction or time averaging. This averaging is also +included in the ⟨⟩ averaging operator that has been used so far. +8 + +2.4. Finding the optimal grid-spacing +Once the estimated residuals for test-filtering in each direction are com- +puted, the optimal element size in each direction can be found from an op- +timization problem and an assumed model for how the residuals vary under +grid-refinement. +Following Toosi and Larsson [1], the directional residual G(⃗x,⃗n) is as- +sumed to vary as +G(⃗x,⃗n) = g(⃗x,⃗n) ∆(⃗x,⃗n)α . +(5) +In the asymptotic limit of convergence, the power α would be the order- +of-accuracy of the numerical method. In the context of LES which is, by +definition, far from the asymptotic limit of convergence, the power α is in- +stead related to the spectral behavior of turbulence near the grid cut-off. +Since the spectral slope of inertial range turbulence varies across flow types +and directions, it is clear that α should in theory also vary analogously. The +more important point is that the value of α comes entirely from turbulence +physics and not from numerical analysis. Rather than try to find the “cor- +rect” α field, we follow the suggestion of Toosi and Larsson [18] and simply +assume a constant value of α = 2 throughout the flow domain. This is almost +certainly larger than the “correct” value in most flow scenarios; erring on the +side of a larger value produces lower variations in the optimal grid-spacing, +which can be beneficial in practice. With this model for the residual scaling, +the “residual density” g(⃗x,⃗n) can be computed by evaluating Eqn. (5) for the +residual estimated on the current grid, in each direction. +We then define the cost functional +J [∆(⃗x,⃗n)] = +��� +V +� +� +�� +m +g(⃗x,⃗nm)β∆(⃗x,⃗nm)βα +�1/β ++ +λ +� +m +∆(⃗x,⃗nm) +� +� dV +where the first term accounts for the sum of residuals in all directions in a +β-norm (note that all factors are positive) and the second term accounts for +the computational complexity (defined here as the number of elements) of the +new grid, with λ being a Lagrange multiplier. The solution to this particular +calculus-of-variations problem is that the variation of the integrand with +respect to the ∆(⃗x,⃗ni) field is zero for each hexahedral direction i = 1, 2, 3. +9 + +Using the abbreviated notation ∆i = ∆(⃗x,⃗ni) and gi = g(⃗x,⃗ni), this is +αgβ +i ∆βα−1 +i,opt +�� +m +gβ +m∆βα +m,opt +�1/β−1 +− +λ +∆i,opt +� +m +∆m,opt += 0 , +i = 1, 2, 3 , +where the subscript “opt” has been added to indicate that this is the ∆(⃗x,⃗n) +field that minimizes the cost functional J . Some rearrangement yields +� +gi∆α +i,opt +�β = +λ/α +� +m +∆m,opt +�� +m +gβ +m∆βα +m,opt +�1−1/β +, +i = 1, 2, 3 . +This is valid for all i = 1, 2, 3 but the right-hand-side is the same for all i; thus +the optimal residual Gi,opt = gi,opt∆α +i,opt is the same in all three directions. +This can then be exploited to find that +gi,opt∆α +i,opt +� +m +∆m,opt +� +�� +� +Vopt += Λ , +i = 1, 2, 3 , +(6) +where Λ is a re-defined Lagrange multiplier and Vopt is the volume of the +optimal element; this equation shows that the element-integrated residual +should be equi-distributed in both space and direction. +The optimal grid-spacing is then found by using a root-finding proce- +dure to find the value of the modified Lagrange multiplier Λ for which the +computational complexity (=the number of elements) is as desired. In each +step, Eqn. (6) is used to find the optimal ∆(⃗x,⃗n) field. The process was +documented originally in Toosi and Larsson [1] and is provided here for com- +pleteness. +We note that the final result (Eqn. (6)) is independent of the value of β, +and thus the choice of norm in the cost functional is immaterial. +2.5. Overall process +The grid-adaptation process is implemented entirely as a post-processing +approach that is performed between LES runs, with no adaptation occurring +on-the-fly during a run. After a simulation, the residual field is computed +from multiple instantaneous snapshots of the solution, with averaging in +10 + +time and possibly suitable spatial directions. We then compute the residual +“density” field g(⃗x,⃗n) from the residual scaling model, and the use iterative +root-finding on the Lagrange multiplier Λ to find the optimal grid-spacing +field ∆(⃗x,⃗n) with the desired computational complexity. From this point +we have multiple choices for the actual creation of the adapted grid. One +approach is to modify the existing grid, most simply by splitting individ- +ual elements in those directions for which the element-integrated directional +residual exceeds some threshold. A second choice is to re-generate the grid +to more closely follow the optimal target. This latter approach has mainly +been followed for simplex elements in the literature [19, 20]. Since we desire +meshes with hexahedral cells in this work, we resort to human intervention +in the grid-generation process: we build the grids manually, but aimed to +mimic the optimal grid-spacing field as closely as possible. In future work +we hope to automate this aspect. +We note that the estimated residuals do not say anything about whether +the grid is sufficiently fine to be considered converged; instead, any judgment +about convergence must be made for specific quantities-of-interest (e.g., mean +profiles, drag or lift coefficients, etc). This judgment must be made by the +user, prior to deciding whether to create an adapted grid. +3. Application to channel flows +In this and the following section, we use the abbreviated notation ∆i = +∆(⃗x,⃗ni) where ⃗ni is a unit vector in one of the three natural directions of a +hexahedral element. +As the first step the algorithm is applied to the canonical test case of +the plane turbulent channel flow. The appeal of channel flow is that it is +one of very few problems where the “optimal” grid is known to some degree; +it is then a good test of a grid-adaptation method to see whether it can +arrive at something close to the known “optimum”. The domain is taken +as (10h, 2h, 3h) where h is the channel half-height. Two different Reynolds +numbers of Reτ = 550 and Reτ = 2000 are targeted, with the lower used to +compare the adapted grids to those by Toosi and Larsson [1] and the higher +used to test the grid-adaptation process when a wall-model is used and when +mortar elements are used in the adapted grid. +11 + +3.1. Channel flow at Reτ = 550, validation of the original results +The point of this case is to reproduce the original results of [1], establish- +ing that the discontinuous Galerkin method is suitable for the grid-adaptation +process this way. We start with a uniform mesh of 2 cells along the channel +half-width h, 10 cells along the streamwise direction and 3 cells along the +spanwise direction with each cell containing a basis of 5th order polynomials. +This is a similar setup to [1] (although a tad “finer”), where they start with +grid spacings (0.2h, 0.1h, 0.2h). +For each iteration, the grid-adaptation algorithm is used as described, +with each new grid having 4 times the previous number of elements. It is +constructed such that it is smoothly stretched along the wall-normal direc- +tion, agreeing with the near wall ∆2 and the center line ∆2, and has uniform +cells along the streamwise x and spanwise z directions with the minimum +suggested ∆1 and ∆3, respectively. The wall-normal stretching is realized +using a bell shaped +DR(s) = 1 + +� ∆2,min +∆2,max +− 2 +� +· e−(s·f)2 − e−f2 +1 − e−f2 +(7) +element distribution with s ∈ [−1, 1]. The ration ∆2,min +∆2,max describes the differ- +ence between the smallest and the largest element in y direction, f denotes +the scaling factor and the result DR defines the element local weight which +is later normalized to get the element size distribution in s ∈ [−1, 1]. To +ensure compatibility with Toosi and Larsson [1], we use the Vreman subgrid +scale model for LES closure [17] with the same model constant 0.03. The +grid-adaptation process is terminated when both the mean velocity profile +and the Reynolds stresses change by at most a few percent between adaptive +iterations. +The resulting grid-spacing distributions after each adaptive iteration are +shown in Fig. 1, with some quantities listed in Tab. 1. The staircase pattern in +Fig. 1 corresponds to individual elements; recall that the residual is averaged +over each element, and thus the optimal grid-spacing field inherits this step- +wise nature. +The sequence of grid-spacings found are similar to those by Toosi and +Larsson [1] using the same overall algorithm and residual estimation method +but a totally different code (a finite-difference code). Table 1 indicates that +12 + +0 +200 400 600 +0 +200 +400 +600 +y+ +∆+ +1,opt +0 +200 400 600 +0 +50 +100 +150 +200 +y+ +∆+ +2,opt +0 +200 400 600 +0 +100 +200 +300 +400 +y+ +∆+ +3,opt +l = 0 +l = 1 +l = 2 +l = 3 +l = 4 +Figure 1: Optimal element sizes for the channel with Reτ = 550 after adaptive iteration l. +Table 1: Key properties of the Reτ = 550 channel flow case for different adaptive iterations +l. +Iteration +Ni +∆i(y = h)/h +∆+ +i (y = 0)/(P + 1) +#Elems +#DOF +Reτ +l = 0 +10 +2 +3 +1.00 +0.50 +1.00 +91.67 +45.83 +91.67 +60 +12960 +178.50 +l = 1 +12 +4 +5 +0.83 +0.25 +0.60 +76.39 +13.93 +55.00 +240 +51840 +547.80 +l = 2 +20 +5 +10 +0.50 +0.20 +0.30 +45.83 +5.13 +27.50 +1000 +216000 +498.11 +l = 3 +31 +6 +21 +0.32 +0.17 +0.14 +29.57 +2.57 +13.10 +3906 +843696 +511.73 +l = 4 +50 +9 +36 +0.20 +0.11 +0.08 +18.33 +1.28 +7.64 +16200 +3499200 +556.57 +we are sufficiently resolved after iteration l = 4, with ∆(y = 0,⃗ni)+ ≈ +(18, 1, 8) being viewed as an acceptable (albeit a bit fine) grid by the LES +community. The computed wall shear stress, here shared as Reτ, converges +to the correct value. +An interesting observation in Fig. 1 is that the suggested grid-spacings +along x and z are higher at the viscous sublayer than in the buffer layer. This +is actually the correct behavior since the wall-parallel length scales in the +viscous sublayer become larger than in the buffer layer (e.g., Jimenez [21]), +and it is interesting to see that the residual estimator correctly reflects this. +The mean velocity profiles in inner units are shown in Fig. 2, and show +good agreement and convergence towards the DNS data of Lee and Moser +[22] at l = 4. The Reynolds stresses, shown in Fig. 3, also have good con- +13 + +100 +101 +102 +0 +5 +10 +15 +20 +25 +y+ +u+ +l = 0 +l = 1 +l = 2 +l = 3 +l = 4 +DNS +Figure 2: Mean velocity profiles for the Reτ = 550 channel for different adaptive iterations +l. +vergence on the DNS data. Different from the mean velocity profiles though, +the Reynolds stresses have already converged by l = 3. For the first iteration +l = 0, we can see the unsteady characteristics of the discontinuous Galerkin +scheme, that the element boundaries are clearly visible for u′v′. Additionally, +the flow field is severely underresolved and we observe the typical overshoots +of DG schemes in these cases at the element boundaries as well as the in- +ability to fulfill the weakly enforced no-slip condition. Both behaviors are to +be expected and are a tell-tale sign of insufficient resolution. These oscilla- +tions in the Reynolds stresses, however, are mitigated at higher resolutions +and therefore convergence towards the “smooth” reference DNS solution is +obtained. The overall results for this case show that this grid-adaptation +algorithm is applicable to the discontinuous Galerkin framework, and allow +us to move further with configurations and flows. +14 + +0 +200 +400 +600 +0 +5 +10 +y+ +u′u′ +0 +200 +400 +600 +0 +0.5 +1 +1.5 +y+ +v′v′ +0 +200 +400 +600 +0 +2 +4 +6 +y+ +w′w′ +0 +200 +400 +600 +−1 +−0.8 +−0.6 +−0.4 +−0.2 +0 +y+ +u′v′ +l = 0 +l = 1 +l = 2 +l = 3 +l = 4 +DNS +Figure 3: Reynolds stresses for the Reτ = 550 channel for different iteration l. +3.2. Channel flow at Reτ = 2000, application with meshes containing mortar +elements +Next, we test the full unstructured mesh ability of the discontinuous +Galerkin method in conjunction with the grid-adaptation using mortar ele- +ments. Mortar elements are elements whose faces do not match up one-to-one +with their neighbors, instead a single face of an element may have more than +one neighbor. A representation of this kind of a mesh can be seen in Figure 4, +showing mortars “along” x (teal line) and z (light green line) directions. Note +that these are actually planes with y direction as normal vectors, shown here +as lines to increase the clarity of the representation. The advantage of these +elements is that they allow for substantial cost savings compared to fully +structured grids for flows with massively different length scales. The ben- +efit of mortars comes when the optimal wall-parallel grid-spacing changes +by large amounts across the channel. We therefore increase the Reynolds +15 + +x +y +z +Figure 4: An example of a mortared mesh, with mortars “along” different directions marked +with different colors, teal for x direction and light green for z direction. +number to Reτ = 2000 to get a larger ratio of length scales. +The output of the grid-adaptation optimization problem ∆opt(⃗x,⃗n) is +shown in Fig. 5 after every adaptive iteration. The mortar implementation in +FLEXI is limited to 2-to-1 interfaces, and therefore we decide where to place +the mortar interfaces as follows. We first find the smallest required element +spacing in a direction (generally very close to the wall). We then maintain +this element spacing until we reach a wall-distance at which the optimal +grid-spacing has at least doubled: we then insert a 2-to-1 mortar interface +at that wall-distance, and continue. This results in the actual grid-spacing +distribution in Fig. 6. We note that the mortar interfaces can (and actually +do) occur at different wall-distances for the x- and z-directions. Finally, the +grid is stretched in the y-direction in the same way as for the Reτ = 550 +case. +We use the same domain size and numerical parameters for the Reτ = +2000 test case as already introduced in the Reτ = 550 case. The suggested +element sizes are again visualized in Fig. 5 and the resulting meshes are +described in Fig. 6. It clearly indicates that starting at l = 2, we can use +mortar interfaces in both x and z because the suggested grid spacing starts to +16 + +500 +1,500 +0 +1,000 +2,000 +3,000 +y+ +∆+ +1,opt +500 +1,500 +0 +200 +400 +600 +y+ +∆+ +2,opt +500 +1,500 +0 +500 +1,000 +1,500 +y+ +∆+ +3,opt +l = 0 +l = 1 +l = 2 +l = 3 +l = 4 +l = 5 +Figure 5: Optimal gridspacing according to G for the Reτ = 2000 channel for different +iteration l. +Table 2: Number of elements for the Reτ = 2000 channel with mortar meshes. +Iteration +N1 +N2 +N3 +∆1/h +(∆2/h)min +∆3/h +l = 0 +10 +2 +3 +1.00 +0.50 +1.00 +l = 1 +12 +4 +6 +0.83 +0.25 +0.50 +l = 2 +20 +10 +4 +12 +0.50 +1.00 +0.17 +0.25 +l = 3 +32 +16 +11.5 +26 +13 +0.31 +0.63 +0.09 +0.12 +0.23 +l = 4 +56 +28 +14 +13.5 +60 +30 +15 +0.18 +0.36 +0.71 +0.07 +0.05 +0.10 +0.20 +l = 5 +124 +62 +31 +21 +104 +52 +26 +0.08 +0.16 +0.32 +0.05 +0.03 +0.06 +0.12 +increase by a factor of more than two compared to the minimum grid spacing +in this direction. For l = 2 the interfaces in x and z are located at the same +wall-normal distance. Thus, the resulting mesh l = 3 will contain one mortar +interface. At l = 3 these locations are shifted and are no longer identical as +indicated in Fig. 4. For l = 4 we even get a second mortar interface and +therefore have a largest length ratio of four along the wall-parallel directions +between the wall elements and the elements in the channel center. The grid +for l = 3 is visualized in Fig. 4 showing the different mortar locations in the +z and x plane. +The details of the mesh including the resolution and the number of el- +ements saved is listed in Tab. 2. For Ni and ∆i/h we listed the respective +values at each iteration for each mortar interface. Tab. 2 clearly indicates +17 + +0 +0.5 +1 +10−1 +100 +y/h +∆1,actual +0 +0.5 +1 +10−1 +100 +y/h +∆3,actual +l = 0 +l = 1 +l = 2 +l = 3 +l = 4 +l = 5 +(a) The actual grid spacings for the streamwise and spanwise +direction mortar elements of the Reτ += +2000 channel, +non- +dimensionalized by the largest cell along respective direction. +0 +0.5 +1 +0 +0.5 +1 +y/h +∆1 +actual spacing +optimal spacing +(b) Comparison of actual and op- +timal grid-spacing for l = 2 mesh +using l = 1 as reference. +Figure 6: Overview over the actual and optimal gridspacing for the channel containing +mortar elements. +that using mortars saves us up to approx. 50% of elements in the simulation +compared to a structured mesh with equivalent mortar-free spacing. How- +ever, this does not affect the timestep since it is dictated by the smallest cell. +For an equivalent load per processor, the overall number of processors of the +simulation can however be reduced accordingly. It looks that the higher the +iteration, the more we save, since the grid near the wall requires more refine- +ment every time compared to grid in the center of the channel. This case +converges to the desired Reτ = 2000 at l = 5, if judged only by this metric, +with mortars saving almost 50% of the computational cost. +Table 3: Grid-spacing and element counts for the Reτ = 2000 channel with mortar meshes. +Iteration +∆+ +i (y = 0)/(P + 1) +#Elems +nDOF +Reτ +total +saved +l = 0 +333.33 +166.67 +333.33 +60 +12960 +569.17 +l = 1 +277.78 +46.67 +166.67 +288 +62208 +373.16 +l = 2 +166.67 +12.67 +83.33 +1320 +120 +285120 +1841.85 +l = 3 +104.17 +4.67 +38.46 +7592 +1976 +1639872 +1723.20 +l = 4 +59.52 +2.67 +16.67 +27195 +18165 +5874120 +1875.99 +l = 5 +26.88 +2.00 +9.62 +142662 +128154 +30814992 +1967.37 +18 + +100 +101 +102 +103 +0 +5 +10 +15 +20 +25 +30 +35 +y+ +u+ +l = 0 +l = 1 +l = 2 +l = 3 +l = 4 +l = 5 +DNS +Figure 7: Mean velocity profile for the Reτ = 2000 channel for each iteration l. +To evaluate the convergence and the quality of the simulations, we again +look at the mean velocity and the Reynolds stresses profiles of the channel. +The DNS data is found at Lee and Moser [22], just like the previous case. The +mean velocities in Fig. 7 show the expected converging behavior. The velocity +profile for l = 0 is more shifted towards higher friction velocities compared to +the Reτ = 550 results in Fig. 2. This can be reasoned with the fact that the +Reτ = 2000 channel is way more underresolved with the same initial mesh +l = 0, and the weak implementation of the boundary conditions therefore +cannot generate a good estimation for the τw. At iterations l = [2, 3, 4, 5] we +see convergence towards the reference data, with l = 5 matching the DNS +velocity profile accurately. +The results for the Reynolds stresses are visualized in Fig. 8. +Again, +we match the reference solution well for the latest iteration of the grid- +adaptation algorithm. The large oscillations for the first iterations are now +also more pronounced in l = 1 in addition to l = 0 due to the Re being +higher compared to the previous case. While w′w′ and u′v′ converge well, +19 + +0 +500 +1,000 1,500 2,000 +0 +10 +20 +30 +y+ +u′u′ +0 +500 +1,000 1,500 2,000 +0 +0.5 +1 +1.5 +y+ +v′v′ +0 +500 +1,000 1,500 2,000 +0 +5 +10 +y+ +w′w′ +0 +500 +1,000 1,500 2,000 +−1.5 +−1 +−0.5 +0 +0.5 +y+ +u′v′ +l = 0 +l = 1 +l = 2 +l = 3 +l = 4 +l = 5 +DNS +Figure 8: Reynolds stresses for the Reτ = 2000 channel for different iteration l. +the results of v′v′ lack behind. This behavior is expected, since the mortars +directly affect the wall-normal directions and thus are visible in the corre- +sponding fluctuations in velocity. Despite that the v′v′ still shows convergence +towards the Lee and Moser DNS data, with minimal (but still finite) signs +of the mortars at l = 5. +3.3. Channel flow at Reτ = 2000, application with WMLES +To conclude the application on channel flow, we run the same test case at +Reτ = 2000 as a wall-modeled simulation. This aims to prove the capabilities +of the algorithm to provide a good grid spacing for wall-modeled simulations. +We again allow mortar elements in the meshes. Previous experience with the +code has shown that a higher C is beneficial in the wall-modeled region, +therefore we increase the Vreman constant to C = 0.11 in Vreman subgrid- +scale stress model. We use Spaldings law of the wall as an algebraic wall +20 + +model, and for its solution we use Newton’s method in every Runge-Kutta +timestep. The interface height hwm is fixed at 10% of the channel half height +h and is depicted as a red dashed line in the following figures. +500 +1,500 +0 +1,000 +2,000 +y+ +∆+ +1,opt +500 +1,500 +0 +200 +400 +600 +y+ +∆+ +2,opt +500 +1,500 +0 +500 +1,000 +y+ +∆+ +3,opt +l = 0 +l = 1 +l = 2 +l = 3 +l = 4 +Figure 9: Optimal gridspacing according to G for the wall-modeled Reτ = 2000 channel +for different iteration l. Dashed red line indicates hwm. +First we assess the suggested gridspacings ∆+ +i,opt, shown in Fig. 9. The +spacing in wall-normal direction is again realized by the same method as for +the previous subsection of WRLES containing mortars. However, we have +adapted the suggested grid spacings to the WMLES methodology. The pure +grid-adaptation finds that the underresolution of the relevant scales is harsh- +est near the wall, thus suggesting the most refinement there. However, the +point of wall-modeling is to model the smaller but flow-relevant eddies below +hwm, so the grid-adaptation process needs to be told that severe underresolu- +tion in that region is a choice and not an error. To achieve this, we introduce +a restriction on ∆+ +i,opt according to Toosi (personal communication). We do +not allow any smaller ∆+ +i,opt below hwm, meaning that we reset the values of +∆+ +i,opt for the cells that lie between hwm and the wall to the values of their +counterparts at hwm. This applies to the grid spacings of all directions in +space. Fig. 9 also indicates that the wall model interface is not restricted to +the wall neighboring grid cell, but can also be placed in the second cell, e.g. +l = 4. For l = 4 we also introduce a mortar in x and z which is located at +the same wall-normal positions for both dimensions. In comparison to the +21 + +Table 4: Simulation properties of the wall-modeled Reτ = 550 channel for different itera- +tion l. +Iteration +N1 +N2 +N3 +∆1/h +(∆2/h)min +∆3/h +∆+ +i (y = 0)/(P + 1) +#Elems +#DOF +Reτ +total +saved +l = 0 +10 +2 +3 +1.00 +0.50 +1.00 +333.33 +166.67 +333.33 +60 +12960 +627.91 +l = 1 +12 +4 +5 +0.83 +0.25 +0.60 +277.78 +54.67 +200.00 +240 +51840 +872.29 +l = 2 +19 +5 +11 +0.53 +0.20 +0.27 +175.44 +21.33 +90.91 +1045 +225720 +1241.07 +l = 3 +32 +6 +23 +0.31 +0.17 +0.13 +104.17 +34.67 +43.48 +4416 +953856 +1220.01 +l = 4 +62 +31 +14 +38 +19 +0.16 +0.32 +0.07 +0.08 +0.16 +53.76 +10.00 +26.32 +31217 +1767 +6742872 +1555.60 +wall-resolved case this only happens at the fourth iteration, since the scale +differences between wall and the center of the channel are much smaller, since +wall-modeled LES does not resolve the small scales of the wall turbulence. +The resolution and resulting viscous spacings at the wall are listed in Tab. 4. +100 +101 +102 +103 +0 +5 +10 +15 +20 +25 +y+ +u+ +l = 0 +l = 1 +l = 2 +l = 3 +l = 4 +DNS +Figure 10: Mean velocity profile for the wall-modeled Reτ = 2000 channel for each iteration +l. +To verify the results, we again compare them against the same DNS +data by Lee and Moser. In Fig. 10 even for l = 0 we already have good +agreement, especially at the interface location at h+ +wm = 200. This is because +the initial grid is much closer to the WMLES requirements than the WRLES +22 + +requirements. +A clear convergence towards the DNS data is also visible. +Iteration l = 3 agrees with the DNS data for the whole channel above hwm, +including the center of the channel which was slightly off for the previous +iterations. +0 +500 +1,000 1,500 2,000 +0 +5 +10 +15 +y+ +u′u′ +0 +500 +1,000 1,500 2,000 +0 +0.5 +1 +1.5 +y+ +v′v′ +0 +500 +1,000 1,500 2,000 +0 +1 +2 +3 +4 +y+ +w′w′ +0 +500 +1,000 1,500 2,000 +−1 +−0.5 +0 +y+ +u′v′ +l = 0 +l = 1 +l = 2 +l = 3 +l = 4 +DNS +Figure 11: Reynolds stresses for the wall-modeled Reτ = 2000 channel for different itera- +tion l. +The Reynolds stresses are depicted in Fig. 11. Unlike the velocity pro- +files, the fourth iteration is needed to match the Reynolds stresses better. +This is due to the WMLES approach, not the grid-adaptation. Even then, +the Reynolds stresses, especially u′u′ does not exactly converge on the DNS +data, but become very acceptable for a WMLES. The v′v′ also deserves some +attention, where we can clearly see the converging behavior of the fluctu- +ations; but in the center of the channel the mortar in the mesh of l = 4 +gets visible again. The other Reynolds stress components show good enough +agreement, proving the applicability of the algorithm to WMLES. +23 + +4. Application to the flow over an airfoil +To conclude this work, we investigate the flow over a transonic NACA +64A-110 airfoil at an angle of attack of α = 0 deg. This test case allows us +to apply all the functionalities that we introduced in the channel flow onto a +more practical application. The airfoil grid is generated by extruding a 2D +geometry (xy-plane) in z. The xy-plane grid is generated using an unstruc- +tured 2D mesh tool. Due to the extrusion the mesh will be structured in z. +To allow the grid-adaptation algorithm to show its full potential, we allow +a homogeneous mortar element distribution along z by specifying regions in +the xy-plane with different number of elements in z. This way, we generate +a fully unstructured, yet still hexahedral only, three dimensional grid. The +characteristic length of the airfoil simulation c is set to the chord length and +defined as c = 1. The spanwise extension in z is Lz = 0.05c. +The airfoil is placed in a wind tunnel, therefore we include Euler walls +above and below the profile. The dimensionless simulation parameters are +Ma = 0.72 and Rec = 930 000. We perform it as a WMLES. The reliable +transition to turbulence is achieved through the flow being tripped with a +numerical trip at x = 0.05c using the method introduces by Schlatter and +Örlu [23] on the suction and pressure side. +The interface height is fixed +at hwm = 0.0134 and constant over the surface for all iterations and has +been determined with the interface height adaptation algorithm introduced +by Kahraman et al. [24] on the WRLES reference data. +We eventually compare the results to a wall-resolved LES simulation that +has been performed previously. We start with a coarse mesh and perform +the iterative approach, just like the channel test case. We iterate until we +match the pressure coefficient cp, the skin friction τw and the boundary layer +thickness δ99. +The initial mesh l = 0 is built to be extremely cheap and without any +special treatment of the boundary layer or any other flow properties. The +only constraint on the mesh is to fit the geometry. Thus in l = 0 the mesh +has only 6 elements per cordlength in streamwise direction, which results in +a maximum y+ ≈ 1100. This initial mesh is depicted in Fig. 12a. Since we +use high-order methods we must use curved elements, which allows to take +more details of the geometry into account even for as coarse meshes as in +l = 0. +24 + +(a) initial mesh l = 0 around the airfoil +(b) first refinement level l = 1 +5.0 × 10−1 +2.0 × 10−1 +1.0 × 10−1 +5.0 × 10−2 +2.0 × 10−2 +1.0 × 10−2 +5.0 × 10−3 +2.0 × 10−3 +1.0 × 10−3 +(c) Qualitative contour plot of ∆i,opt with logarithmic scaling after l = 0. All plots are scaled equiva- +lently. From left to right: wall parallel s, wall normal n and spanwise direction z. +Figure 12: Comparison between the velocity field and mesh the initial and the first itera- +tion. +To get the next mesh, of iteration l = 1, we evaluate the grid error +indicator densities g(⃗x,⃗ni) to get ∆i,opt, which is qualitatively visualized in +Fig. 12c. +It shows the same behavior that was already observed for the +channel, that we refine the most in wall normal direction n, followed by the +spanwise direction z and the tangential direction s. Interesting to note is that +the spanwise direction has very coarse suggested grid spacing at the leading +edge. This is expected, and nice to see for the performance of the algorithm, +since that the flow is laminar which means the lengthscale approaches infinity +along that direction. The turbulent wake also is refined in more detail, since +this coarse grid is still able to sustain some turbulence there. The mesh l = 1 +is also visualized in Fig. 12b, showing significantly more elements near the +airfoil, which is very well expected. Fig 12 thus shows the complete process +from l = 0 to l = 1. +The whole development of the relevant quantities defined earlier is visu- +alized in Fig. 13. The graphs indicate a clearly converging behavior and also +shows the different rates of convergence of the quantities. The pressure coef- +ficient cp converges the fastest. The skin friction coefficient cf only converges +after l = 2 and the boundary layer thickness, which is most sensitive to the +grid resolution, needs four iterations to converge. The difference between +l = 2 and l = 3 is only the resolution in x and z, which is listed in Tab. 5 and +25 + +0 +2 +4 +6 +0 +−0.5 +−1 +0 +2 +4 +0 +0.2 0.4 0.6 0.8 +0 +2 +4 +6 ·10−3 +x/c +τw,SS +·10−3 +τw,PS +0 +0.2 0.4 0.6 0.8 +0 +−0.5 +−1 +x/c +cp,SS +cp,PS +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +0 +2 +4 +·10−2 +x/c +δ99,SS +·10−2 +δ99,PS +l = 0 +l = 1 +l = 2 +l = 3 +WRLES +Figure 13: Convergence of pressure coefficient cp and skin friction τw and boundary layer +thickness δ99 on the pressure (PS) and suction (SS) side of the NACA 64A-110 airfoil. +Tab. 6 along other grid characteristics. Especially at the trailing edge the +algorithm suggested mesh refinement. This is also where the discrepancy be- +tween WRLES reference data and the iterations l = [0, 1, 2] was the largest. +The red dashed line in Fig. 13 shows the wall model interface height over x/c +which is roughly 10% of δ99,mean. +Table 6 also shows the mortar information for l = [1, 2, 3] as well as the +minimal and maximum grid spacings at the airfoil boundaries. The values in +26 + +Table 5: Grid spacings of the wall-modeled NACA 64A-110 airfoil for different iteration l. +Iteration +∆+ +SS(x = 0.05c, s)/(P + 1) +∆i/c +min. +max. +min. +max. +min. +max. +l = 0 +1267.21 +219.82 +146.54 +1.21e-1 +2.16e-1 +3.75e-2 +3.75e-2 +2.50e-2 +l = 1 +316.92 +52.76 +48.85 +3.30e-2 +5.40e-2 +9.00e-3 +9.00e-3 +8.33e-3 +1.67e-2 +l = 2 +146.54 +5.86 +12.21 +2.10e-2 +3.00e-2 +1.00e-3 +2.00e-3 +2.08e-3 +1.67e-2 +l = 3 +58.62 +5.86 +12.21 +5.00e-3 +1.50e-2 +1.00e-3 +2.50e-3 +1.04e-3 +1.67e-2 +Table 6: Number of cells of the wall-modeled NACA 64A-110 airfoil for different iteration +l. +Iteration +Ni +#Elems +#DOF +Ns · Nn +Nz +total +saved +l = 0 +395 +(2) +790 +404480 +l = 1 +1472 +(3, 6) +5946 +2886 +3044352 +l = 2 +2590 +(3, 6, 12, 24) +36072 +26088 +18468864 +l = 3 +9111 +(3, 6, 12, 24, 48) +222426 +214902 +113882112 +between can be easily calculated by assuming the 2-1 mortar interfaces. The +observation made in Tab. 2 are confirmed here. The usage of the unstructured +capabilities of the discontinuous Galerkin framework allow us to save up to +approximately 50% of the elements, which directly translates to the same +amount of savings in compute time. +In Fig. 14 the mortar zones in z are visualized, each denoting a different +number of elements in z. The colors are based on the minimum number +of cells along spanwise direction Nz, which are Nz(l = 0) = 2 and Nz(l = +[1, 2, 3]) = 3. The zones are designed in a way to meet the target values of the +indicator in an efficient way with our limitation to 2-1 mortar interfaces, even +though the indicator suggests even more coarsening in some locations and +much more rapid refinement at other locations, e.g. from Nz ·24 to Nz ·20. We +insert buffer layers in between to make the mesh compatible and to enable +a transition between the mortar zones. Still, the number of elements saved +justifies this approach. +5. Summary +In this work we presented turbulent channels and a transonic airfoil in +discontinuous Galerkin framework using different grids proposed by using the +high-order optimized version of the grid-adaptation algorithm based on the +27 + +(a) Initial mesh l = 0 around the airfoil. +(b) First refinement iteration l = 1. +(c) Second refinement iteration l = 2. +(d) Third refinement iteration l = 3. +Nz · 20 +Nz · 21 +Nz · 22 +Nz · 23 +Nz · 24 +Figure 14: Comparison between the z mortar zones of the airfoil iterations, visualized with +non-curved elements. +works from Toosi and Larsson [1, 7]. The boundary layers were either wall- +resolved or wall-modeled to show the ability of the algorithm to handle two +major cases of large eddy simulation. For all test cases we showed convergence +and good agreement with the results in literature. +The grid-adaptation indicator in general for high-order methods has turned +out to be most efficient when applied elementwise and not per collocation +point due to the reduced robustness of the numerical scheme in underresolved +settings. Having the indicator values on the element, splitting and remeshing +with the proposed recommended gridspacing is straight forward. The appli- +cation of the grid-adaptation machinery to a p-refined approach is yet to be +investigated. +Another important aspect of the simulation run is the usage of the un- +structured capabilities of the discontinuous Galerkin implementation in the +FLEXI framework. Unstructured meshes in three dimensions were realized +using mortar interfaces between elements and the usage of these elements +saved up to 50% of compute time without any significant loss in accuracy. +28 + +Especially for the airfoil case, we were able to clearly distinguish different +mortar zones in spanwise direction, even after few iterations. Therefore, we +have shown that the grid-adaptation framework is able to deliver a problem- +tailored and cost-optimized mesh with converging behavior. +Thus, the grid-adaptation algorithms can be applied to arbitrary geome- +tries without any prior knowledge to the flow field and still create a problem- +tailored and cost optimized mesh, which does not rely on the experience of +the user. +Acknowledgments +MB and AB gratefully acknowledge the Deutsche Forschungsgemeinschaft +DFG (German Research Foundation) for funding this work in the frame- +work of the research unit FOR2895, and thank the Gauss Centre for Su- +percomputing e.V. (www.gauss-centre.eu) for funding this project (GCS- +lesdg) by providing computing time on the GCS Supercomputer HAWK at +Höchstleistungsrechenzentrum Stuttgart (www.hlrs.de). +AK and JL were +supported by the Department of Energy PSAAP III program (grant DE- +NA0003993) and the NASA Transformational Tools and Technologies project +(grant 80NSSC18M0148). +References +[1] S. Toosi, J. Larsson, Towards systematic grid selection in LES: identify- +ing the optimal spatial resolution by minimizing the solution sensitivity, +Computers & Fluids (2020) 104488doi:https://doi.org/10.1016/j. +compfluid.2020.104488. +[2] B. J. Geurts, J. Fröhlich, A framework for predicting accuracy limita- +tions in large-eddy simulation, Physics of Fluids 14 (2002) L41. +[3] I. B. Celik, Z. N. Cehreli, I. Yavuz, Index of resolution quality for large +eddy simulations, J. Fluids Engr. 127 (2005) 939–958. +[4] J. Jimenez, R. D. 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Jimenez, Cascades in wall-bounded turbulence, Annual Review of +Fluid Mechanics 44 (2012) 27–45. +[22] M. Lee, R. D. Moser, Direct numerical simulation of turbulent channel +flow up to Reτ ≈ 5200, Journal of Fluid Mechanics 774 (2015) 395–415. +[23] P. Schlatter, R. Örlü, Turbulent boundary layers at moderate reynolds +numbers: inflow length and tripping effects, Journal of Fluid Mechanics +710 (2012) 5–34. doi:10.1017/jfm.2012.324. +[24] A. B. Kahraman, J. Larsson, Adaptive determination of the optimal +exchange location in wall-modeled large-eddy simulation, AIAA Journal +(2022) 1–12doi:10.2514/1.J061347. +31 + diff --git a/c9E1T4oBgHgl3EQfdwSQ/content/tmp_files/load_file.txt b/c9E1T4oBgHgl3EQfdwSQ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..de5c926dc3beed37685cf87163f2a932a62174c1 --- /dev/null +++ b/c9E1T4oBgHgl3EQfdwSQ/content/tmp_files/load_file.txt @@ -0,0 +1,877 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf,len=876 +page_content='Grid-Adaptation for Wall-Modeled Large Eddy Simulation Using Unstructured High-Order Methods Marcel Blinda,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Ali Berk Kahramanb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Johan Larssonb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Andrea Becka,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='c aInstitute for Aerodynamics and Gas Dynamics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' University of Stuttgart,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Pfaffenwaldring 21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Stuttgart,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 70569,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Germany bDepartment of Mechanical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' University of Maryland,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' College Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 20742,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' MD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' USA cInstitute of Fluid Dynamics and Thermodynamics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Otto-von-Guericke-University Magdeburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Universitätsplatz 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Magdeburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 39106,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Germany Abstract The accuracy and computational cost of a large eddy simulation are highly dependent on the computational grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Building optimal grids manually from a priori knowledge is not feasible in most practical use cases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' instead, solution-adaptive strategies can provide a robust and cost-efficient method to generate a grid with the desired accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' We adapt the grid-adaptation algorithm developed by Toosi and Larsson [1] to a Discontinuous Galerkin Spectral Elements Method (DGSEM) and show its potential on fully un- structured grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The core of the method is the computation of the estimated modeling residual using the polynomial basis functions used in DGSEM, and the averaging of the estimated residual over each element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The final method is assessed in multiple channel flow test cases and for the transonic flow over an airfoil, in both cases making use of mortar interfaces between elements with hanging nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The method is found to be robust and reliable, and to provide solutions at up to 50% lower cost at comparable accuracy compared to when using human-generated grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Keywords: high-order methods, WMLES, non-conforming grids, grid error indicator ∗corresponding author Email address: blind@iag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='uni-stuttgart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='de (Marcel Blind) Preprint submitted to arXiv January 10, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='03199v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='flu-dyn] 9 Jan 2023 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Introduction As the large eddy simulation (LES) technique gathers increased attention from the engineering community, the open research questions for the method are also changing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' While much of the focus in the past was on the devel- opment of subgrid scale stress models and numerical methods with limited dissipation, today other issues in the LES method demand more attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' First and foremost among those is the question of how to design a good grid, especially for more complicated geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The grid-generation process is still mainly an art that relies on the experience of the user, who needs to incorporate knowledge of numerics and the resolution requirements of flow physics into his/her judgment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' This can even lead to a point where differ- ent users can create different grids for the same geometry and get different results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Creating an objectively good mesh is crucial for an efficient and resource saving simulation, as well as increasing the dependability of the solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The ultimate objective of this work is to provide a workflow to generate a problem-tailored mesh that is compatible with unstructured high- order methods starting from a very coarse initial mesh generated without any problem-specific insight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' There have been several attempts at devising ways to quantify how well resolved an LES is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Early arguments were based on the ratio of subgrid to viscous dissipation or viscosity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' [2, 3]), but these concepts are meaningful only in the buffer layer of wall-bounded turbulence since LES should be ap- plicable to free shear flows at any Reynolds number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Others have suggested that the sufficiency of a grid in LES should be measured by the ratio of modeled to resolved (or total) turbulence kinetic energy (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' [4, 5]), but this has been found to correlate poorly to the known behavior of length scales in wall-bounded flows [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Methods that approximate a local turbulent spec- trum have some basis for isotropic flows, but fail for more relevant cases [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The most well-grounded approach to date is that by Toosi and Larsson [1] which can be viewed as an estimate of the LES modeling residual, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=', the source term in an error transport equation [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The objective of the present work is to extend the residual estimate of [1, 7] to high-order codes and to test the ability of the resulting grid-adaptation method to produce efficient (high accuracy at low cost) grids in a high-order type solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' In this paper we use the Discontinuous Galerkin (DG) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Implementation in a DG-type code differs from that in finite-difference or 2 finite-volume codes (where the residual estimate has been tested before) in several ways, including how one performs the low-pass filtering, the sensitivity to aliasing errors, and how one locally averages the results to be meaningful for grid-adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The method is mainly assessed in several channel flows where the behavior of LES is relatively well known and where we therefore can judge the optimality of the resulting grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The method is finally applied to the transonic flow around an airfoil to verify that it works for more realistic and complex problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Methodology Any grid-adaptation method necessarily starts by estimating the spatial distribution of the error generation process (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=', the residual, or source term for the solution error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' It then proceeds by generating a new mesh that would minimize the residual field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The main focus of this paper is on the first step, specifically on how to compute the estimated residual field in the context of a Discontinuous Galerkin Spectral Element Method (DGSEM) code for grids with primarily hexahedral elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The DGSEM code FLEXI The simulations are run using the Discontinuous Galerkin Spectral Ele- ment Method (DGSEM) framework FLEXI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The code is developed in the Numerics Research Group at the University of Stuttgart [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' It is open source and can be downloaded from http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='flexi-project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='org/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The DGSEM [8, 9, 10] uses piecewise smooth functions in every element, but al- lows for discontinuities at element interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' However, the residual estima- tion procedure (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='2) and subsequent grid-adaptation can be transferred to any other DG or related scheme with minimal changes, as long as an element-local filter operation can be specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Thus, the details regarding implementation given below serve to elucidate the procedure for this specific DG variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The computational domain Ω is partitioned in non-overlapping elements C ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The solution in each element is described using a polynomial basis of degree P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' We discretize the compressible Navier-Stokes equation using the same polynomial basis function as ansatz function in each three-dimensional element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The polynomials in each direction are represented using nodal 1D 3 Lagrange basis functions on P + 1 Legendre-Gauss-Lobatto points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Other node choices are possible, here we stick with these to be able to formulate the so-called split forms for the advective operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' To apply common integra- tion rules and to build an efficient scheme, we define the three-dimensional hexahedral reference element [−1, 1]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' We create the three-dimensional basis as the tensor product of three one-dimensional polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' We integrate the Navier-Stokes equation in space using the associated P + 1 Legendre- Gauss-Lobatto quadrature to compute the projection integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' This results in the very efficient DGSEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' At the element boundaries we use a Riemann solver e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' To take the parabolic terms into account, the BR1 lifting procedure [12] is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' According to the method of lines we advance the resulting ODE in time by applying an explicit-in-time low storage Runge- Kutta method with optimized stability region [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' We rely on skew-symmetric splitting of the advective terms of the com- pressible Navier-Stokes equations for stability [14], since in implicitly filtered LES the simulations are typically under-resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' To obtain a dissipation- free and kinetic-energy preserving semi-discrete system we use the skew- symmetric split fluxes by Pirozzoli [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The Roe numerical flux is used to solve the Riemann problem at the cell interfaces [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' We use the Vreman model for explicit sub grid scale modeling in the following simulation [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Estimating the LES Residual Consider a general evolution equation ∂q/∂t = R(q) solved numerically on a grid with spacing ∆, the solution of which can be denoted by N∆(q∆) ≡ R∆(q∆) − ∂q∆ ∂t = 0 , (1) where R∆ implies that R is approximated at grid-spacing ∆ and q∆ means the solution to the discrete problem at this grid-spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The error in this equation is then q∆ − q, which satisfies the error equation ∂N∆ ∂q (q∆ − q) ≈ N∆(q∆) � �� � =0 − N∆(q) � �� � F , where F is the residual, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=', the source term for error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The residual is defined based on the exact solution q which is, of course, not known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' All grid- adaptation methods therefore involve some type of process for estimating the 4 residual from the numerical solution q∆, for example using the leading terms in Taylor expansions of the numerical operators or by directly approximating q by interpolating q∆ onto a refined mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Neither approach works in large eddy simulation (LES) which by definition seeks a solution to the coarse- grained Navier-Stokes equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' While one could interpolate the solution q∆ onto a finer grid, in reality the solution on this finer grid should have developed smaller scales due to the broadband nature of turbulence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' the resulting residual estimate would therefore be incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' For the same reason, the fact that an LES solution is, by definition, a rough solution far from the asymptotic range of numerical convergence means that many terms in a Taylor expansion should be expected to be large, and thus one could not approximate the error behavior from the leading term only as is traditionally done in numerical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Toosi and Larsson [1, 7] proposed that the modeling residual (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=', the residual due to imperfect resolution/modeling of the small scale turbulence, excluding the residual due to numerical errors) in LES can be estimated in a post-processing step using low-pass test-filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Specifically, they argued that the residual must be estimated at an imagined coarser resolution �∆ than the resolution ∆ used in the actual LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Their argument went as follows: Assume that the LES equations at resolution ∆ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=', the LES solved in the code) can be written as in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The residual at the test-filtered (or additionally coarse-grained) level is then N�∆(�q), where �q is the exact solution restricted (or test-filtered) to the resolution �∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Note that one could not define the residual with q directly, since this would contain smaller scales that would then be double-counted due to the possible subgrid-model in the LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Toosi and Larsson [1, 7] then suggested that �q can be approximated by � q∆, and thus defined the approximate modeling residual as F�∆ = N�∆ (� q∆) = R�∆ (� q∆) − ∂� q∆ ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Assuming that the test-filter commutes with the time-derivative and using Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' (1) then yields F�∆ = R�∆ (� q∆) − R∆(q∆) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' (2) In the context of subsonic flows without shocks or other multi-physics effects (heat release, multi-phase, etc), the modeling residual for the momen- 5 tum equation then becomes Fi,�∆ = ∂ ∂xj � ρuiuj + pδij + τ mod ij (ρ, ui, ∆) − σij � � (3) − ∂ ∂xj � �ρ�ui�uj + �pδij + τ mod ij (�ρ, �ui, �∆) − � σij � , where the test-filtered velocity should be understood as a Favre-weighted filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' All operations above were defined based on the instantaneous flow fields, and thus the estimated modeling residual Fi,�∆ is necessarily a chaotic field in space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Assuming that one seeks a stationary grid (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=', that one runs a full LES and then adapts the grid before running another full LES), one must then reduce the residual field in time (and possibly in any spatially homogeneous directions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' While there is no theoretical reason for any specific choice of reduction, Toosi and Larsson [1, 7] suggested using the L2 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' In addition, they argued that the test-filter should be chosen as a uni-directional one, providing filtering in one direction at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' For hexahedral elements, this implies that one should compute 3 different residual fields (using test-filtering in each of the 3 directions of a hexahedral element) which then provides information about the lack of resolution in each direction separately;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' this then allows for anisotropic grid-adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Putting these things together, the final time-averaged modeling residual is written as G(⃗x,⃗n) = � ⟨Fi(⃗x,⃗n)Fi(⃗x,⃗n)⟩ , (4) where ⃗x denotes the spatial location, ⃗n denotes the direction of the uni- directional test-filter, and the angular brackets denote averaging in time, spatially homogeneous directions, and possibly locally in space as well (to be discussed below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' We emphasize that the subgrid model contribution in the residual (3) must be computed with the correct length scale: when evaluated for the test-filtered field, it must use the length scale of the imagined �∆ resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' This length scale must be estimated from the properties of the test filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' For implicit LES run without an explicit subgrid model, the subgrid model terms are simply zero in the residual estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Computation of the LES modeling residual in a DGSEM code The implementation of the LES residual estimator described above in a Discontinuous Galerkin Spectral Element Method (DGSEM) requires some care and specific implementation details, which are described in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The low-pass test-filtering operator �· is realized as a uni-directional modal cut-off filter, which removes the highest modes of the polynomial rep- resentation in one direction only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Throughout this work we used 5th order polynomials, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' P = 5 where P represents the polynomial order, to repre- sent the solution in each direction resulting in a 6th order accurate scheme, and chose to low-pass filter by setting the coefficients of the 4th and 5th order polynomials to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Importantly, this was done in only one direction, with the high-order polynomial coefficients in the other directions remaining un- changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' A nice feature of the DGSEM method is that the filtering in one element is independent from all other elements, and thus it works equally well in an unstructured grid (of hexahedral elements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The LES residual F�∆ could be computed using either the generic Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' (2) or the more flow-specific Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The most convenient and easy-to-implement option is the former one, as most codes (especially ones with explicit time- stepping) have functions to compute R∆ for a given solution field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Given an instantaneous LES field q∆, it is then trivial to compute R∆(q∆) using a sin- gle function-call in the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The resulting field is then low-pass test-filtered three times, one for each natural direction of each hexahedral element, to form three different instantiations of the second part of Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' In a separate process, the LES solution itself is test-filtered to form � q∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' In the present work, we apply the test-filter to the conserved variables which then implies that the filtered velocity is Favre-weighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' This test-filtered LES solution is then fed into the computation of R�∆(� q∆) to complete Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' This last step requires the most care, as the function in the LES code will (for the given grid and polynomial order) compute R∆(� q∆) rather than the correct R�∆(� q∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' If we are interested only in the LES modeling residual rather than the numerical errors (as was the case in Toosi and Larsson [1, 7]), then the only modification required is to ensure that the length scale in the subgrid model is reflective of the coarse-grained state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' This is the option chosen in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' In principle one could also use a lower polynomial order when evaluating R�∆(� q∆) as a way to also approximately account for the numerical error, but this is left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 7 The present simulations use the Vreman subgrid model [17] but in a modified form in which the length scale is taken as ∆ = V1/3 P + 1 , where V is the element volume and P is the polynomial order, taken as P = 5 in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Since we use a uni-directional filter that keeps modes up to the Mth order (taken as M = 3 in this work), the consistent length scale after test-filtering is �∆ = � P + 1 M + 1 �1/3 ∆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Since the model implemented in the FLEXI code uses an isotropic length scale, this implies that the eddy viscosity at the test-filtered level is simply a factor of [(P +1)/(M +1)]2/3 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='59 larger than the eddy viscosity computed by the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The norm operation in equation Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' (4) can be tricky because the solu- tion is represented as a polynomial in an appropriate number of collocation points, but the included squaring operator in the norm operation needs a higher order of polynomial, thus more collocation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The approach we took is to map the F�∆ from the original P degree of polynomial to 2P degree of polynomial and its associated collocation points, and then take the square of the polynomial at that order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' This way we got an exact representation of the term Fi(⃗x,⃗n)Fi(⃗x,⃗n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Another issue has risen in the computation, which is the cell edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Since the DG solution is discontinuous at the element boundaries and local projec- tions tend to produce large gradients at these boundaries, the local gradients at the cell boundaries in an underresolved setting can overshoot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' This ba- sically results in the residual estimator showing the element boundaries as high error regions, when this is in fact a numerical effect that is excluded from its goal, to find high error regions in the modeling of turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' To mitigate this issue, and to stay true to the cell based nature of the discontin- uous Galerkin method, we average the Fi(⃗x,⃗n)Fi(⃗x,⃗n) term over the whole cell before the periodic direction or time averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' This averaging is also included in the ⟨⟩ averaging operator that has been used so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Finding the optimal grid-spacing Once the estimated residuals for test-filtering in each direction are com- puted, the optimal element size in each direction can be found from an op- timization problem and an assumed model for how the residuals vary under grid-refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Following Toosi and Larsson [1], the directional residual G(⃗x,⃗n) is as- sumed to vary as G(⃗x,⃗n) = g(⃗x,⃗n) ∆(⃗x,⃗n)α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' (5) In the asymptotic limit of convergence, the power α would be the order- of-accuracy of the numerical method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' In the context of LES which is, by definition, far from the asymptotic limit of convergence, the power α is in- stead related to the spectral behavior of turbulence near the grid cut-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Since the spectral slope of inertial range turbulence varies across flow types and directions, it is clear that α should in theory also vary analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The more important point is that the value of α comes entirely from turbulence physics and not from numerical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Rather than try to find the “cor- rect” α field, we follow the suggestion of Toosi and Larsson [18] and simply assume a constant value of α = 2 throughout the flow domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' This is almost certainly larger than the “correct” value in most flow scenarios;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' erring on the side of a larger value produces lower variations in the optimal grid-spacing, which can be beneficial in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' With this model for the residual scaling, the “residual density” g(⃗x,⃗n) can be computed by evaluating Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' (5) for the residual estimated on the current grid, in each direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' We then define the cost functional J [∆(⃗x,⃗n)] = ��� V � � �� m g(⃗x,⃗nm)β∆(⃗x,⃗nm)βα �1/β + λ � m ∆(⃗x,⃗nm) � � dV where the first term accounts for the sum of residuals in all directions in a β-norm (note that all factors are positive) and the second term accounts for the computational complexity (defined here as the number of elements) of the new grid, with λ being a Lagrange multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The solution to this particular calculus-of-variations problem is that the variation of the integrand with respect to the ∆(⃗x,⃗ni) field is zero for each hexahedral direction i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 9 Using the abbreviated notation ∆i = ∆(⃗x,⃗ni) and gi = g(⃗x,⃗ni), this is αgβ i ∆βα−1 i,opt �� m gβ m∆βα m,opt �1/β−1 − λ ∆i,opt � m ∆m,opt = 0 , i = 1, 2, 3 , where the subscript “opt” has been added to indicate that this is the ∆(⃗x,⃗n) field that minimizes the cost functional J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Some rearrangement yields � gi∆α i,opt �β = λ/α � m ∆m,opt �� m gβ m∆βα m,opt �1−1/β , i = 1, 2, 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' This is valid for all i = 1, 2, 3 but the right-hand-side is the same for all i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' thus the optimal residual Gi,opt = gi,opt∆α i,opt is the same in all three directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' This can then be exploited to find that gi,opt∆α i,opt � m ∆m,opt � �� � Vopt = Λ , i = 1, 2, 3 , (6) where Λ is a re-defined Lagrange multiplier and Vopt is the volume of the optimal element;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' this equation shows that the element-integrated residual should be equi-distributed in both space and direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The optimal grid-spacing is then found by using a root-finding proce- dure to find the value of the modified Lagrange multiplier Λ for which the computational complexity (=the number of elements) is as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' In each step, Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' (6) is used to find the optimal ∆(⃗x,⃗n) field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The process was documented originally in Toosi and Larsson [1] and is provided here for com- pleteness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' We note that the final result (Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' (6)) is independent of the value of β, and thus the choice of norm in the cost functional is immaterial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Overall process The grid-adaptation process is implemented entirely as a post-processing approach that is performed between LES runs, with no adaptation occurring on-the-fly during a run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' After a simulation, the residual field is computed from multiple instantaneous snapshots of the solution, with averaging in 10 time and possibly suitable spatial directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' We then compute the residual “density” field g(⃗x,⃗n) from the residual scaling model, and the use iterative root-finding on the Lagrange multiplier Λ to find the optimal grid-spacing field ∆(⃗x,⃗n) with the desired computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' From this point we have multiple choices for the actual creation of the adapted grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' One approach is to modify the existing grid, most simply by splitting individ- ual elements in those directions for which the element-integrated directional residual exceeds some threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' A second choice is to re-generate the grid to more closely follow the optimal target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' This latter approach has mainly been followed for simplex elements in the literature [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Since we desire meshes with hexahedral cells in this work, we resort to human intervention in the grid-generation process: we build the grids manually, but aimed to mimic the optimal grid-spacing field as closely as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' In future work we hope to automate this aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' We note that the estimated residuals do not say anything about whether the grid is sufficiently fine to be considered converged;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' instead, any judgment about convergence must be made for specific quantities-of-interest (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=', mean profiles, drag or lift coefficients, etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' This judgment must be made by the user, prior to deciding whether to create an adapted grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Application to channel flows In this and the following section, we use the abbreviated notation ∆i = ∆(⃗x,⃗ni) where ⃗ni is a unit vector in one of the three natural directions of a hexahedral element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' As the first step the algorithm is applied to the canonical test case of the plane turbulent channel flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The appeal of channel flow is that it is one of very few problems where the “optimal” grid is known to some degree;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' it is then a good test of a grid-adaptation method to see whether it can arrive at something close to the known “optimum”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The domain is taken as (10h, 2h, 3h) where h is the channel half-height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Two different Reynolds numbers of Reτ = 550 and Reτ = 2000 are targeted, with the lower used to compare the adapted grids to those by Toosi and Larsson [1] and the higher used to test the grid-adaptation process when a wall-model is used and when mortar elements are used in the adapted grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Channel flow at Reτ = 550, validation of the original results The point of this case is to reproduce the original results of [1], establish- ing that the discontinuous Galerkin method is suitable for the grid-adaptation process this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' We start with a uniform mesh of 2 cells along the channel half-width h, 10 cells along the streamwise direction and 3 cells along the spanwise direction with each cell containing a basis of 5th order polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' This is a similar setup to [1] (although a tad “finer”), where they start with grid spacings (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='2h, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='1h, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='2h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' For each iteration, the grid-adaptation algorithm is used as described, with each new grid having 4 times the previous number of elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' It is constructed such that it is smoothly stretched along the wall-normal direc- tion, agreeing with the near wall ∆2 and the center line ∆2, and has uniform cells along the streamwise x and spanwise z directions with the minimum suggested ∆1 and ∆3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The wall-normal stretching is realized using a bell shaped DR(s) = 1 + � ∆2,min ∆2,max − 2 � e−(s·f)2 − e−f2 1 − e−f2 (7) element distribution with s ∈ [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The ration ∆2,min ∆2,max describes the differ- ence between the smallest and the largest element in y direction, f denotes the scaling factor and the result DR defines the element local weight which is later normalized to get the element size distribution in s ∈ [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' To ensure compatibility with Toosi and Larsson [1], we use the Vreman subgrid scale model for LES closure [17] with the same model constant 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The grid-adaptation process is terminated when both the mean velocity profile and the Reynolds stresses change by at most a few percent between adaptive iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The resulting grid-spacing distributions after each adaptive iteration are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 1, with some quantities listed in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The staircase pattern in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 1 corresponds to individual elements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' recall that the residual is averaged over each element, and thus the optimal grid-spacing field inherits this step- wise nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The sequence of grid-spacings found are similar to those by Toosi and Larsson [1] using the same overall algorithm and residual estimation method but a totally different code (a finite-difference code).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Table 1 indicates that 12 0 200 400 600 0 200 400 600 y+ ∆+ 1,opt 0 200 400 600 0 50 100 150 200 y+ ∆+ 2,opt 0 200 400 600 0 100 200 300 400 y+ ∆+ 3,opt l = 0 l = 1 l = 2 l = 3 l = 4 Figure 1: Optimal element sizes for the channel with Reτ = 550 after adaptive iteration l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Table 1: Key properties of the Reτ = 550 channel flow case for different adaptive iterations l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Iteration Ni ∆i(y = h)/h ∆+ i (y = 0)/(P + 1) #Elems #DOF Reτ l = 0 10 2 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='00 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='67 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='83 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='67 60 12960 178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='50 l = 1 12 4 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='60 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='39 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='93 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='00 240 51840 547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='80 l = 2 20 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='30 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='83 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='13 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='50 1000 216000 498.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='11 l = 3 31 6 21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='14 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='57 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='57 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='10 3906 843696 511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='73 l = 4 50 9 36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='08 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='33 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='28 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='64 16200 3499200 556.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='57 we are sufficiently resolved after iteration l = 4, with ∆(y = 0,⃗ni)+ ≈ (18, 1, 8) being viewed as an acceptable (albeit a bit fine) grid by the LES community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The computed wall shear stress, here shared as Reτ, converges to the correct value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' An interesting observation in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 1 is that the suggested grid-spacings along x and z are higher at the viscous sublayer than in the buffer layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' This is actually the correct behavior since the wall-parallel length scales in the viscous sublayer become larger than in the buffer layer (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=', Jimenez [21]), and it is interesting to see that the residual estimator correctly reflects this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The mean velocity profiles in inner units are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 2, and show good agreement and convergence towards the DNS data of Lee and Moser [22] at l = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The Reynolds stresses, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 3, also have good con- 13 100 101 102 0 5 10 15 20 25 y+ u+ l = 0 l = 1 l = 2 l = 3 l = 4 DNS Figure 2: Mean velocity profiles for the Reτ = 550 channel for different adaptive iterations l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' vergence on the DNS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Different from the mean velocity profiles though, the Reynolds stresses have already converged by l = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' For the first iteration l = 0, we can see the unsteady characteristics of the discontinuous Galerkin scheme, that the element boundaries are clearly visible for u′v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Additionally, the flow field is severely underresolved and we observe the typical overshoots of DG schemes in these cases at the element boundaries as well as the in- ability to fulfill the weakly enforced no-slip condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Both behaviors are to be expected and are a tell-tale sign of insufficient resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' These oscilla- tions in the Reynolds stresses, however, are mitigated at higher resolutions and therefore convergence towards the “smooth” reference DNS solution is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The overall results for this case show that this grid-adaptation algorithm is applicable to the discontinuous Galerkin framework, and allow us to move further with configurations and flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 14 0 200 400 600 0 5 10 y+ u′u′ 0 200 400 600 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='5 y+ v′v′ 0 200 400 600 0 2 4 6 y+ w′w′ 0 200 400 600 −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='2 0 y+ u′v′ l = 0 l = 1 l = 2 l = 3 l = 4 DNS Figure 3: Reynolds stresses for the Reτ = 550 channel for different iteration l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Channel flow at Reτ = 2000, application with meshes containing mortar elements Next, we test the full unstructured mesh ability of the discontinuous Galerkin method in conjunction with the grid-adaptation using mortar ele- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Mortar elements are elements whose faces do not match up one-to-one with their neighbors, instead a single face of an element may have more than one neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' A representation of this kind of a mesh can be seen in Figure 4, showing mortars “along” x (teal line) and z (light green line) directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Note that these are actually planes with y direction as normal vectors, shown here as lines to increase the clarity of the representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The advantage of these elements is that they allow for substantial cost savings compared to fully structured grids for flows with massively different length scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The ben- efit of mortars comes when the optimal wall-parallel grid-spacing changes by large amounts across the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' We therefore increase the Reynolds 15 x y z Figure 4: An example of a mortared mesh, with mortars “along” different directions marked with different colors, teal for x direction and light green for z direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' number to Reτ = 2000 to get a larger ratio of length scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The output of the grid-adaptation optimization problem ∆opt(⃗x,⃗n) is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 5 after every adaptive iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The mortar implementation in FLEXI is limited to 2-to-1 interfaces, and therefore we decide where to place the mortar interfaces as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' We first find the smallest required element spacing in a direction (generally very close to the wall).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' We then maintain this element spacing until we reach a wall-distance at which the optimal grid-spacing has at least doubled: we then insert a 2-to-1 mortar interface at that wall-distance, and continue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' This results in the actual grid-spacing distribution in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' We note that the mortar interfaces can (and actually do) occur at different wall-distances for the x- and z-directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Finally, the grid is stretched in the y-direction in the same way as for the Reτ = 550 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' We use the same domain size and numerical parameters for the Reτ = 2000 test case as already introduced in the Reτ = 550 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The suggested element sizes are again visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 5 and the resulting meshes are described in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' It clearly indicates that starting at l = 2, we can use mortar interfaces in both x and z because the suggested grid spacing starts to 16 500 1,500 0 1,000 2,000 3,000 y+ ∆+ 1,opt 500 1,500 0 200 400 600 y+ ∆+ 2,opt 500 1,500 0 500 1,000 1,500 y+ ∆+ 3,opt l = 0 l = 1 l = 2 l = 3 l = 4 l = 5 Figure 5: Optimal gridspacing according to G for the Reτ = 2000 channel for different iteration l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Table 2: Number of elements for the Reτ = 2000 channel with mortar meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Iteration N1 N2 N3 ∆1/h (∆2/h)min ∆3/h l = 0 10 2 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='00 l = 1 12 4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='50 l = 2 20 10 4 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='25 l = 3 32 16 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='5 26 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='23 l = 4 56 28 14 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='5 60 30 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='20 l = 5 124 62 31 21 104 52 26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='12 increase by a factor of more than two compared to the minimum grid spacing in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' For l = 2 the interfaces in x and z are located at the same wall-normal distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Thus, the resulting mesh l = 3 will contain one mortar interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' At l = 3 these locations are shifted and are no longer identical as indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' For l = 4 we even get a second mortar interface and therefore have a largest length ratio of four along the wall-parallel directions between the wall elements and the elements in the channel center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The grid for l = 3 is visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 4 showing the different mortar locations in the z and x plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The details of the mesh including the resolution and the number of el- ements saved is listed in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' For Ni and ∆i/h we listed the respective values at each iteration for each mortar interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 2 clearly indicates 17 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='5 1 10−1 100 y/h ∆1,actual 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='5 1 10−1 100 y/h ∆3,actual l = 0 l = 1 l = 2 l = 3 l = 4 l = 5 (a) The actual grid spacings for the streamwise and spanwise direction mortar elements of the Reτ = 2000 channel, non- dimensionalized by the largest cell along respective direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='5 1 y/h ∆1 actual spacing optimal spacing (b) Comparison of actual and op- timal grid-spacing for l = 2 mesh using l = 1 as reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Figure 6: Overview over the actual and optimal gridspacing for the channel containing mortar elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' that using mortars saves us up to approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 50% of elements in the simulation compared to a structured mesh with equivalent mortar-free spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' How- ever, this does not affect the timestep since it is dictated by the smallest cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' For an equivalent load per processor, the overall number of processors of the simulation can however be reduced accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' It looks that the higher the iteration, the more we save, since the grid near the wall requires more refine- ment every time compared to grid in the center of the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' This case converges to the desired Reτ = 2000 at l = 5, if judged only by this metric, with mortars saving almost 50% of the computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Table 3: Grid-spacing and element counts for the Reτ = 2000 channel with mortar meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Iteration ∆+ i (y = 0)/(P + 1) #Elems nDOF Reτ total saved l = 0 333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='33 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='67 333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='33 60 12960 569.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='17 l = 1 277.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='78 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='67 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='67 288 62208 373.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='16 l = 2 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='67 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='67 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='33 1320 120 285120 1841.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='85 l = 3 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='67 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='46 7592 1976 1639872 1723.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='20 l = 4 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='52 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='67 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='67 27195 18165 5874120 1875.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='99 l = 5 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='88 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='00 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='62 142662 128154 30814992 1967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='37 18 100 101 102 103 0 5 10 15 20 25 30 35 y+ u+ l = 0 l = 1 l = 2 l = 3 l = 4 l = 5 DNS Figure 7: Mean velocity profile for the Reτ = 2000 channel for each iteration l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' To evaluate the convergence and the quality of the simulations, we again look at the mean velocity and the Reynolds stresses profiles of the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The DNS data is found at Lee and Moser [22], just like the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The mean velocities in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 7 show the expected converging behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The velocity profile for l = 0 is more shifted towards higher friction velocities compared to the Reτ = 550 results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' This can be reasoned with the fact that the Reτ = 2000 channel is way more underresolved with the same initial mesh l = 0, and the weak implementation of the boundary conditions therefore cannot generate a good estimation for the τw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' At iterations l = [2, 3, 4, 5] we see convergence towards the reference data, with l = 5 matching the DNS velocity profile accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The results for the Reynolds stresses are visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Again, we match the reference solution well for the latest iteration of the grid- adaptation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The large oscillations for the first iterations are now also more pronounced in l = 1 in addition to l = 0 due to the Re being higher compared to the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' While w′w′ and u′v′ converge well, 19 0 500 1,000 1,500 2,000 0 10 20 30 y+ u′u′ 0 500 1,000 1,500 2,000 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='5 y+ v′v′ 0 500 1,000 1,500 2,000 0 5 10 y+ w′w′ 0 500 1,000 1,500 2,000 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='5 −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='5 y+ u′v′ l = 0 l = 1 l = 2 l = 3 l = 4 l = 5 DNS Figure 8: Reynolds stresses for the Reτ = 2000 channel for different iteration l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' the results of v′v′ lack behind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' This behavior is expected, since the mortars directly affect the wall-normal directions and thus are visible in the corre- sponding fluctuations in velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Despite that the v′v′ still shows convergence towards the Lee and Moser DNS data, with minimal (but still finite) signs of the mortars at l = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Channel flow at Reτ = 2000, application with WMLES To conclude the application on channel flow, we run the same test case at Reτ = 2000 as a wall-modeled simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' This aims to prove the capabilities of the algorithm to provide a good grid spacing for wall-modeled simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' We again allow mortar elements in the meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Previous experience with the code has shown that a higher C is beneficial in the wall-modeled region, therefore we increase the Vreman constant to C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='11 in Vreman subgrid- scale stress model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' We use Spaldings law of the wall as an algebraic wall 20 model, and for its solution we use Newton’s method in every Runge-Kutta timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The interface height hwm is fixed at 10% of the channel half height h and is depicted as a red dashed line in the following figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 500 1,500 0 1,000 2,000 y+ ∆+ 1,opt 500 1,500 0 200 400 600 y+ ∆+ 2,opt 500 1,500 0 500 1,000 y+ ∆+ 3,opt l = 0 l = 1 l = 2 l = 3 l = 4 Figure 9: Optimal gridspacing according to G for the wall-modeled Reτ = 2000 channel for different iteration l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Dashed red line indicates hwm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' First we assess the suggested gridspacings ∆+ i,opt, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The spacing in wall-normal direction is again realized by the same method as for the previous subsection of WRLES containing mortars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' However, we have adapted the suggested grid spacings to the WMLES methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The pure grid-adaptation finds that the underresolution of the relevant scales is harsh- est near the wall, thus suggesting the most refinement there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' However, the point of wall-modeling is to model the smaller but flow-relevant eddies below hwm, so the grid-adaptation process needs to be told that severe underresolu- tion in that region is a choice and not an error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' To achieve this, we introduce a restriction on ∆+ i,opt according to Toosi (personal communication).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' We do not allow any smaller ∆+ i,opt below hwm, meaning that we reset the values of ∆+ i,opt for the cells that lie between hwm and the wall to the values of their counterparts at hwm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' This applies to the grid spacings of all directions in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 9 also indicates that the wall model interface is not restricted to the wall neighboring grid cell, but can also be placed in the second cell, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' l = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' For l = 4 we also introduce a mortar in x and z which is located at the same wall-normal positions for both dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' In comparison to the 21 Table 4: Simulation properties of the wall-modeled Reτ = 550 channel for different itera- tion l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Iteration N1 N2 N3 ∆1/h (∆2/h)min ∆3/h ∆+ i (y = 0)/(P + 1) #Elems #DOF Reτ total saved l = 0 10 2 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='00 333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='33 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='67 333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='33 60 12960 627.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='91 l = 1 12 4 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='60 277.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='78 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='67 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='00 240 51840 872.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='29 l = 2 19 5 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='27 175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='44 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='33 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='91 1045 225720 1241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='07 l = 3 32 6 23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='13 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='17 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='67 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='48 4416 953856 1220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='01 l = 4 62 31 14 38 19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='16 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='76 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='00 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='32 31217 1767 6742872 1555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='60 wall-resolved case this only happens at the fourth iteration, since the scale differences between wall and the center of the channel are much smaller, since wall-modeled LES does not resolve the small scales of the wall turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The resolution and resulting viscous spacings at the wall are listed in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 100 101 102 103 0 5 10 15 20 25 y+ u+ l = 0 l = 1 l = 2 l = 3 l = 4 DNS Figure 10: Mean velocity profile for the wall-modeled Reτ = 2000 channel for each iteration l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' To verify the results, we again compare them against the same DNS data by Lee and Moser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 10 even for l = 0 we already have good agreement, especially at the interface location at h+ wm = 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' This is because the initial grid is much closer to the WMLES requirements than the WRLES 22 requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' A clear convergence towards the DNS data is also visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Iteration l = 3 agrees with the DNS data for the whole channel above hwm, including the center of the channel which was slightly off for the previous iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 0 500 1,000 1,500 2,000 0 5 10 15 y+ u′u′ 0 500 1,000 1,500 2,000 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='5 y+ v′v′ 0 500 1,000 1,500 2,000 0 1 2 3 4 y+ w′w′ 0 500 1,000 1,500 2,000 −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='5 0 y+ u′v′ l = 0 l = 1 l = 2 l = 3 l = 4 DNS Figure 11: Reynolds stresses for the wall-modeled Reτ = 2000 channel for different itera- tion l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The Reynolds stresses are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Unlike the velocity pro- files, the fourth iteration is needed to match the Reynolds stresses better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' This is due to the WMLES approach, not the grid-adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Even then, the Reynolds stresses, especially u′u′ does not exactly converge on the DNS data, but become very acceptable for a WMLES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The v′v′ also deserves some attention, where we can clearly see the converging behavior of the fluctu- ations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' but in the center of the channel the mortar in the mesh of l = 4 gets visible again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The other Reynolds stress components show good enough agreement, proving the applicability of the algorithm to WMLES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 23 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Application to the flow over an airfoil To conclude this work, we investigate the flow over a transonic NACA 64A-110 airfoil at an angle of attack of α = 0 deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' This test case allows us to apply all the functionalities that we introduced in the channel flow onto a more practical application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The airfoil grid is generated by extruding a 2D geometry (xy-plane) in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The xy-plane grid is generated using an unstruc- tured 2D mesh tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Due to the extrusion the mesh will be structured in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' To allow the grid-adaptation algorithm to show its full potential, we allow a homogeneous mortar element distribution along z by specifying regions in the xy-plane with different number of elements in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' This way, we generate a fully unstructured, yet still hexahedral only, three dimensional grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The characteristic length of the airfoil simulation c is set to the chord length and defined as c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The spanwise extension in z is Lz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='05c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The airfoil is placed in a wind tunnel, therefore we include Euler walls above and below the profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The dimensionless simulation parameters are Ma = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='72 and Rec = 930 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' We perform it as a WMLES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The reliable transition to turbulence is achieved through the flow being tripped with a numerical trip at x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='05c using the method introduces by Schlatter and Örlu [23] on the suction and pressure side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The interface height is fixed at hwm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='0134 and constant over the surface for all iterations and has been determined with the interface height adaptation algorithm introduced by Kahraman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' [24] on the WRLES reference data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' We eventually compare the results to a wall-resolved LES simulation that has been performed previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' We start with a coarse mesh and perform the iterative approach, just like the channel test case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' We iterate until we match the pressure coefficient cp, the skin friction τw and the boundary layer thickness δ99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The initial mesh l = 0 is built to be extremely cheap and without any special treatment of the boundary layer or any other flow properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The only constraint on the mesh is to fit the geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Thus in l = 0 the mesh has only 6 elements per cordlength in streamwise direction, which results in a maximum y+ ≈ 1100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' This initial mesh is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 12a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Since we use high-order methods we must use curved elements, which allows to take more details of the geometry into account even for as coarse meshes as in l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 24 (a) initial mesh l = 0 around the airfoil (b) first refinement level l = 1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='0 × 10−1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='0 × 10−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='0 × 10−1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='0 × 10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='0 × 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='0 × 10−2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='0 × 10−3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='0 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='0 × 10−3 (c) Qualitative contour plot of ∆i,opt with logarithmic scaling after l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' All plots are scaled equiva- lently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' From left to right: wall parallel s, wall normal n and spanwise direction z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Figure 12: Comparison between the velocity field and mesh the initial and the first itera- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' To get the next mesh, of iteration l = 1, we evaluate the grid error indicator densities g(⃗x,⃗ni) to get ∆i,opt, which is qualitatively visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 12c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' It shows the same behavior that was already observed for the channel, that we refine the most in wall normal direction n, followed by the spanwise direction z and the tangential direction s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Interesting to note is that the spanwise direction has very coarse suggested grid spacing at the leading edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' This is expected, and nice to see for the performance of the algorithm, since that the flow is laminar which means the lengthscale approaches infinity along that direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The turbulent wake also is refined in more detail, since this coarse grid is still able to sustain some turbulence there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The mesh l = 1 is also visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 12b, showing significantly more elements near the airfoil, which is very well expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Fig 12 thus shows the complete process from l = 0 to l = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The whole development of the relevant quantities defined earlier is visu- alized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The graphs indicate a clearly converging behavior and also shows the different rates of convergence of the quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The pressure coef- ficient cp converges the fastest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The skin friction coefficient cf only converges after l = 2 and the boundary layer thickness, which is most sensitive to the grid resolution, needs four iterations to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The difference between l = 2 and l = 3 is only the resolution in x and z, which is listed in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 5 and 25 0 2 4 6 0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='5 −1 0 2 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='8 0 2 4 6 ·10−3 x/c τw,SS 10−3 τw,PS 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='8 0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='5 −1 x/c cp,SS cp,PS 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='9 0 2 4 10−2 x/c δ99,SS 10−2 δ99,PS l = 0 l = 1 l = 2 l = 3 WRLES Figure 13: Convergence of pressure coefficient cp and skin friction τw and boundary layer thickness δ99 on the pressure (PS) and suction (SS) side of the NACA 64A-110 airfoil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 6 along other grid characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Especially at the trailing edge the algorithm suggested mesh refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' This is also where the discrepancy be- tween WRLES reference data and the iterations l = [0, 1, 2] was the largest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The red dashed line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 13 shows the wall model interface height over x/c which is roughly 10% of δ99,mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Table 6 also shows the mortar information for l = [1, 2, 3] as well as the minimal and maximum grid spacings at the airfoil boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The values in 26 Table 5: Grid spacings of the wall-modeled NACA 64A-110 airfoil for different iteration l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Iteration ∆+ SS(x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='05c, s)/(P + 1) ∆i/c min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' l = 0 1267.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='21 219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='82 146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='54 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='21e-1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='16e-1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='75e-2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='75e-2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='50e-2 l = 1 316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='92 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='76 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='85 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='30e-2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='40e-2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='00e-3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='00e-3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='33e-3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='67e-2 l = 2 146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='54 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='86 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='21 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='10e-2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='00e-2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='00e-3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='00e-3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='08e-3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='67e-2 l = 3 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='62 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='86 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='21 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='00e-3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='50e-2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='00e-3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='50e-3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='04e-3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='67e-2 Table 6: Number of cells of the wall-modeled NACA 64A-110 airfoil for different iteration l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Iteration Ni #Elems #DOF Ns · Nn Nz total saved l = 0 395 (2) 790 404480 l = 1 1472 (3, 6) 5946 2886 3044352 l = 2 2590 (3, 6, 12, 24) 36072 26088 18468864 l = 3 9111 (3, 6, 12, 24, 48) 222426 214902 113882112 between can be easily calculated by assuming the 2-1 mortar interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The observation made in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 2 are confirmed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The usage of the unstructured capabilities of the discontinuous Galerkin framework allow us to save up to approximately 50% of the elements, which directly translates to the same amount of savings in compute time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 14 the mortar zones in z are visualized, each denoting a different number of elements in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The colors are based on the minimum number of cells along spanwise direction Nz, which are Nz(l = 0) = 2 and Nz(l = [1, 2, 3]) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The zones are designed in a way to meet the target values of the indicator in an efficient way with our limitation to 2-1 mortar interfaces, even though the indicator suggests even more coarsening in some locations and much more rapid refinement at other locations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' from Nz ·24 to Nz ·20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' We insert buffer layers in between to make the mesh compatible and to enable a transition between the mortar zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Still, the number of elements saved justifies this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Summary In this work we presented turbulent channels and a transonic airfoil in discontinuous Galerkin framework using different grids proposed by using the high-order optimized version of the grid-adaptation algorithm based on the 27 (a) Initial mesh l = 0 around the airfoil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' (b) First refinement iteration l = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' (c) Second refinement iteration l = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' (d) Third refinement iteration l = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Nz · 20 Nz · 21 Nz · 22 Nz · 23 Nz · 24 Figure 14: Comparison between the z mortar zones of the airfoil iterations, visualized with non-curved elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' works from Toosi and Larsson [1, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The boundary layers were either wall- resolved or wall-modeled to show the ability of the algorithm to handle two major cases of large eddy simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' For all test cases we showed convergence and good agreement with the results in literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The grid-adaptation indicator in general for high-order methods has turned out to be most efficient when applied elementwise and not per collocation point due to the reduced robustness of the numerical scheme in underresolved settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Having the indicator values on the element, splitting and remeshing with the proposed recommended gridspacing is straight forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' The appli- cation of the grid-adaptation machinery to a p-refined approach is yet to be investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Another important aspect of the simulation run is the usage of the un- structured capabilities of the discontinuous Galerkin implementation in the FLEXI framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Unstructured meshes in three dimensions were realized using mortar interfaces between elements and the usage of these elements saved up to 50% of compute time without any significant loss in accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' 28 Especially for the airfoil case, we were able to clearly distinguish different mortar zones in spanwise direction, even after few iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Therefore, we have shown that the grid-adaptation framework is able to deliver a problem- tailored and cost-optimized mesh with converging behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Thus, the grid-adaptation algorithms can be applied to arbitrary geome- tries without any prior knowledge to the flow field and still create a problem- tailored and cost optimized mesh, which does not rely on the experience of the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' Acknowledgments MB and AB gratefully acknowledge the Deutsche Forschungsgemeinschaft DFG (German Research Foundation) for funding this work in the frame- work of the research unit FOR2895, and thank the Gauss Centre for Su- percomputing e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='gauss-centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='eu) for funding this project (GCS- lesdg) by providing computing time on the GCS Supercomputer HAWK at Höchstleistungsrechenzentrum Stuttgart (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='hlrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content='de).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' AK and JL were supported by the Department of Energy PSAAP III program (grant DE- NA0003993) and the NASA Transformational Tools and Technologies project (grant 80NSSC18M0148).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E1T4oBgHgl3EQfdwSQ/content/2301.03199v1.pdf'} +page_content=' References [1] S.' 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b/gtAzT4oBgHgl3EQfavzP/content/tmp_files/2301.01375v1.pdf.txt @@ -0,0 +1,3693 @@ +Draft version January 5, 2023 +Typeset using LATEX twocolumn style in AASTeX631 +Brightest Cluster Galaxy Formation in the z=4.3 Protocluster SPT 2349-56: Discovery of a +Radio-Loud AGN. +Scott C. Chapman,1, 2, 3, 4 Ryley Hill,3 Manuel Aravena,5 Melanie Archipley,6, 7 Arif Babul,8 James Burgoyne,9 +Rebecca E. A. Canning,10 Carlos De Breuck,11 Anthony H. Gonzalez,12 Christopher C. Hayward,13 +Seon Woo Kim,6 Matt Malkan,14 Dan P. Marrone,15 Vincent McIntyre,16 Eric Murphy,17 Emily Pass,18, 1 +Ryan W. Perry,1 Kedar A. Phadke,6, 7 Douglas Rennehan,13 Cassie Reuter,6 Kaja M. Rotermund,19 +Douglas Scott,3 Nick Seymour,16 Manuel Solimano,5 Justin Spilker,20 Anthony A. Stark,18 +Nikolaus Sulzenauer,21 Nick Tothill,22 Joaquin D. Vieira,6, 7 David Vizgan,6 George Wang,3 and Axel Weiss21 +1Department of Physics and Atmospheric Science, Dalhousie University, Halifax, NS, B3H 4R2, Canada +2NRC Herzberg Astronomy and Astrophysics, 5071 West Saanich Rd, Victoria, BC, V9E 2E7, Canada +3Department of Physics and Astronomy, University of British Columbia, Vancouver, BC, V6T1Z1, Canada +4Eureka Scientific Inc, Oakland, CA 94602, USA +5N´ucleo de Astronomia, Facultad de Ingenieria y Ciencias, Universidad Diego Portales, Av. Ej´ercito 441, Santiago, Chile +6Department of Astronomy, University of Illinois, 1002 West Green St., Urbana, IL 61801, USA +7Center for AstroPhysical Surveys, National Center for Supercomputing Applications, 1205 West Clark Street, Urbana, IL 61801, USA +8Department of Physics and Astronomy, University of Victoria, Victoria, BC V8P 1A1, Canada +9Department of Physics and Astronomy, University of British Columbia, Vancouver, BC, Canada +10Institute of Cosmology and Gravitation, University of Portsmouth, Dennis Sciama Building, Portsmouth, PO1 3FX, UK +11European Southern Observatory, Karl Schwarzschild Strasse 2, 85748 Garching, Germany +12Department of Astronomy, University of Florida, 211 Bryant Space Science Center, Gainesville, FL 32611-2055, USA +13Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA +14Department of Physics and Astronomy, University of California, Los Angeles, CA 90095, USA +15Steward Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721, USA +16International Center for Radio Astronomy Research, Curtin University, GPO Box U1987, 6102 Perth, Australia +17NRAO National Radio Astronomy Observatory 520 Edgemont Road Charlottesville, VA 22903 +18Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA +19Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA +20Department of Physics and Astronomy and George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy, +Texas A&M University, 4242 TAMU, College Station, TX 77843-4242, US +21Max-Planck-Institut f¨ur Radioastronomie, Auf dem Hugel 69, Bonn, D-53121, Germany +22School of Science, Western Sydney University, Locked Bag 1797, Penrith NSW 2751, Australia +(Received XXX; Revised YYY; Accepted ZZZ) +Submitted to ApJ +ABSTRACT +We have observed the z = 4.3 protocluster SPT2349−56 with the Australia Telescope Compact Array +(ATCA) with the aim of detecting radio-loud active galactic nuclei (AGN) amongst the ∼ 30 submil- +limeter (submm) galaxies (SMGs) identified in the structure. We detect the central complex of submm +sources at 2.2 GHz with a luminosity of L2.2 = (4.42±0.56) × 1025 W Hz−1. +The Australian Square +Kilometre Array Pathfinder (ASKAP) also detects the source at 888 MHz, constraining the radio spec- +tral index to α = −1.6±0.3, consistent with ATCA non-detections at 5.5 and 9 GHz, and implying +L1.4, rest = (2.4±0.3) × 1026 W Hz−1. This radio luminosity is about 100 times higher than expected +from star formation, assuming the usual far-infrared (FIR)-radio correlation, which is a clear indica- +tion of an AGN driven by a forming brightest cluster galaxy (BCG). None of the SMGs in SPT2349−56 +show signs of AGN in any other diagnostics available to us (notably 12CO out to J = 16, OH163µm, +Corresponding author: Scott C. Chapman +scott.chapman@dal.ca +arXiv:2301.01375v1 [astro-ph.GA] 3 Jan 2023 + +2 +Chapman et al. +[C ii]/IR, and optical spectra), highlighting the radio continuum as a powerful probe of obscured AGN +in high-z protoclusters. No other significant radio detections are found amongst the cluster members, +with stacking on either all members or just the ten most luminous members yielding non-detections +consistent with the FIR-radio correlation for star-forming galaxies. We compare these results to field +samples of radio sources and SMGs, along with the 22 SPT-SMG gravitational lenses also observed +in the ATCA program, as well as powerful radio galaxies at high redshifts. Our results allow us to +better understand the effects of this gas-rich, overdense environment on early supermassive black hole +(SMBH) growth and cluster feedback. We estimate that (3.3±0.7) × 1038 W of power are injected into +the growing intra-cluster medium (ICM) by the radio-loud AGN, whose energy over 100 Myr is com- +parable to the binding energy of the gas mass of the central halo. The AGN power is also comparable +to the instantaneous energy injection from supernova feedback from the 23 catalogued SMGs in the +core region of 120 kpc projected radius. The SPT2349−56 radio-loud AGN may be providing strong +feedback on a nascent ICM. +Keywords: Submillimeter astronomy (1647) — Galaxy evolution (594) +1. INTRODUCTION +Submillimeter (submm) galaxies (SMGs) are impor- +tant sites of stellar mass build-up at cosmic noon and +earlier (e.g., Chapman et al. 2003, 2005; Smail et al. +2004), with star-formation rates (SFRs) as high as hun- +dreds to thousands of solar masses per year. +Several +studies have also suggested that SMGs may be good +tracers of dark matter halos at early cosmic time (e.g., +Blain et al. 2004; Chen et al. 2016; Dudzeviˇci¯ut˙e et al. +2020). +Simulations conducted by Miller et al. (2015) +found that while many dark matter halos at z = 2–4 do +not contain any SMGs, large and rare associations of five +or more SMGs do trace massive overdensities of dark +matter that have the potential of evolving into present- +day massive clusters. Supporting this, in the recent past, +several high-redshift protoclusters have been identified +entirely through their submm emission (e.g., Chapman +et al. 2009; Daddi et al. 2009; Capak et al. 2011; Casey +et al. 2015; Miller et al. 2018; Oteo et al. 2018; G´omez- +Guijarro et al. 2019; Wang et al. 2021). +AGN and star-formation processes in galaxy evolution +are clearly related (e.g. Kormendy & Ho 2013). +En- +hanced AGN activity relative to the field environment +has been found in massive protoclusters at z = 2–3 (e.g., +Pentericci et al. 2002; Lehmer et al. 2009; Digby-North +et al. 2010), which is likely related to the enhancement +of star formation in galaxy protocluster members (e.g. +Elbaz et al. 2007; Chapman et al. 2009; Brodwin et al. +2013; Casey et al. 2015; Gilli et al. 2019). The efficient +quenching of star formation in clusters requires mechan- +ical and radiative feedback, which is naturally generated +by AGN. The resulting shocked hot gas detected as “cav- +ities” in the cluster is constrained by resolved, extended +X-ray emission (e.g. Fabian 2012). The Clusters Around +Radio-Loud AGN (CARLA) survey of around 400 high- +redshift radio galaxies (HzRGs) from z = 1–3 (Wyleza- +lek et al. 2013) showed that in the majority of cases, the +radio AGN is located near the center of the galaxy over- +density as traced by their stellar mass (Spitzer-IRAC +emission). +This is strong evidence that radio galaxy +feedback in a growing brighest cluster galaxy (BCG) is +important for the evolution of massive galaxy clusters. +Overdense regions at high redshift have likely not yet +virialized. Their abundant reservoirs of cold gas and the +ongoing mergers and galaxy encounters expected in the +dense environments will trigger star formation and drive +gas down to the supermassive black hole (SMBH) poten- +tial well. AGN require this nuclear accretion as a power +source. This is in contrast to low redshifts, where struc- +tures are virialized, which prevents AGN from forming +(e.g. van Breukelen et al. 2009). Studies of AGN in pro- +toclusters has recently become a viable endeavor, with +relatively deep Chandra and XMM-Newton observations +at z = 1.5–3 (e.g., Digby-North et al. 2010; Wang et al. +2013; Travascio et al. 2020). Continuing to study the +rich variety of protoclusters and extending these studies +to earlier times can inform how host galaxies are af- +fected by their SMBHs, as well as the connection to the +surrounding environment. In the z = 3.09 SSA22 pro- +tocluster, 50% of the SMGs were found to host X-ray +luminous AGN (Umehata et al. 2019) – a clear excess +over the 15% found for field SMGs (e.g. Wang et al. +2013). At larger distances, an overdensity of ten SMGs +found by the Hershel Space Telescope at z = 4.0 (Oteo +et al. 2018) has been studied by Chandra in the X-ray +(Vito et al. 2020) and in the radio (Oteo et al. 2018), +revealing no significant excess of AGN activity in the +system over field SMGs (22% versus 15%, respectively). +The 2,500 deg2 survey conducted by the South Pole +Telescope (SPT – Vieira et al. 2010; Everett et al. + +A radio-loud AGN in SPT2349 +3 +Table 1. ATCA-selected sources within a radius of 1 Mpc +in projection (140′′) of SPT2349−56. +ID +RA +Dec +Freq. +Flux +(GHz) +(µJy) +ID1 +– +– +8.98 +< 159†† +ID1 +– +– +5.47 +< 120†† +ID1 +23:49:42.760 +−56:38:25.05 +2.17 +214±27 +ID1† +23:49:42.55 +−56:38:19.4 +0.888 +867±189 +ID2 +23:49:38.838 +−56:37:09.63 +2.17 +547±36 +ID2† +23:49:38.750 +−56:37:06.09 +0.888 +1324±182 +ID3 +23:49:43.692 +−56:38:01.82 +2.17 +135±26 +† ASKAP measurement +†† 3σ ATCA limit +2020) at 3.0 mm, 2.0 mm and 1.4 mm has uncovered +a small population of nine millimeter sources ranging +from z = 3–7, which are extremely luminous, yet ap- +parently not gravitationally lensed (e.g. Spilker et al. +2016; Reuter et al. 2020; Wang et al. 2021). +A well +characterized example of this is SPT2349−56, a proto- +cluster system at z = 4.303 (Miller et al. 2018). +Ob- +servations at 870 µm using the Large APEX BOlome- +ter CAmera (LABOCA; Siringo et al. 2009) on the At- +acama Pathfinder Experiment (APEX; G¨usten et al. +2006) telescope (with a 19-arcsec beam size) first re- +vealed an extended structure with two distinct lobes +connected by a bridge with a combined flux density +of S870 µm = (106 ± 8) mJy (Miller et al. 2018; Wang +et al. 2021). Follow-up observations with the Atacama +Large Millimeter-submillimeter Array (ALMA; Woot- +ten & Thompson 2009) measured the redshift of its +brightest central source through 12CO lines (Strandet +et al. 2016), and then resolved the structure into over 30 +submm-luminous sources (Miller et al. 2018; Hill et al. +2020; Rotermund et al. 2021), with a velocity disper- +sion suggesting a central halo mass of around 1013 M⊙. +A VLT/MUSE observation reveals the presence of a +Lyα blob (LAB), with a linear size of about 60 kpc, +close to the core of SPT2349−56 (Y. Apostolovski et +al. in prep.). None of the other protocluster SMGs were +detected as Lyα emitters (LAEs) in the MUSE data. +Lyα halos are commonplace in most HzRG protoclusters +(Venemans et al. 2007). Similar objects are often found +in protoclusters identified through other means, for ex- +ample optical galaxy overdensities (Overzier 2016), and +indicate the presence of significant amounts of neutral +gas in the assembling cluster. +This paper presents a search for radio detections of +members of the SPT2349−56 cluster. +Section 2 de- +scribes the radio and (sub)millimeter observations. Sec- +Table 2. +ALMA observing programs used for follow-up +analysis. Details on additional ALMA Band-7 observations +used in this paper can be found in (Hill et al. 2020). Here +the frequency is the central frequency between the upper and +lower sidebands, the continuum sensitivity is calculated at +the center of the primary beam and averaged over the upper +and lower sidebands, and the beam is the average circular +synthesized beam FWHM. +ID +Date +Freq. +σctm +Beam +Array +(GHz) +(µJy) +(′′) +2015.1.01543.T +03/20/16 +148.3 +10 +0.88 +C36-2/3 +2018.1.00058.S +10/03/18 +146.8 +12 +0.28 +C43-6 +2021.1.01313.S +07/27/22 +146.3 +21 +0.27 +C-6 +2021.1.01313.S +09/01/22 +231.9 +31 +0.47 +C-4 +tion 3 presents the results derived from the source ex- +traction and analysis. In Section 4 we discuss the de- +tected central radio source, the energy injected into +a growing ICM, and the implications for radio-loud +AGN in protoclusters. +We conclude in Section 5. +Throughout our analysis, a Hubble constant of H0 = 70 +km s−1 Mpc−1 and density parameters of ΩΛ = 0.7 and +Ωm = 0.3 are assumed, resulting in a proper angular +scale of 6.88 kpc/′′ at z = 4.3. +2. DATA +2.1. ATCA observations +SPT2349−56 was observed by the Australia Tele- +scope Compact Array (ATCA) at 2.2, 5.5, and 9.0 GHz +between 2012, January 23 to 27, as part of a pro- +gram (C1563) to observe 23 SPT-SMGs (described in +Appendix C). We used the Compact Array Broad- +band Backend (CABB) configured in the 1M-0.5k mode, +which leads to a bandwidth of 2 GHz per correlator +window with 1 MHz per channel of spectral resolution. +The observations were performed in the most extended +ATCA configuration, 6A, with six working 22 m anten- +nas. +The on source time was 34 min, which was typ- +ical for all SPT-SMGs observed (see Table 5). +The +data were edited, calibrated, and imaged using the +Miriad package. +Data affected by known radio fre- +quency interference (RFI) or with bad visibility ranges +were flagged accordingly. We estimate an absolute cal- +ibration uncertainty of 5% at 2.2 and 5.5 GHz, and +10% at 9.0 GHz. We inverted the visibilities using nat- +ural weighting, leading to beam sizes of 7.7′′ × 4.2′′, +3.2′′ × 2.1′′, and 2.0′′ × 1.3′′ at 2.2, 5.5, and 9.0 GHz, +respectively, with associated RMS noise values of 27, +40, and 53 µJy beam−1, respectively. Figure 1 displays +the ATCA 2.2 GHz map surrounding SPT2349−56, re- + +4 +Chapman et al. +23:49:45 +40 +35 +30 +-56:37:00 +30 +38:00 +30 +39:00 +RA +Dec +SPT2349 − 56 +ATCA 2.2GHz +ALMA 350GHz +K +A +B +C +G +F +L +H +J +I +N +E +D +Figure 1. +Background: ATCA 2.2 GHz imaging of the SPT2349−56 region, with gold contours highlighting the 106 mJy +extended LABOCA source at 870µm. The linear ATCA feature east of SPT2349−56 is from the synthesized beam structure +of a bright 20 mJy source to the south (see Appendix A). An ATCA radio source is identified near the LABOCA core, with an +ASKAP source (white contours) overlapping. The bright ATCA+ASKAP source to the northwest is identified with a Milky +Way star (ID2 in Table 1). Inset: A 20′′ × 20′′ zoom-in of ALMA 350 GHz continuum imaging (Hill et al. 2022) with overlays +of ATCA 2.2 GHz (cyan), ASKAP 888 MHz (white). ATCA contours start at 3.7σ revealing the FWHM of the source (4′′ × 8′′). +White contours (ASKAP) start at 3σ, and the FWHM of the source is 16′′ × 25′′. ALMA sources are named from Miller et al. +(2018) in order of their 850 µm flux density. The ATCA radio detection of the B-C-G complex of galaxies is evident. + +A radio-loud AGN in SPT2349 +5 +100 +101 +102 +103 +104 +105 +Wavelength (microns) +10 +2 +10 +1 +100 +101 +Flux Density (mJy) +C +B +Arp220 +Figure 2. +Spectral energy distribution showing the ATCA +and ASKAP radio detections at rest wavelengths, and the +flux densities of the brightest two of the three central SMGs +(B – red circles; C – black circles). Also shown is the rest- +frame optical photometry of ALMA source C (as in Fig. 1), +which was modelled with a 3 × 1011 M⊙ stellar mass fit +(Rotermund et al. 2021). Source B is undetected at these +wavelengths. The Arp220 SED (dashed blue line) is normal- +ized to the submm photometry, revealing that the SPT2349 +BCG galaxy complex has significant excess in radio above the +far-infrared (FIR)-radio correlation for star-forming galax- +ies. The 5.5 GHz and 9 GHz limits are shown at 3σ. The +radio spectral index is constrained to α = − 1.58 ± 0.31 (fit- +ted line), primarily by the ASKAP detection, and consistent +with the upper limits. The grey shadings show the 1 and 2σ +uncertainties in the fit (Appendix C); an Arp220 α = − 0.8 +spectral index is ruled out at the 3σ level by the 5.5 GHz +non-detection. +vealing a well-detected (8σ) source near the core of +SPT2349−56. No sources at 5.5 or 9.0 GHz are found in +the vicinity of SPT2349−56. ATCA sources surround- +ing SPT2349−56 out to 1 Mpc in projection are listed +in Table 1, and the wider-field ATCA map is shown in +Appendix A. +The shortest baseline is 30 m and the images should +be sensitive to emission on angular scales up to a few +arcminutes. In principle, these data should not be miss- +ing any flux on the scales covering both the ATCA and +ASKAP (see below) sources, although the ATCA data +will be less sensitive to lower surface brightness emis- +sion. However the short 34 min integration, with quite +limited SNR, may still be missing some structure due to +sparse uv coverage. Similar issues were discussed in an +ATCA snapshot survey of distant HzRGs (De Breuck +et al. 2000) and are elaborated in section 4. +2.2. ASKAP observations +The Australian Square Kilometre Array Pathfinder +(ASKAP) comprises 36 twelve-metre dishes located in +the Inyarrimanha Ilgari Bundra1 at the CSIRO Murchi- +son Radio-astronomy Observatory (MRO) in Western +Australia, observing between 700 MHz and 1.8 GHz, +with an instantaneous bandwidth of up to 288 MHz. +ASKAP is equipped with phased-array feeds (PAF; +Hotan et al. 2014; McConnell et al. 2016), capable of +simultaneously forming up to 36 independent beams, +covering some 30 deg2. +SPT2349−56, along with all 22 of the lensed SPT- +SMGs in the ATCA program, were observed by the +Rapid ASKAP Continuum Survey (RACS, McConnell +et al. 2020), covering the sky south of +41 deg dec- +lination at a central frequency of 887.5 MHz, using +903 individual pointings with 15-minute observations. +The beam size at the location of SPT2349−56 is +24′′ × 13′′. We retrieved the ASKAP image surrounding +SPT2349−56 using the cutout server. At the declination +of SPT2349−56 the achieved RMS sensitivity is 189 µJy. +The RMS is similar in the ASKAP images around the +other 22 lensed SPT-SMGs, although the actual sensi- +tivity depends on proximity to other nearby bright ra- +dio sources (see Appendix C). The SPT2349−56 ATCA- +detected source is not cataloged in the RACS, but we +find a 4.6σ peak approximately 5′′ from the ATCA +source (shown in Fig. 1). +2.3. ALMA observations +Extensive ALMA properties of SPT2349−56 sources +B, C, and G have already been published (Miller et al. +2018; Hill et al. 2020; Rotermund et al. 2021). Here we +present several new ALMA observations (Table 2), sup- +porting our measurements of line emission in the context +of searching for AGN. +ALMA Band-4 imaging (150 GHz) was obtained un- +der three different programs in Cycles 3, 6, and 8, all +targeting the brightest peak of the LABOCA source, +and tuned to place CO(7–6) (νrest = 806.652 GHz) and +[C i](2–1) (νrest = 809.34 GHz) in the upper sideband, +and para-H2O(211–202) (νrest = 752.033 GHz) in the +lower sideband. +The Cycle 3 program 2015.1.01543.T (PI: K. Lacaille) +was observed on March 20, 2016. The array was in the +C36-2/3 configuration with baselines ranging from 15 to +1 The name means ‘shared skies and stars’ in the local indigenous +language, Wajarri Yamatji. + +6 +Chapman et al. +460 m, and provided a naturally-weighted synthesized +beam size of 0.88′′. Pallas and J2343−5626 were used +to calibrate the amplitude and phase, respectively. The +Cycle 6 program (2018.1.00058.S; PI: S. Chapman) ob- +servations were obtained on 2018, October 3rd in the +C43-6 array configuration with baseline lengths of 15 to +2500 m, giving a naturally-weighted synthesized beam +size of 0.28′′. +J2056−4714 was used to calibrate the +amplitude, while J2357−5311 was used to calibrate the +phase. Lastly, the Cycle 8 program (2021.1.01313.S; PI: +R. Canning) observations were obtained on 2022, July +27. These observations used the C-6 array configura- +tion with baselines of 15 to 2500 m, giving a naturally- +weighted synthesized beam size of 0.27′′. J2357−5311 +was used to calibrate the amplitude, while J2336−5236 +was used to calibrate the phase. +The Cycle 8 program (2021.1.01313.S) also observed +CO(11–10) (νrest = 1267.01 GHz) and continuum at +about 230 GHz in Band 6. These observations, carried +out on 2022, September 1, used the C-4 array configu- +ration with baselines of 15 to 784 m, giving a naturally- +weighted synthesized beam size of 0.47′′. J2357−5311 +and J2258−2758 were used to calibrate the amplitude, +while J2357−5311 and J2336−5236 were used to cali- +brate the phase. +We also make use of previously-published Band 7 +(345 GHz) ALMA Cycle 5 and 6 observations (Hill +et al. 2020). +The deep 0.5′′-resolution (i.e. +synthe- +sized beam) Cycle 5 data contain the CO(16–15) line +(νrest = 1841.35 GHz) and an OH doublet; each of the +doublets is actually composed of a triplet whose fre- +quencies are about 0.01 GHz separated, which is com- +pletely unresolved by our spectral resolution, so we +consider the OH line to be a doublet. +The mean +frequencies of the doublet are νrest = 1837.80 GHz and +νrest = 1834.74 GHz). These lines are present in the up- +per sideband, which was not previously analyzed or pub- +lished. The high-resolution Cycle 6 data described by +(Hill et al. 2020) has a synthesized beam of about 0.2′′ +and is here used to further analyze kinematics through +a moment analysis of the [C ii] line (Section 3.3). +All the data were calibrated using the standard +observatory-supplied calibration script. +Imaging was +done using the CASA task tclean, using Briggs weighting +with a robust parameter of 0.5, and in all cases channel +widths were averaged down to a common 15.625 MHz. +The Cycle 6 and 8 observations covering the CO(7–6), +[C i](2–1), and H2O lines were combined in uv space and +then imaged together, while the Cycle 3 observation was +imaged separately. We chose this approach as the two +data sets did not overlap entirely in frequency, which led +to artefacts in the imaging step. The higher-resolution +Cycle 6 and 8 data cubes were then convolved to match +the resolution of the Cycle 3 data (about 0.88′′). The +continuum was subtracted using the task imcontsub af- +ter flagging all channels expected to contain line emis- +sion based on previously-detected [C ii] lines given in Hill +et al. (2020). At each spatial pixel, imcontsub extracts +a one-dimensional spectrum and calculates the average +over all channels not flagged by the user, then subtracts +this average and returns a continuum-subtracted data +cube. +The same apertures used by Hill et al. (2020) to ex- +tract [C ii] line strengths and 350-GHz continuum flux +densities were applied to sources B, C, and G in order +to extract one-dimensional spectra for each line. The +Cycle 3 and Cycles 6+8 CO(7–6), [C i](2–1), and H2O +spectra were averaged to produce a final spectrum. De- +tails on how line strengths and continuum flux densities +(including our procedure for deblending lines) are given +in Appendix B, and the spectra are shown in Figs. 9 – +11. All new continuum flux densities and line strengths +are listed in Table 3, and the new continuum measure- +ments are also shown in Fig. 2. +3. RESULTS +3.1. Identifying and characterizing radio sources +We first searched for radio sources at the positions +of known ALMA and optically-identified members of +the SPT2349−56 protocluster. There is one strong ra- +dio detection at 2.2 GHz (S2.2 = 214 µJy) found near +the SPT2349−56 core with ATCA (detected at 8σ), +which corresponds to a less robust (4.6σ) detection +with ASKAP at 888 MHz (Fig. 1 and Table 1). +The +ATCA source with a much smaller beam encompasses +the bright central ALMA sources, named B, C, and G +based on their rank-ordered 850 µm flux densities Miller +et al. (2018).2 It is unclear from positional uncertainty +and beam size whether the emission comes from all three +galaxies or just a single source. +Irrespective of this, +the strong radio emission would be in excess from that +expected from the far-infrared (FIR)-radio correlation +(Helou et al. 1985). We analyse these issues in detail in +Section 3.2. +There are no other significant (>3σ) ATCA or +ASKAP detections of any known protocluster members +(Fig. 1). +The FIR-radio correlation for star-forming +galaxies (Helou et al. 1985; Ivison et al. 2010) would +imply S2.2 ≈ 12 µJy for a S850 = 5 mJy source at z = 4.3. +The ten brightest SPT2349−56 SMGs (excluding B, C, +2 These three sources are named C3, C6, and C13 in Hill et al. +(2020) based on their rank-ordered [C ii] line strength. + +A radio-loud AGN in SPT2349 +7 +1 +2 +3 +4 +5 +Redshift +1031 +1032 +1033 +1034 +1035 +1036 +L(1.4, rest) (erg s +1 Hz +1) +ALESS14 +HDF850.1 +SPT2349 +8C 1435 +TN J1338 +MRC1138 +1 +2 +3 +4 +5 +Redshift +102 +103 +104 +L1.4 \ L350 +SPT2349 +ALESS14 +ALESS66 +AGN +SMG +Figure 3. +Left: Redshift vs. 1.4 GHz radio power using the GOODS-N sample (Barger et al. 2017) and radio-excess candidates +from the ALESS sample (Thomson et al. 2014). The detection threshold for the GOODS-N radio sample is shown with the blue +curve. Red circles show sources detected above the 3σ level at 850 µm, while black circles show sources not detected at this level. +SPT2349−56 has about 10 times more radio power than any radio source found in GOODS-N, although it has more than 500 +times lower radio power than other well-studied radio-loud galaxies that were used to identify high redshift protoclusters. Right: +1.4 GHz luminosity over 350 GHz luminosity vs. redshift for the submm sources with spectroscopic redshifts in GOODS-N (red +circles), and lower limits on radio sources undetected in the submm (red triangles). Also shown is the ATCA survey of lensed +SPT-SMGs described in Appendix C (purple squares). The blue dashed line region shows where the submillimeter luminosities +and radio luminosities produce consistent estimates of SFRs. None of the GOODS-N SMGs show any excess radio emission +over the FIR-radio relation, while two of the ALESS sources do have a clear excess. Some of the higher redshift SPT-SMGs +show a marginal radio excess, discussed in Appendix C. SPT2349−56 is about 100 times higher than the median relation at +rest 1.4 GHz. The HzRGs lie about 500 times higher with their measured S850 = 6 − 12 mJy. +and G) span 0.8–15 mJy, with an average of 4.7 mJy. +Thus even the brightest SMGs would only be expected +to be at the 1σ level in our ATCA map. A radio stack- +ing analysis on these remaining ten brightest SMGs finds +(11.0±10.0) µJy, which is completely consistent with the +average 2.2 GHz emission expected from the FIR-radio +correlation, ⟨S2.2⟩ = 12 µJy. Stacking on all 40 known +cluster members yields −5.0±5.8 µJy. +We then consider if there might be other radio sources +in SPT2349−56 that could be cluster members. +We +searched for robustly-detected radio sources in the sur- +roundings of SPT2349−56 out to 1 Mpc in projection +(140′′ in radius) from the core, roughly the region stud- +ied with ALMA by Hill et al. (2020). We find two ATCA +sources above 5σ, ID2 and ID3 in Table 1. ID2 is identi- +fied to a bright star, and is also detected by ASKAP. +ID3 has a clear optical counterpart, which does not +have properties (especially non-detections in the g-band) +of optical sources likely to be near z = 4.3 (Rotermund +et al. 2021). +We thus focus on the properties of the central ID1 +radio source, starting with the positional uncertainty, +∆α. From Condon (1997), we can derive the synthe- +sized beam positional uncertainty for the ATCA and +ASKAP detections, assuming that the beam is a single +2D Gaussian with an RMS ‘width’ σ = FWHM/2.354 +in each coordinate. +In the limit where centroiding +uncertainty dominates over systematic astrometry er- +rors and for uncorrelated Gaussian noise, we have +∆α = 0.6 (SNR)−1 FWHM. +For both the ATCA and +ASKAP sources in SPT2349−56, we have confirmed +that the source size and position angle is indistinguish- +able from other brighter, unresolved sources in the field, +in agreement with the synthesized beam. We conclude +that the SPT2349−56 radio source is unresolved with +our current data. +For the ATCA source (ID1) detected at SNR=7.9 and +a beam size of 4′′ × 8′′ (PA = 27 deg east of north), the +positional uncertainty is therefore 0.3′′ × 0.6′′. For the +ASKAP source detected with SNR of 4.6 and a beam +size of 24′′ × 13′′ (PA = 89 deg east of north) the po- +sitional uncertainty is therefore 3.0′′ × 1.7′′. +There is +a 5.1′′ roughly northern offset between the ATCA and +ASKAP sources, which is consistent at the joint 2σ level. + +8 +Chapman et al. +The ASKAP centroid is most consistent with the ALMA +source A. Comparison of our wider field ATCA map +and the ASKAP RACS map reveals that the major- +ity of the sources show excellent astrometric alignment, +but we also identify a few other ATCA sources with +ASKAP counterparts with several arcsecond offsets (see +Appendix A). In two cases, there is a robust association +of the ATCA position to other cataloged objects (from +2MASS), suggesting the offset to the ASKAP position +is likely due to measurement error. For ID1, the more +robust ATCA position and association to the B, C, and +G galaxies in SPT2349−56 is the most likely interpreta- +tion, with the ASKAP source being assumed to be en- +tirely related to the ATCA source for the purposes of de- +riving a radio spectral index. The 5′′ offset is not entirely +unexpected, but may be significant enough to require a +physical interpretation rather than just measurement er- +ror (Appendix A). It could for instance be related to a +radio core-jet morphology. However, as noted, the 30m +minimum baselines of ATCA would not resolve out flux +on scales smaller than several arcmin. +While it’s not +clear why the sources are offset, it appears more likely +to be instrumental than physical based on the analysis +in Appendix A. +3.2. Physical interpretation of ID1 +We first constrain the radio spectral index to esti- +mate and compare luminosities between sources. The +radio source ID1 has a steep spectrum with an index of +α = − 1.58 ± 0.28, constrained by the ASKAP 888 MHz +detection, and the non-detections at 5 and 9 GHz. The +uncertainty can be estimated by propagation of errors +on the two frequencies as follows: +∆α = +� +SNR−2 +2.2 + SNR−2 +888 +ln(2.2/0.89) +. +(1) +In Appendix C, we describe a MCMC method to as- +sess the uncertainty for any number of spectral measure- +ments, and show this distribution in Figure 2. The spec- +trum is too steep to be consistent with synchrotron ra- +diation due to shock acceleration of cosmic ray electrons +from supernovae (i.e., star formation), where Thomson +et al. (2014) recently constrained α = −0.79±0.06 specif- +ically for high-z SMGs. The steep SPT2349−56 spec- +trum seems to demand an AGN interpretation. +The radio luminosity can then be assessed by assum- +ing it is associated with one of the central SPT2349−56 +galaxies at z = 4.3. +With a specific luminosity of +L2.2 = (4.4±0.3) × 1025 W Hz−1, it is far larger than ex- +pected from star formation through the FIR-radio cor- +relation. +For reference, the FIR-radio correlation for +star-forming galaxies (Ivison et al. 2010) would im- +ply L2.2 = 2.4 × 1024 W Hz−1 for a similar S850 = 5 mJy +source at z = 4.3. +Adopting the measured spectral +index above, the radio excess increases to greater +than a factor 100 at a rest-frame of 1.4 GHz, with +L1.4, rest = (2.4±0.3) × 1026 W Hz−1. +This strong radio +excess suggests the presence of an AGN (e.g. Guidetti +et al. 2017); however, the radio emission is still distinctly +less luminous than powerful radio galaxies, like those re- +siding in other structures studied at these redshifts, by +a few orders of magnitude (Fig. 3). MRC 1138, for in- +stance, is almost 1000 times more powerful in radio, and +it is also hosted by the obvious BCG of the protocluster +(e.g. Hatch et al. 2009). +We then compare SPT2349−56 to radio sources from +the literature. In Fig. 3, the redshift versus radio power +is shown using the 0.3 deg2 GOODS-N VLA sample +(Barger et al. 2017), which is highly complete in spec- +troscopic redshift. We compute the rest-frame radio lu- +minosity using the equation +L1.4 = +� +4πd2 +L S1.4/1029� +(1 + z)−(α+1) erg s−1 Hz−1, +(2) +where dL is the luminosity distance (in cm) and +S1.4 is the flux density in units of µJy observed at +1.4 GHz. This equation assumes Sν ∝ να, and we adopt +a radio spectral index of α = −0.8 (Ibar et al. 2010) +for the GOODS-N sources, and the measured α for +SPT2349−56 and the literature HzRG sources (in fact +all very close to −1.6). +Shown for comparison are +several well-studied HzRGs that were used as beacons +to uncover massive galaxy overdensities: +MRC 1138 +(Large et al. 1981; Seymour et al. 2012); TN J1338 (De +Breuck et al. 1999); and 8C 1435 (Lacy et al. 1994). +SPT2349−56 has around 10 times more radio power +than any radio source found in GOODS-N, but it has +less than 500 times the radio power of these HzRGs. +Figure 3 also directly assesses the departure of +SPT2349−56 from the radio-FIR correlation by plot- +ting the luminosity ratio of 1.4 GHz to 350 GHz versus +redshift for all GOODS-N submm sources with spec- +troscopic redshifts (Barger et al. 2014; A. Huber in +prep.). +All of the submm sources in GOODS-N are +radio-detected, even at z = 5.2, and the submm luminos- +ity and radio luminosity produce consistent estimates +of the SFRs for all sources – there is no sign of AGN +from their radio emission. A similar analysis of the sub- +set of gravitationally lensed SPT-SMGs also observed +from ATCA in this program (Appendix C) suggests the +majority (87%) also follow this relation; however, there +are three very significant outliers in this sample which +is most likely attributed to an AGN contribution from +the foreground lensing galaxy (discussed further in Ap- +pendix C). SPT2349−56 is an outlier by a factor of + +A radio-loud AGN in SPT2349 +9 +Figure 4. +Moment maps of B, C, and G. Left: 850-µm continuum (red contours) from high-resolution (0.3′′) combined Cycle +5 and Cycle 6 ALMA data (Hill et al. 2020), shown overlaid over 3-orbit HST F160W imaging (Hill et al. 2022). Mid-left: +[C ii] moment-0 maps from Cycle 6 high-resolution ALMA data. Mid-right: [C ii] moment-1 maps (in velocity units) from +Cycle 6 ALMA data, with the zero velocity centered at the peak of each galaxy’s [C ii] emission. All three sources show a +clear velocity gradient (listed in Table 4, along with dynamical mass comparisons). Right: [C ii] moment-2 maps (in velocity +units) are shown from lower-resolution data to increase the SNR, revealing centrally concentrated dispersions. In all panels, the +synthesized beam FWHM is shown in the bottom-left corner. +about 100 from this envelope (assuming the radio emis- +sion is coming exclusively from ALMA source C). The +HzRGs shown in the left panel of figure 3 have com- +parable S850 = 6 − 12 mJy to other SMGs shown (e.g., +Dannerbauer et al. 2014; De Breuck et al. 1999), and +would remain about 500 times above SPT2349−56 in +the radio/submm ratio plot in the right panel. By con- +trast, the GOODS-N radio sources without submm de- +tection rise significantly above this envelope, into the +AGN regime. +Thomson et al. (2014) have used deep JVLA (1.4 GHz) +and GMRT (610 MHz) to study the 76 ALMA-identified +SMGs in the CDFS field (the ALESS survey – e.g., +Simpson et al. 2014). They find four SMGs whose radio- +FIR values are > 2σ above the sample median, which +they classify as potential AGN. The most robust of these +(ALESS 066.1) is a strong X-ray source with an inverted +radio spectrum (α > 0.51). Of the remaining three, one +(ALESS 014.1) has a flat radio spectrum (α > −0.1) +and an obviously high radio luminosity, while the other +two (ALESS 094.1 and ALESS 118.1) have spectral in- +dex limits consistent with star formation (α ∼ −0.8). +We show these four SMGs in figure 3, where it is clear +that none are comparable to SPT2349−56 in radio lumi- +nosity or departure from the FIR-radio relation. In fact, +two of the four are not at all unusual in their properties +relative to the other samples. +Radio emission provides an extinction-free probe of +AGN (which even X-ray cannot claim, since practical +sensitivity limits preclude the detection of the most ob- +scured, Compton-thick AGN with NH > 1024 cm−2). +Traditionally radio AGN are divided into two sub- +sets (Padovani 2017): +(i) radio-loud AGN L1.4 +> +1024 W Hz−1, which exhibit steep spectrum radio jets +and lobes on kpc scales (Yun et al. 1999); and (ii) radio- +quiet AGN, with flat-spectrum, lower luminosity radio +emission, typically contained within a compact, several +pc, core (Blundell & Kuncic 2007). +SPT2349−56 is +solidly a radio-loud AGN, whereas most of the other +candidate AGN found in the surveys described above +(GOODS-N and ALESS) cannot clearly be defined as +such. +3.3. Resolved properties of the ‘BCG’ sources +Given the radio detection in SPT2349−56, it is of +interest to assess the properties of the B, C, and G +ALMA sources, and to compare them to other proto- +cluster members. As noted, these are three very submm- +luminous sources in the core region (S850 = 6.7 mJy for +B, 4.7 mJy for C, and 1.3 mJy for G), with only source +A being brighter, although two even more luminous +sources are present in the northern extension (sources +N1 and N2; Hill et al. 2020). +The most distinguishing features of this trio (beyond +their flux-ordered source names serendipitously spelling +out ‘BCG’) are their locations near the center-of-mass +of the cluster core, and their immediate environment. +They are very close neighbours (they lie within an arc- +second of each other), and are likely to be interacting. +Further, there is a notable arc seen in [C ii] surrounding +the three galaxies (Hill et al. 2020; N. Sulzanauer, in +prep). Source C does distinguish itself with an anoma- +lously narrow [C ii] and CO(4–3) line width for its lumi- +nosity (Hill et al. 2020). Rotermund et al. (2021) iden- +tified C as a significant outlier from the SPT2349−56 +galaxy sample in its Mdyn/Mgas ratio inferred from the +narrow CO(4–3) line width and large luminosity, similar +to many high-z QSOs (e.g., Narayanan et al. 2008; Wal- +ter et al. 2009; Hill et al. 2019), where selection effects +favoring face-on orientation offer viable explanations. It +is also noteworthy that source C has by far the largest +stellar mass of any cluster member (> 1011 M⊙ Roter- +mund et al. 2021; Hill et al. 2022). It is associated with a + +1.0 +300 +200 +D +200 +1" +1" +0.7 +150 9 +100 +[Jy kms- +[km +[kms +0" +0 +S +0.4 +100 +200 +0.1 +-300 +50 +0" +1" +_1" +0" +1" +-1" +0" +1" +Aa +Aa +Aa10 +Chapman et al. +Table 3. Continuum and line properties of B, C, and G. S147 and S231 are the continuum flux densities at 147 and 231 GHz, +respectively, while the other columns provide various line strengths (the line is indicated by the subscript). The OH doublet +arises from blended hyper-fine triplets centered at 1835 and 1838 GHz, and the H2O line is the para-211–202 line. +ID +RA,Dec +S147 +S231 +FCO(16−15) +FCO(11−10) +FCO(7−6) +FH2O +FOH +F[CI](2−1) +µJy +µJy +Jy km s−1 +Jy km s−1 +Jy km s−1 +Jy km s−1 +Jy km s−1 +Jy km s−1 +B +23:49:42.79, -56:38:24.0 +589±15 +3322±156 +0.12±0.05 +0.26±0.09 +0.73±0.05 +0.22±0.02 +0.93±0.11 +0.46±0.03 +C +23:49:42.84, -56:38:25.1 +336±11 +1810±118 +0.10±0.06 +0.17±0.05 +0.61±0.03 +0.15±0.01 +0.83±0.08 +0.35±0.03 +G +23:49:42.74, -56:38:25.1 +181±23 +136±9 +0.10±0.05 +0.12±0.05 +0.18±0.02 +0.02±0.01 +0.22±0.06 +0.10±0.02 +Table 4. Physical properties of B, C, and G. SFRH2O is the SFR estimated using Eq. 5, S850/FH2O is the ratio of 850 µm +continuum flux density (from Hill et al. 2020) to H2O line strength, and SFRLIR/SFRH2O is the ratio of the FIR-derived SFR +(from Hill et al. 2020) to the H2O-derived SFR. Vp−p is the peak-to-peak velocity from moment-1 maps, while FWHMcen is the +central velocity dispersion (multiplied by 2 +√ +2 ln 2) from moment-2 maps (see Fig. 4), and FWHMint is the width of the [Cii] +line after fitting a single Gaussian to the lines shown in Fig. 9–11. Mdyn, disk is the dynamical mass derived using Vp−p and a +disk model (Eq. 3), while Mdyn, cen and Mdyn, int are dynamical masses derived using the velocity dispersion measurements and +Eq. 4. +ID +SFRH2O +S850/FH2O +SFRLIR/SFRH2O +Vp−p +Mdyn, disk +FWHMcen +Mdyn, cen +FWHMint +Mdyn, int +M⊙ yr−1 +10−3 km−1 s +km s−1 +1010 M⊙ +km s−1 +1010 M⊙ +km s−1 +1010 M⊙ +B +1100±410 +31±3 +0.8+0.5 +−0.4 +600±50 +18.2±2.2 +540±20 +11.0±0.8 +612±10 +14.0±0.5 +C +750±280 +31±2 +0.8+0.5 +−0.4 +240±50 +2.9±0.8 +280±20 +2.9±0.4 +358±5 +4.7±0.2 +G +80±60 +65±23 +2.3+2.0 +−1.8 +690±50 +18.1±2.5 +520±20 +7.6±0.8 +901±54 +22.8±2.8 +bright and very compact HST F160W source (Hill et al. +2022), as shown in Figs. 1 and 4, and it has been sug- +gested to be the seed of a growing BCG galaxy in this +ongoing mega-merger (Rennehan et al. 2020). +3.3.1. [C ii] kinematics +We consider here a more detailed analysis of the kine- +matic properties of the B, C, and G galaxies. Using high- +resolution Cycle 6 [C ii] data (Hill et al. 2020), which has +a synthesized beam of about 0.2′′, we construct moment +0, 1, and 2 maps of the B, C, and G sources and ana- +lyze the resolved velocity and dispersion fields. We use +the CASA task immoments, focusing on channels between +±3σ of the best-fit [C ii] line, and masking pixels < 4 +times the RMS per channel. Since second moments are +particularly sensitive to noise (being a squared term), we +use uv-combined Cycle 5 and 6 data cubes (described in +Hill et al. 2020) to calculate the moment 2 maps; for +reference, the resolution of the combined data is about +0.3′′. The results (moments 0, 1, and 2) are shown in +Fig. 4. +All three sources show a clear velocity gradient and re- +solved, centrally-concentrated dispersion, characteristic +of rotationally-supported disks. From these velocity gra- +dients and velocity dispersion maps we extract peak-to- +peak velocities, Vp−p, and central velocity dispersions, +FWHMcen. We draw a line along the semi-major axis +of each galaxy, then from the moment 1 map calculate +the velocity difference between the two ends, and from +the moment 2 map extract the velocity dispersion at the +midpoint of the line. We find that moving the position +angle of the line by ±10 deg and moving the midpoint +of the line by 5 pixels results in a peak-to-peak velocity +change of ±50 km s−1 and a central velocity dispersion +change of ±10 km s−1 (±20 km s−1 in FWHM), so we +quote these as our uncertainties. The results are given +in Table 4, multiplied by a factor of 2 +√ +2 ln 2 to estimate +a FWHM. +We use these peak-to-peak velocities and central dis- +persions to estimate masses assuming a disk model, with +the enclosed dynamical mass given by +Mdyn, disk[M⊙] = 2.35 × 105 [Vp−p/⟨sin(i)⟩]2 R, +(3) +where Vp−p is the peak-to-peak velocity in km s−1, +R is the radius in kpc, and i is the inclination an- +gle of the galaxy. +We adopt a mean inclination suit- +able for a collection of randomly oriented disks of +⟨sin(i)⟩ = π / 4 ≃ 0.79 (see Law et al. 2009), and we use +the half-light radii from Hill et al. (2022), estimated by +fitting S´ersic profiles to the high-resolution ALMA [C ii] +moment 0 images. The results are given in Table 4. + +A radio-loud AGN in SPT2349 +11 +The dynamical masses were derived previously (Roter- +mund et al. 2021) from the unresolved velocity disper- +sions, using the width of the integrated [C ii] lines shown +in Figs. 9–10, with an assumption about the structure +of the source based on the virial theorem, using the re- +lation +Mdyn[M⊙] = 2.81 × 105 FWHM2R, +(4) +where FWHM is a one-dimensional velocity dispersion +(multiplied by a factor of 2 +√ +2 ln 2) in km s−1, and R is +the radius of the virialised structure. First, we use the +resolved central velocity dispersion, FWHMcen, adopt- +ing the [C ii] size measurements from Hill et al. (2022) +and the central resolved velocity dispersions from the +moment 2 maps (Table 4). Next we use the width of +the integrated [C ii] line, FWHMint, obtained by fitting +a single Gaussian model to the [C ii] spectra shown in +Figs. 9–11 and given in Table 4, again using Eq. 4 and +the same [C ii] size measurements. +The resulting dy- +namical masses are provided in Table 4. +Considered in the context of a disk model, source C +does show a similar dynamical mass comparing both its +central and integrated velocity dispersion (Table 4, and +Rotermund et al. 2021); however, it still appears to have +substantially lower mass (six times lower) than B from +any kinematics analysis. Inclination is reasonably con- +strained, since the aspect ratio of these galaxies is re- +solved by ALMA. While it remains an uncertainty in +any mass modelling, the aspect ratios of B and C are +similar at ∼1.8 (major to minor axis). +Sources B and G have similarly large inferred disk +masses (18 × 1011 M⊙). However, the distinct double- +horned profile of source G (Appendix B) is direct evi- +dence for a rotating disk or bar-like structure at high +inclination (explaining the broad velocity profile), while +the profile for B is possibly due to a tidal torque in re- +sponse to the interaction with C. Source G also has a +higher aspect ratio (2.3, major/minor axes) in moment-1 +than B and C, suggesting the disk is seen closer to edge- +on. Explicitly using this higher implied inclination in +Eq. 3 brings down the disk mass estimate by 25%, more +consistent with the much lower gas mass of G compared +with B and C. +It is noteworthy that in projection at least, B is +counter-rotating relative to C. Several studies have pre- +dicted that mergers configured with counter-rotating gas +disks should lead to the most intense starbursts, and +conditions for fueling the SMBHs (e.g., Mihos & Hern- +quist 1994, 1996; Di Matteo et al. 2007; Salom´e et al. +2012). +3.3.2. Submm line properties +We then consider line diagnostics to elucidate which +of the three might be most likely to host the radio-AGN. +We first assess the [C ii]/FIR ratio, which has been +shown to highlight AGN with a deficit compared with +star-forming galaxies (e.g. Stacey et al. 2010). However +at high luminosities, both AGN and SMGs (without ob- +vious AGN) exhibit similar deficits in the ratio. +Hill +et al. (2020) have shown that all three of B, C, and G +are ‘deficit sources’ in [C ii]/FIR, inhabiting similar re- +gions in the [C ii]/FIR-to-FIR plot as many luminous +AGN. However, this work also showed that all 12 of the +most luminous SMGs in SPT2349−56 have comparable +[C ii]/FIR ratios, and none of these are obviously AGN +from any available diagnostics. +One possibility to consider is that the FIR estimates +are being affected by an AGN in one of B, C, or G. +Since the shortest wavelength measured by ALMA is +160µm in the rest frame, the peak of the SED is not +sampled, and there is little constraint on whether the +dust might be substantially hotter than the Td ≈40 K +estimated in Hill et al. (2020). To test this we make use +of the para-H2O(211–202) lines observed in the ALMA +Band 4 dataset (Table 3 and Figs. 9–11). +H2O is +strongly coupled to the FIR radiation field whether it +is being produced by star-formation or AGN (Omont +et al. 2013). Jarugula et al. (2021) compiled a sample +of low- and high-z submm galaxies with para-H2O(211– +202) measurements (including two sources, SPT0346−52 +and SPT0311−58, from the same parent sample as +SPT2349−56), and found that a simple single-parameter +scaling relation described the correlation between LH20 +and SFR (derived from FIR) of the form +SFR [M⊙ yr−1] = (2.07 ± 0.75) × 10−5LH2O [L⊙]. +(5) +. +A simple test is to first take the ratio of 850µm contin- +uum flux density to H2O line strength, where measure- +ment errors are mostly small. These values are given in +Table 4, where we have used S850 values from Hill et al. +(2020). Using the same modified blackbody SED as in +Hill et al. (2020) to model the continuum flux density +emission, a dust temperature of 40 K at a redshift of 4.3 +means that S850 = 1 mJy corresponds to 115 M⊙ yr−1, +and so Eq. 5 implies S850/FH2O = (42 × 10−3) km−1 s; B +and C sit significantly below this value, implying they +might have higher FIR than currently estimated. In con- +trast, G is significantly above the relation, which could +imply cooler dust and lower FIR than previously esti- +mated. This would make G less of a deficit source in +[C ii]/FIR, and less likely to be considered an AGN by +this criterion. + +12 +Chapman et al. +5 +10 +15 +12CO Jup +106 +107 +108 +Lline (L +) +Mrk231 +M82 +o|vo +o +o +|v +|v|v|v +| +| +G +C +B +5 +10 +15 +12CO Jup +10 +1 +100 +Normalized Lline (L +) +Mrk231 +M82 +o|v +o +o +o +|v +|v +|v +|v +| +| +G +C +B +Figure 5. +SLED for B, C, and G in the CO Jupper = 2, 4, 7, 11, and 16 transitions (new data for the Jupper = 7, 11, 16 lines are +shown in Appendix B). The ATCA detection of CO(2–1) (Miller et al. 2018) is shown as an upper limit as it is an unresolved +measurement over the core region including at least B, C, and G. Open circles illustrate the J = 2 division if the luminosities +scale from J = 4. We compare to the LFIR = 3 × 1012 L⊙ AGN-dominated galaxy Mrk231 (van der Werf et al. 2010), and the +LFIR = 3 × 1010 L⊙ starburst M82 (Kamenetzky et al. 2012), here normalized to Mrk231 at CO(7–6). In the right panel the +SPT2349−56 galaxies are also normalized to the CO(7–6) luminosity of Mrk231 for comparison of the excitation curves. Source +G is undetected in the Jupper = 11 and 16 lines, while B and C (shown as 2σ upper limits) are marginally detected in CO(16–15). +Using Eq. 5, we can also estimate SFRs directly for +B, C, and G using our measured para-H2O(211–202) line +strengths (Table 4), and compare these with the SFRs +from the FIR (taken from Hill et al. 2020). These re- +lations show the same behaviour as our ratio of mea- +surements above, that B and C both may have higher +FIR (from hotter dust) than estimated from our current +ALMA data. We note that systematic errors in convert- +ing to these physical quantities are large (as listed in +4). +We next consider the CO spectral line energy distribu- +tion (SLED), which can distinguish AGN with high ex- +citation lines driven by X-ray dominated regions (XDRs, +e.g. van der Werf et al. 2010). Here we present higher- +J CO transitions (Section 2) than have previously been +published in Miller et al. (2018). All of B, C, and G are +well-detected in CO(7–6) observations, while B and C +are detected in CO(11–10). Remarkably B and C may +be marginally detected in CO(16–15) due to the sensi- +tivity of the deep Band-7 ALMA data presented in Hill +et al. (2020), although strictly they are upper limits. All +the new line channel maps and one-dimensional spectra +are shown in Appendix B. SLEDs are shown in Fig. 5 +from the available J =2, 4, 7, 11, and 16 transitions, and +compared to AGN and starburst templates. The ATCA +detection of CO(2-–1) (Miller et al. 2018) is shown as +an upper limit as it is an unresolved measurement over +the core region including at least B, C, and G. However +for illustration, we show the division into B, C, and G +of the integrated J = 2 luminosity assuming they scale +with the J = 4 fluxes. None of B, C, or G appear to have +high excitation SLEDs similar to AGN like Mrk231 (e.g. +van der Werf et al. 2010), and are all similar to or less +excited than M82 at high-J (Kamenetzky et al. 2012), +with the caveat that G has only upper limits beyond +J = 7. +Source B has the highest excitation SLED confirmed +of the three, lying near the M82 SLED. Source B also has +a stronger cool/warm gas component similar to Mrk231. +However, all three are reasonably characterized with a +combination of cool and warm star-forming Photo Dis- +sociation Regions (PDR) components, and without sig- +nificant XDR contributions. Detailed SLED modelling +of SPT2349−56 sources will appear in a future contri- +bution. +Finally, the 163µm OH doublet in B, C, and G can be +compared to Runco et al. (2020), who studied 178 local +galaxies in six of the 14 OH transitions in the FIR range. +They found the highest frequency OH163µm (detected in +25 galaxies) is the only OH doublet which is always in + +A radio-loud AGN in SPT2349 +13 +emission, with most transitions often appearing in ab- +sorption. Runco et al. (2020) presented the correlations +of the equivalent width, EW(OH), with various galaxy +properties and line ratios, finding EW(163µm) is not +well established as a direct AGN indicator. For example, +while galaxies with lower X-ray luminosities exclusively +have low EW(OH), the full range of EW is seen for the +highest X-ray luminosities (Runco et al. 2020). How- +ever, a strong correlation is found for EW(OH) with the +ratio of AGN activity to SFR, suggesting this is a bet- +ter predictor of EW(OH) than the total AGN power. In +figure 6, we compare the equivalent width, EW(163µm), +to local starbursts, LINERs and Seyfert galaxies from +Runco et al. (2020). +The EW(163µm)=0.076µm and +0.095µm measured for B and C respectively are amongst +the highest found locally. (G is similarly high, but is +only marginally detected in OH). Their EW(163µm) +are more similar to values in local Seyfert galaxies than +starburst galaxies, the latter having EW(163µm)∼0.02– +0.05µm. It is not yet clear at z > 4 what is “normal” +for EW(163µm), since the line has only before been de- +tected locally. However, these results may provide ini- +tial evidence that B and C do in fact present as AGN +through some submm-wave diagnostics. +3.3.3. Optical and near-infrared properties +Finally, we summarize optical and near-infrared spec- +tra taken with the Gemini and VLT observatories. A +VLT XSHOOTER spectrum (λobs = 0.35–2.4 µm) was +obtained which targeted B and C (Rotermund et al. +2021), covering redshifted Lyα through [OII]3727. No +lines were detected. A VLT-MUSE spectral cube was +used to extract one-dimensional spectra at the locations +of each of B, C, and G (Y. Apostolovski in prep.), but +no lines are detected in any of these galaxies (nor any of +the SPT2349−56 SMGs). +Non-detections are not particularly surprising given +the faintness of the galaxies at 6450 ˚A, the wavelength +of redshifted Lyα, where C has S0.63 µm = 0.061 µJy, +while B and G are undetected to S0.63 µm < 0.01 µJy +(Rotermund et al. 2021; Hill et al. 2022). +The diffi- +culty of spectroscopy in the near-infrared at 19,768 ˚A +in the vicinity of the redshifted [OII]3727 means these +limits on line equivalent widths are also not particularly +constraining. Nonetheless, strong AGN often exhibit de- +tectable high excitation lines in optically faint, obscured +SMG hosts (Chapman et al. 2003; Chapman et al. 2004; +Chapman et al. 2005; Danielson et al. 2017), and it is +surprising that this AGN in SPT2349−56 eludes all op- +tical and near-infrared spectral detection. +3.3.4. Concluding remarks +1010 +1011 +1012 +LIR (L +) +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +0.16 +0.18 +EW(OH163) ( m) +B +C +G +Figure 6. +The equivalent width of the OH163µm doublet +versus IR luminosity. B, C, and G (blue circles) have similar +values, although the error in G is large. +We compare to +all local galaxies detected in OH163µm from the compilation +in Runco et al. (2020). Galaxies are colour coded by their +classification as starbursts (lime), LINERs (green), Seyfert-1 +(red), and Seyfert-2 and intermediate types (orange). B, C, +and G all appear well above local starbursts, although all +three AGN types have a few examples this high. +Thus while the radio observations do not have suffi- +cient spatial resolution to uniquely identify one of the +three galaxies as the AGN, the source properties them- +selves suggest source C could be a likely host, consid- +ering mainly its large stellar mass, along with narrow +emission lines, and high EW(OH). Sources with sim- +ilar radio luminosities in the local Universe are typi- +cally found in massive hosts. However, the more FIR- +luminous and much more dust-obscured source B might +also be a possible host for the AGN, given the large dy- +namical mass from kinematic modelling and the higher +excitation SLED. Of course all three SMGs could have +AGN components at the same time. The fact that they +are likely strongly interacting dispels the typical duty- +cycle arguments that would disfavor this scenario. To +make progress, we will need deeper radio data with +better resolution, and sensitive infrared spectroscopic +observations now possible with the James Webb Space +Telescope. +4. DISCUSSION +4.1. Inferring AGN properties from radio power +4.1.1. Jet power and energy input to the ICM + +14 +Chapman et al. +Radio jets are thought to provide an important feed- +back mode in galaxy clusters by preventing the cooling +of hot (X-ray) gas surrounding central galaxies (e.g., Mc- +Namara & Nulsen (2012)). This is named “jet-mode” +feedback and is associated to radio sources characterised +by radiatively-inefficient accretion. However, radio jets +can also drive massive gas outflows on galactic scales, +another signature of AGN feedback. +A theoretical relation between radio luminosity and +radio jet power was determined by Willott et al. (1999), +and can be used to estimate the kinetic energy output of +AGN (e.g., Hardcastle et al. 2007). The jet power can be +estimated by assuming that the mechanical power of the +jet can be approximated as the energy of the detected +radio cavity averaged over some timescale (e.g., Bˆırzan +et al. 2004). The X-ray-detectable “cavities” that result +from AGN jet activity (O’Sullivan et al. 2011) allow us +to quantify the heating experienced by the intra-cluster +medium (ICM). The energy contained in these cavities +comes from the product of the pressure and volume (pV ) +over the cavity. This is the work done by the jet to cre- +ate the cavity, and the internal energy of the radio lobes. +Under the assumption that the cavity is dominated by +relativistic plasma, this becomes 4pV . Dividing the en- +ergy of the cavity by the cavity age gives the power, +Pcav. +Thus the most direct inference we can make from +the radio properties of SPT2349−56 adopts a relatively +tight correlation observed between radio power and cav- +ity power (Cavagnolo et al. 2010; O’Sullivan et al. 2011; +Panessa et al. 2015), where a fitted relation follows: +log Pcav = (0.35 ± 0.07) log L1.4 + (1.85 ± 0.10), +(6) +yielding Pcav = (3.3 ± 0.7) × 1038 W. This is strictly a +lower limit to the jet power, and therefore energy injec- +tion into the ICM. The true jet power depends on how +the radio cavity is inflated (as described in Nusser et al. +2006), with some energy from the jet being carried away +by shocks. The relation of L1.4 to Pcav is still affected +by uncertainties due to the assumption that the cavity is +dominated by relativistic plasma and the detectability +of cavities within the sample used in O’Sullivan et al. +(2011), as discussed in their work. +This jet power is a sizeable amount of energy, given +the potential well of the ∼1013 M⊙ SPT2349−56 halo +(see below) constrained from the central velocity disper- +sion and radial distribution of cluster members (Miller +et al. 2018; Hill et al. 2020). This is also a significant +addition to the already abundant energy injection from +the 6600 M⊙ yr−1 of star formation being experienced +by the core of SPT2349−56 from summing the SFRs of +all member galaxies found in these works (Miller et al. +2018; Hill et al. 2020; Rotermund et al. 2021). We take +the instantaneous injection of energy at z = 4.3 as +˙Ekin = 1 +2 +˙Moutv2, +(7) +where +˙Mout is the total amount of gas ejected per unit +time by galaxies and v is the outflow velocity. While v +is not measured in SPT2349−56 galaxies, the average +outflow in high−z SMGs and other starforming galax- +ies has been constrained with increasingly large samples +(e.g., Banerji et al. 2011; F¨orster Schreiber et al. 2014). +We adopt a typical 500 km s−1 wind speed for the SN- +driven outflows in each SPT2349−56 galaxy. +A mass +outflow rate can then be found by converting SFRs into +mass outflow rates +˙Mout by multiplying by a conser- +vative mass loading factor η = ˙Mout/SFR = 1. η could +even be greater than one based on observational (i.e., +Newman et al. 2012) and theoretical work (i.e., Hop- +kins et al. 2012). However, the same amount of metals +is found in stars and the ICM, which suggests equal- +ity, +˙Mout ≈ SFR, (e.g., Renzini & Andreon 2014). We +therefore obtain ˙Ekin = (3.3 ± 1.1) × 1038 W, where the +uncertainty reflects both the range in SFR estimates and +the range of likely wind velocities. This energy injection +is remarkably similar to that found from the radio-loud +AGN above from equation 6. +Estimating the total mechanical energy injected by +the radio jets requires an estimate of the radio source +lifetime. +Brienza et al. (2017) and Hardcastle et al. +(2019) have suggested that remnant sources fade rapidly, +with most of the observed remnant radio galaxies being +relatively young, with ages between 50 to 100 Myr. +With +τ = 100 Myr +for +SPT2349−56, +we +find +Emech = Pcav × τ = (1.0 ± 0.2) × 1054 J, assuming only +the uncertainty in the Pcav scaling relation. +4.1.2. Binding energy of the halo gas +We then turn to estimating the binding energy of the +gas in the SPT2349−56 halo. +Giodini et al. (2010) +demonstrated that the mechanical energy from jets is +comparable to the binding energy (Ebinding) in galaxy +groups, while it is lower by a factor of 102–103 in clus- +ters. Since the SPT2349−56 halo mass is comparable +to a large group today, and the entire protocluster is +expected to form a massive cluster by z = 0, it is thus +of interest to investigate how Ebinding compares to our +estimate of Emech. +We define the binding energy as the total potential +energy needed to push the ICM gas within R500 (the +radius where the mean dark matter halo density drops +to 500 times the critical density) beyond R200 (the radius +where the mean dark matter halo density drops to 200 + +A radio-loud AGN in SPT2349 +15 +times the critical density, which we assume to be equal to +the virial radius). Hill et al. (2022) estimated M200 (the +mass contained within R200) to be (9 ± 5) × 1012 M⊙, +corresponding to R200 = (120 ± 70) kpc at z = 4.3, and +R500 can be computed if one assumes a density profile +for the dark matter. +Following Giodini et al. (2010), the binding energy is +computed as +Ebinding = +� Mgas,500 +0 +[φ(r) − φ(R200)] dMgas += 4 π +� R500 +0 +φ(r) ρgas(r) r2 dr, +(8) +where the constant term φ(R200) is small compared to +the potential within R500 and can be ignored, and ρgas +is the gas mass density. +Assuming the gas mass density follows the dark matter +density but scaled by a single gas-mass fraction param- +eter, fgas, we can adopt an NFW dark matter profile to +write the binding energy as (see Giodini et al. 2010 for +details) +Ebinding = fgas4 π ρcrit δc A r3 +s +� c500 +0 +ln(1 + x) +(1 + x)2 dx, +(9) +where x = r/rs, with rs being the characteristic ra- +dius related to the halo concentration parameter by +c = R200/rs, δc is a numerical factor that depends only +on the halo concentration parameter c, A scales with +M200 and also depends on c, and ρcrit is the criti- +cal density at the redshift of interest (here 4.3). +We +compute the concentration parameter using the mass- +dependent relation of Macci`o et al. (2007); they find a +linear trend between log c and log Mvir (which we assume +is equal to M200), and we find c = 7.9, corresponding +to rs = 15 kpc. We note that c = 5 is typically adopted +for massive clusters > 1014 M⊙. +With the concentra- +tion parameter known, we calculate R500 = 80 kpc and +c500 = R500/rs = 5.3. +We cannot estimate the halo gas mass directly in +SPT2349−56, beyond summing the measured cold gas +masses in individual galaxies from the core region and +inferring additional cool and warm gas components in +the halo. +Summing the H2 gas masses from the 23 +SMGs within the cluster core from Hill et al. (2020) +yields Mgas,cool = 3 × 1011 M⊙. The unseen gas compo- +nents in the halo are more uncertain. A trend observed +in groups and clusters is an increase of the fraction of +hot gas with total system mass (Connor et al. 2014), +approximately following fgas ∝ M 0.1−0.2, where 1013 M⊙ +groups typically have fgas of around 10%. +The LX-M relation has been shown to remain approx- +imately self-similar out to z = 2 (Mantz et al. 2018), in- +cluding X-ray-detected clusters at z = 2 (Gobat et al. +2011). However, for low-mass systems the gas mass frac- +tions may evolve with redshift (Connor et al. 2014). Re- +gardless, this in itself does not constrain the ICM gas +fraction, which requires more detailed X-ray properties +than LX to be detected. +Based on the above, we will assume for SPT2349−56 +a gas mass of 10% of the halo mass, or 9 × 1011 M⊙, +which nominally requires that Mgas,hot = 6 × 1011 M⊙, +unless there are substantial cold flows feeding the +submm galaxies (Dekel et al. 2009). +We estimate +Ebinding = (1.5+0.7 +−1.4) × 1054 J, where the uncertainty has +been propagated from M200 and the uncertainty in the +M200-c scaling relation using a Markov chain Monte +Carlo (MCMC) approach. +The radio feedback alone therefore conceivably pro- +vides all of the energy required to unbind the total gas +in the cluster core. The stellar feedback has a compara- +ble energy input, and could also be unbinding the cluster +gas. However, the total energy is a minimum condition; +the energy must also couple efficiently to the ICM. An +energetic jet may not couple to the bulk of the ICM gas +(Babul et al. 2013; Yang & Reynolds 2016; Cielo et al. +2018). +Any hot ICM established at z > 4 may not be in hy- +drostatic equilibrium since cold inflows likely dominate +the flow of gas in protocluster halos (Dekel et al. 2009). +The infalling gas only increases the energy required to +inflate a bubble in the nascent ICM, acting as an addi- +tive term to Ebinding. While L1.4 is fixed, the work done +on an inflowing medium will be higher than for an ambi- +ent static medium. Therefore the Pcav (∝ 4pV ) to L1.4 +relationship might not hold when inflows dominate the +halo. Yajima et al. (2022) and Trebitsch et al. (2021) +have begun to explore some of these issues in hydrody- +namical simulations of protoclusters, aiming to better +understand AGN feedback and the impact of massive +starburst galaxies in forming clusters. We leave more +detailed calculations to future work (D. Rennehan, in +prep.). +4.1.3. Inferred X-ray luminosity and accretion rate +A correlation also exists between radio power and X- +ray luminosity (LX) for radio-loud AGN (Ballo et al. +2012), although there is substantial scatter in this rela- +tion. While the correlation appears to be similar over +a large range (nine orders of magnitude) in X-ray lumi- +nosity, there is a range of over 100 in LX for a given +radio luminosity in well populated areas of the correla- +tion. The relation plotted in Ballo et al. (2012) is char- +acterized at 5 GHz rest frame, which we measure almost +directly (through the ASKAP detection). Using the cor- + +16 +Chapman et al. +relation, we find that L5 = 7 × 1025 W Hz−1 in the radio +corresponds to LX = 1038 W (where the X-ray luminos- +ity is between 2 and 10 keV). We conclude that the X-ray +emission from the central AGN in SPT2349−56 can be +easily detected by XMM-Newton or Chandra under the +full range of possible LX = 1037−39 W suggested by this +correlation. +Finally, taking source C as the most likely host, we +can infer the SMBH mass from the stellar mass that has +been well characterized for C (Rotermund et al. 2021; +Hill et al. 2022). For M∗ = 4 × 1011 M⊙, the SMBH mass +is 7 × 108 M⊙ (e.g. Ding et al. 2020). From this, we can +infer the range of Eddington luminosities with respect to +the range in X-ray luminosity constrained by the radio +power. In other words, how close to the maximal rate of +accretion is the SPT2349−56 AGN if its SMBH is close +to that implied by the stellar mass of source C. In partic- +ular, following Ballo et al. (2012), our measurements of +L5 GHz/MBH constrain LX/LEdd to the range of roughly +0.005 to 0.05, based on their distribution shown (their +figure 10). Directly measuring the X-ray properties of +SPT2349−56 will allow substantial progress in charac- +terizing the system and its environment. +4.2. Implications of the steep spectrum +For an optically-thin synchrotron source, the spec- +trum will steepen in spectral index from low to high +frequencies by ∆α = −0.5 if the source lifetime is greater +than the timescale for energy-loss from the radiating +electrons. This leads to a concave spectral shape with +a characteristic bend frequency, νb (Kellermann et al. +1969). Thus the age of the electron population within +radio jets contributes to the steepness of the spectrum. +Three effects will then decrease νb as the source red- +shift increases (Krolik & Chen 1991): (1) for a fixed +bend frequency ν∗ in the rest frame, the observed bend +νb = ν∗ / (1+z); (2) losses due to inverse Compton scat- +tering off the microwave background rise with redshift +as (1 + z)4, so that for a fixed time electrons spend in +the radiating region, the lowest energy electron that can +cool has a frequency (or energy), which decreases with +increasing redshift; and (3) flux-limited samples result +in a selection effect that favors low ν∗ at high-z. Sources +must have higher emissivity at higher redshift to be in- +cluded in the sample. +They also must have stronger +implied magnetic fields, and therefore more rapid syn- +chrotron losses. +A combination of these effects has been used to explain +the observed trend that higher-redshift radio galaxies +have steeper spectral indices (Carilli & Walter 2013; van +Breukelen et al. 2009). The ultra-steep spectral indices +of HzRGs (up to the α = 1.6 we find in SPT2349−56) is +a main selection criterion for identifying these powerful +radio sources in the distant Universe (De Breuck et al. +2000; Broderick et al. 2007). All three HzRGs shown +in Fig. 2 in fact have α very close to 1.6. It is of note +that SPT2349−56 would have been discovered by these +HzRG surveys over one to two decades ago had the radio +source been 10–100 times more radio luminous, and even +cursory submm followup would then have revealed the +extended S850 = 110 mJy source that belies its nature +as a submm-luminous protocluster. +A steep spectrum generally argues for self-absorbed +synchrotron, and a lack of electron injection (e.g., Rad- +cliffe et al. 2021). +Thus the steep α in SPT2349−56 +could represent a dying radio source. In this case, the +ATCA flux should be extended over the same area as +the ASKAP data. +Thus if there is no physical offset +between ATCA and ASKAP, and this is just a mea- +surement uncertainty, SPT2349−56 could be a young +and completely unresolved compact radio source. +On +the other hand, this might be a “contained” or “frus- +trated” radio source inside a dense medium, sometimes +referred to as a compact steep spectrum source, or CSS +(Padovani 2017), but an issue with this interpretation +is that the luminosity of the source is low relative to +these typical GHz-peaked sources. If self-absorbed syn- +chrotron is contributing to the steep spectrum, the ob- +servational constraints would mean that the break fre- +quency is well below about 5 GHz in the rest frame. In +principle this break frequency can provide a constraint +on the age of the radio source, but since we do not con- +strain this break with the current data, we do not pursue +this further here. However if the radio emission is due +to a CSS then it would have to be older than 500 Myr +to have a break frequency below 0.9 GHz (4.8 GHz rest) +(Padovani 2017). +4.3. Connection to the LAB +The powering sources of Ly-α blobs (LABs) have of- +ten been identified broadly with the photoionizing emis- +sion from a close ionizing source (e.g., a QSO, Geach +et al. 2009; Overzier et al. 2013), shocks (e.g., Taniguchi +& Shioya 2000), or “cooling radiation” during gravita- +tional collapse of the gas (e.g., Haiman et al. 2000). The +SPT2349−56 LAB (shown in figure 7) was originally +hypothesized to be heated by some combination of the +three ALMA sources that reside near or within it (Y. +Apostolovski et al. in prep.). However, given that the +LAB center is only 4.5′′ (31 kpc in projection) offset from +ALMA source C, it could instead be heated by the radio- +loud AGN. The LAB is centered on the weak SMG, N, +which was originally identified through its [C ii] emis- +sion (Miller et al. 2018). N is a luminous infrared galaxy + +A radio-loud AGN in SPT2349 +17 +23:49:43 +42 +-56:38:15 +20 +25 +30 +35 +RA +Dec +M +N +I +B +C +G +Figure 7. Does the radio-loud AGN power the LAB? The +background shows HST F160W imaging with ATCA 2.2 GHz +contours (cyan) and the Y. Apostolovski et al. (in prep.) +MUSE Lyα contours (lime). +The center of the LAB lies +4.5′′ (31 kpc) from the radio source centroid. ALMA 850µm +contours are shown (coral), but sources M and N are too +weak to see in this representation. +The LAB is centered +near the SMG, N, which was originally identified through +its [C ii] emission and undetected in continuum (Miller et al. +2018). N has S1.1mm = 0.18 mJy and an LIR = 4 × 1011 L⊙, +with an implied SFR=35 M⊙ yr−1. +(LIRG) with S850 = 0.27±0.04 mJy, LFIR = 4 × 1011 L⊙, +and a substantial M ∗ = 3 × 1010 M⊙. +It is a plausi- +ble, but somewhat unlikely power source for the lumi- +nous LAB (whose total luminosity is 3 × 1042 erg s−1, or +3 × 1035 W); source N fails to provide the necessary UV +ionizing photons by at least a factor of ten, scaling from +its meager R-band flux density of 0.37 µJy (similar to +the analysis in Y. Apostolovski et al. in prep.). We can +directly estimate the AGN X-ray emission expected for +powering the Ly-α blob following Overzier et al. (2013), +assuming that the fraction of ionizing photons that will +cascade to Lyα is 68% (case B recombination). +This +number likely exceeds the actual amount of ionizing ra- +diation available due to the absorption by dust by a fac- +tor of around 10, which we account for here. We then +assume a radio-quiet QSO spectrum given by Richards +et al. (2006). The predicted observed frame (0.2–12 keV) +X-ray luminosity would be 2 × 1037 W, which is compa- +rable to the low end of the expected range of LX from +the SPT2349−56 radio source, as discussed above. The +radio AGN may therefore be at least as plausible a heat- +ing source as N. +Regarding the Lyα blob being spatially offset from +the AGN position, we note that in the radio source +B3 J2330 at z = 3.1 (Matsuda et al. 2009), the peak of +the Lyα emission was also found to be similarly offset +from the HzRG itself. Even in the z = 4 Distant Red +Core (DRC) LAB, there is a roughly 3′′ (21 kpc) offset +from the X-ray-emitting AGN that is proposed as the +LAB’s power source (Vito et al. 2020). However, these +are rare cases. Venemans et al. (2007) showed that gen- +erally the AGN is very near the center of the Lyα halo, +which grants some geometrical credence to the idea that +the Lyα halo is ionized by the central AGN’s photons. +In SPT2349−56, this is harder to argue, but the Lyα +could be completely absorbed by the copious amounts +of dust in the core. The SPT pre-selection (as with the +Herschel selection of the DRC) may favor finding sources +with such offsets. +4.4. AGN fractions in protoclusters +As described in Section 3.1, +with 27 µJy RMS +at 2.2 GHz, we are sensitive to moderately-luminous +and heavily-obscured z = 4.3 AGN among the 30 sub- +millimeter galaxies identified in the SPT2349−56 struc- +ture. +They need to lie approximately 5 times above +the radio-FIR relation to be significantly (5σ) detected +by ATCA. In GOODS-N (Fig. 3), there are seven radio +sources (all lacking submm detection) that satisfy this +threshold, all of which lie at at redshifts less than 2. +Another seven such radio sources lie 2–3.5 times above +the relation, extending to a redshift of about 4, which +would not be detected by our observations. +The fact +that all submm-detected sources in GOODS-N, and 74 +of 76 SMGs in ALESS, are consistent with the radio- +FIR relation does signify that radio-loud AGN are not +common amongst the submm-luminous population. No +significant radio emission is found from any other (non- +SMG) cluster members or candidates. +With our cur- +rent radio depth, the radio-AGN content among SMGs +in this protocluster is constrained to be less than 10% +(three of 30 members), and most likely 3% (assuming +C is the host of the ATCA radio source). However, the +radio-loud AGN are only about 10% of the total AGN +population in the field (Barger et al. 2007; Radcliffe et al. +2021). The X-ray AGN fraction remains unconstrained, +and given that many X-ray AGN are not radio emit- +ters (Barger et al. 2007), our AGN fraction estimates in +SPT2349−56 are lower limits. +In the z = 4.0 DRC protocluster (Oteo et al. 2018), +a central galaxy is radio-undetected, but is a Compton- +thick X-ray AGN. Only one of the three X-ray-identified +AGN is detected in the radio – DRC6 (S5.5 = 128 µJy, +S9 = 120 µJy), indicating a flat-spectrum source. In this + +18 +Chapman et al. +case, the radio-AGN in the DRC lies towards the edge of +the projected distribution of SMGs (offset from the core +of the cluster). +Thus without X-ray data, we cannot +tell if the total AGN fraction of SPT2349−56 is differ- +ent from that in the DRC (23%, Vito et al. 2020). As +another example, in the core of the z = 3.09 SSA22 pro- +tocluster, the SMGs have a 50% X-ray AGN fraction, +with four of eight SMGs detected by Chandra (Umehata +et al. 2019), significantly larger than the DRC. +4.5. Radio sources and cluster evolution +Given that SPT2349−56 is conceivably the most mas- +sive and active halo we know of at z > 4, an open ques- +tion concerns the feedback or radio mode that this AGN +is operating in, and how it is shaping the early core evo- +lution of the cluster. +With the current data, having +only the two photometric points characterizing the ra- +dio emission, and not even localizing it uniquely to one +galaxy, we cannot definitively address these issues. Most +radio-loud AGN appear to be hosted in recent or ongo- +ing mergers (e.g., Ramos Almeida et al. 2012; Chiaberge +et al. 2015). In this light it may not be too surprising +to find a radio-loud AGN in the core of SPT2349−56. +Given that the radio luminosity of SPT2349−56 is mod- +est for an HzRG, we may be seeing a radio-loud AGN +fueled via radiatively-inefficient flows with low accretion +rates (Best & Heckman 2012). In this picture, the gas +supplying the radio galaxy is frequently associated with +hot X-ray halos surrounding massive galaxies, groups +and clusters, as part of a radio-AGN feedback loop. +This contrasts with more luminous radio sources (e.g. +TN J1338) thought to be fuelled at higher rates through +radiatively efficient standard accretion disks by cold gas +(Best & Heckman 2012). +These more luminous radio +sources are hypothesized to have fuel brought in through +mergers and interactions, which are in fact abundant +in SPT2349−56. The debate thus remains open as to +whether we are seeing a decaying radio source, or a radio +source quickly building in luminosity. By better speci- +fying the radio emission and its origin, we could learn +about the build-up and state of the ICM that may al- +ready be present at z = 4.3. +5. CONCLUSIONS +We have presented ATCA radio observations of +SPT2349−56, a starbursting and gas-rich protoclus- +ter, consisting of over 30 SMGs at z = 4.3. We placed +SPT2349−56 in context with µJy radio sources in the +GOODS-N and ALESS fields, and with the other 22 +gravitationally-lensed SPT SMGs also observed with +ATCA in our program. We also studied in detail the cen- +tral galaxies identified by ALMA in SPT2349−56 near +this strong radio detection. +– +We +detected +a +single +source +at +2.2 GHz +in +SPT2349−56, spatially coincident with the central three +luminous members of the protocluster, denoted B, C, +and G in Miller et al. (2018). While the ATCA radio +centroid lies close to source C, which has the largest +stellar mass in the protocluster, we cannot rule out that +the radio emission is coming from B or G, or even a +combination of the galaxies. +- Under any of the possibilities above, the 214 µJy flux +density at 2.2 GHz translates to more than 20 times the +radio luminosity expected from the FIR-radio correla- +tion defined by star-forming galaxies, and suggests that +an AGN is driving the radio emission. +- The radio source has a steep spectrum, with an in- +dex of α = − 1.58 ± 0.31, constrained by the ASKAP +888MHz detection, and the non-detections at 5.5 and +9 GHz, consistent with an AGN. +- No other clear signs of AGN activity have yet been +detected in this protocluster using any other diagnostics +available to us (CO SLEDs; EW(OH163µm), [C ii]/FIR +ratios; optical spectra), highlighting the radio contin- +uum as a powerful probe of obscured AGN in high-z +protoclusters. +- The three SMGs likely associated to the radio source +have amongst the highest gas and dynamical mass of the +protocluster members (Rotermund et al. 2021). More- +over, high resolution ALMA imaging resolves this sys- +tem into multiple interacting, star-forming clumps, with +a surrounding arc of [C ii] emission (Hill et al. 2020; +Sulzanauer et al. in prep.). This is consistent with the +idea that the availability of large amounts of gas and +galaxy interactions, both of which are enhanced in gas- +rich overdensities at high redshift, can trigger fast and +obscured SMBH accretion. +– No significant radio emission (nor any other robust +AGN signature) is found from any other cluster member, +constraining the radio-loud AGN content among SMGs +in this protocluster to no more than 10% (three of 30 +members), and likely just 3%. A radio stacking analy- +sis on the remaining ten brightest SPT2349−56 SMGs +finds (11±10) µJy, which is consistent with the average +2.2 GHz emission from star formation via the FIR-radio +correlation. We thus find no evidence that nuclear accre- +tion powering radio emission exists below our detection +threshold in other SMG members of SPT2349−56. How- +ever, radio-loud AGN represent only 10% of all AGN, +and X-ray observations and JWST infrared spectroscopy +would be the next key steps to constrain AGN in this +system and compare to AGN fractions found in other +protoclusters. +- The SPT2349−56 radio-loud AGN has a luminos- +ity density of L2.2 = 4.4 × 1025 W Hz−1, extrapolating to + +A radio-loud AGN in SPT2349 +19 +L1.4, rest = (2.4±0.3) × 1026 W Hz−1 with the measured +α = −1.6, which is still over two orders of magnitude +less luminous than the powerful radio galaxies normally +studied at these redshifts. Many such HzRGs have rich +protocluster environments, however it remains unclear if +the opposite is true, that all massive z > 4 protoclusters +have a central radio galaxy. +- The fact that the radio AGN is detected in the hy- +pothesized central seed of a growing BCG galaxy with +significant stellar mass already in place makes this dis- +covery an important new ingredient in understanding +the formation and evolution of the cluster. +- The radio luminosity was used to infer a radio jet +power of Pcav = (3.3 ± 0.7) × 1038 W, sufficiently large +as to provide a dominant feedback on the cooling gas in +the 1013 M⊙ halo. The radio luminosity also suggests a +strong X-ray source with LX = 1038 W (integrated be- +tween 2 and 10 keV), easily detectable by Chandra or +XMM-Newton. SPT2349−56 therefore has a high lumi- +nosity AGN, even if in the form of a highly obscured +quasar, and JWST will be a powerful tool to uncover +its properties through high ionization infrared emission +lines. +ACKNOWLEDGEMENTS +The +Australia +Telescope +Compact +Array +is +part +of +the +Australia +Telescope +National +Facility +(https://ror.org/05qajvd42), which is funded by the +Australian Government for operation as a National +Facility managed by CSIRO. The Australian SKA +Pathfinder is part of the Australia Telescope National +Facility (https://ror.org/05qajvd42) which is managed +by CSIRO. Operation of ASKAP is funded by the +Australian Government with support from the Na- +tional Collaborative Research Infrastructure Strategy. +ASKAP uses the resources of the Pawsey Supercom- +puting Centre. Establishment of ASKAP, the Murchi- +son Radio-astronomy Observatory and the Pawsey Su- +percomputing Center are initiatives of the Australian +Government, with support from the Government of +Western Australia and the Science and Industry En- +dowment Fund. We acknowledge the Wajarri Yamatji +people as the traditional owners of the Observatory +site. +The National Radio Astronomy Observatory is +a facility of the National Science Foundation oper- +ated under cooperative agreement by Associated Uni- +versities, Inc. +This paper makes use of the follow- +ing ALMA data: +ADS/JAO.ALMA#2015.1.01543.T, +ADS/JAO.ALMA#2018.1.00058.S, and +ADS/JAO.ALMA#2021.1.01010.P. ALMA is a part- +nership of ESO (representing its member states), +NSF (USA) and NINS (Japan), together with NRC +(Canada), MOST and ASIAA (Taiwan), and KASI +(Republic of Korea), in cooperation with the Repub- +lic of Chile. +The Joint ALMA Observatory is op- +erated by ESO, AUI/NRAO and NAOJ. S.C., A.B., +and D.S. gratefully acknowledge support for this re- +search from NSERC. Manuel A. acknowledges support +from FONDECYT grant 1211951, CONICYT + PCI ++ INSTITUTO MAX PLANCK DE ASTRONOMIA +MPG190030. M.A. and M.S. acknowledge support from +CONICYT + PCI + REDES 190194 and ANID BASAL +project FB210003. +K.A.P., Melanie A. are supported +by the Center for AstroPhysical Surveys at the National +Center for Supercomputing Applications as an Illinois +Survey Science Graduate Fellow. +APPENDIX +A. APPENDIX A +In this appendix, we further assess the offsets between source centroids in the 888 MHz ASKAP image and the +ATCA 2 GHz image that were discussed in section 3.1 (Fig 8). We measured peak fluxes in both images for all sources +within a 13′ radius of SPT2349−56, measured their centroids, and calculated radial offsets for each. The offsets appear +random in orientation, with the mean x and y offset being close to zero (0.3′′, −0.2′′). In Fig. 8 we plot the radial +offsets versus ASKAP flux. SPT2349−56 shows the largest offset, which could indicate that its origin may be physical. +It has a 4.6σ deviation from the median (excluding SPT2349−56) offset of 1.4′′. Even restricting the analysis to those +sources with comparable flux densitites and SNRs (S888 < 2 mJy) only increases the median offset to 1.6′′. +B. APPENDIX B +In this appendix and Figs. 9 – 11, we show the CO J = 7, 11, and 16 lines for the central B, C, and G sources, whose +line strengths are plotted in the SLED diagram (Fig. 5). We also show the H2O lines that are used to compare with +the FIR luminosity estimates from Hill et al. (2020). +In order to measure line strengths, the bright and well-detected [C ii] lines provided in Hill et al. (2020) were used +as a template. These [C ii] lines were fit by single and double Gaussian profiles, and we selected the integration range + +20 +Chapman et al. +100 +101 +102 +S(888MHz) (mJy) +1 +2 +3 +4 +5 +Offset (arcsec) +SPT2349 +Figure 8. +Assessing the offsets between ASKAP and ATCA sources. +Top: +The 888 MHz ASKAP image surrounding +SPT2349−56 (green LABOCA contours, showing ALMA sources as green dots) with ATCA 2 GHz contours overlaid (26′ × 19′ +field shown). +Bottom: The radial offsets between the centroids for sources common in both images. +SPT2349−56 shows +the largest offset, which might therefore require a physical interpretation. It has a 4.6σ deviation from the median (excluding +SPT2349−56) offset of 1.4′′ offset. + +宣 +.A radio-loud AGN in SPT2349 +21 +by scaling the [C ii] profile to the rest frequency of the line of interest and then summing channels between −2σ and +2σ (where σ is the standard deviation of the best-fitting linewidth), or for cases where two Gaussians were a better +fit, from −2σL to +2σR, where σL and σR are from the left and right Gaussian fits, respectively. +The CO(7–6) line is blended with the [C i](2–1) line, and the CO(16–15) line is blended with the OH doublet, so +these had to be fit and subtracted before integrating over the CO lines. For the former case, where both CO(7–6) and +[C i](2–1) are both well-detected in B and C, we simultaneously fit single Gaussian profiles at the locations of the two +lines, then subtract the best-fit [C i](2–1) model from the spectrum, and sum over the relevant channels as described +above (then vice-versa to obtain [C i](2–1) line strengths). For source G, we do not see any strong line features around +the expected [C i](2–1) frequency, so we simply sum over the CO(7–6) and [C i](2–1) channels in the raw spectrum. +The CO(16–15) line is not well-detected for any sources but the OH doublet is, so we fit a Gaussian to these OH +lines and subtract the models before summing over the CO(16–15) channels. In the fit we force the amplitude of each +doublet component to be equal, and we fix the width of each doublet component to be equal to the width of the [C ii] +line (described in Sec. 3.3; see Table 4). Since the profile for G is two Gaussians, we include an additional OH doublet +component of equal amplitude and fixed frequency separation/width to match the [C ii] profile. This leaves two free +parameters in all fits: the frequency of the first doublet, and the amplitude of the all the components. In Fig. 9 we +can see that for source B the two OH doublet components are blended with each other due to the large FWHM of the +system, and for G the four components blend into three peaks. +Lastly, Band 4 and Band 6 continuum flux densities were estimated by averaging over all line-free channels in the +original (non continuum-subtracted) data cubes (again using the [C ii] line as a template). We combined channels from +the lower and upper sideband of these observations, meaning they are at observed frequencies of 147 and 231 GHz, +respectively. Band 7 continuum flux densities (around the CO(16–15) and OH lines) are already provided in Hill et al. +(2020). +C. APPENDIX C +Here we describe the ATCA observations of the full sample of 23 SPT-SMGs observed in the survey program, shown +in Fig. 3. These SPT-SMGs were drawn from the complete sample of 81 sources (Reuter et al. 2020), selecting those +that had the best redshift constraints at the time of observations. All but three of the 23 SPT-SMGs are detected +at > 4σ significance at 2.2 GHz. The 2.2 GHz flux densities are measured at peak pixels (Table 5), as in all cases the +sources are unresolved in the 8′′ × 5′′ beam. The restored beam sizes and position angles are also listed in Table 5. +ALMA 850 µm overlays are shown in Fig. 12 (data from Spilker et al. 2016; Reuter et al. 2020). LABOCA 850 µm +fluxes and ALMA-derived redshifts from Reuter et al. (2020) are also listed in Table 5 for completeness. +Most sources are not detected or only marginally detected at 5.5GHz (nine detections at > 3σ) and 9.0 GHz (four +detections at > 3σ). For those sources detected at these higher frequencies with ATCA, we measure flux densities from +peak pixels when the source is unresolved, or as aperture measurements when the source is resolved. We show the +nine sources detected at 5.5 GHz in Fig. 13, the four sources detected at 9.0 GHz in Fig. 14. We have also searched for +detections in the ASKAP 0.888 MHz RACS survey described in section 2.2, listing their flux densities in Table 5. We +find 15 of the 23 sources are significantly detected by ASKAP. +We derive radio spectral indices directly for all sources with at least two radio detections, and list these together with +flux densities in Table 5. The data was fit according to a linear function using a Markov Chain Monte Carlo algorithm +(MCMC) implemented by the emcee package (Foreman-Mackey et al. 2013). This MCMC package samples the posterior +probability function, and is used to determine the error contours shown in Fig. 2, as well as the uncertainties on α in +Table 5. We summarize their radio spectral indices in in Fig. 15 and Table 5. +The lensed SMGs are shown in Fig. 3, where we estimate their rest 1.4 GHz luminosities directly using the measured +α, or with α = −0.8 if only detected at a single radio frequency. In general these SPT-SMGs follow the same FIR-radio +correlation as the other field samples shown. However, three sources are highly significant outliers from the FIR-radio +correlation: SPT0125−50 at z =3.96; SPT0202−61 at z = 5.02; and SPT0550−53 at z = 3.13. Given how rare such +strong outliers are in the field SMG samples (only one of 76 SMGs shows anywhere near this level of radio excess +in the ALESS SMG sample – Thomson et al. 2014), we propose that the lensing galaxy rather than SMG may be +the more likely radio-AGN in these three cases. These radio excess sources exhibit steeper radio indices than typical +star-forming galaxies, comparable to or exceeding SPT2349−56. Without knowing if the lens or source redshift is +correct, we cannot reasonably apply the radio K-correction to estimate the rest 1.4 GHz luminosity, and therefore we +do not include these three in figure 3. + +22 +Chapman et al. +−1′′ +0′′ +1′′ +∆α +−1′′ +0′′ +1′′ +∆δ +153-GHz continuum +CO(7 − 6) +−2000 +−1000 +0 +1000 +2000 +∆v[kms−1] +0 +0.4 +0.8 +1.2 +1.6 +Sν [mJy] +B (C3) +[CI](2 − 1) +CO(7 − 6) +[CII] +−1′′ +0′′ +1′′ +∆α +−1′′ +0′′ +1′′ +∆δ +231-GHz continuum +CO(11 − 10) +−2000 +−1000 +0 +1000 +2000 +∆v[kms−1] +0 +0.4 +0.8 +Sν [mJy] +B (C3) +CO(11 − 10) +[CII] +−1′′ +0′′ +1′′ +∆α +−1′′ +0′′ +1′′ +∆δ +346-GHz continuum +CO(16 − 15) +−1000 +0 +1000 +2000 +∆v[kms−1] +0 +0.4 +0.8 +1.2 +Sν [mJy] +B (C3) +OH +CO(16 − 15) +[CII] +−1′′ +0′′ +1′′ +∆α +−1′′ +0′′ +1′′ +∆δ +141-GHz continuum +H2O +−2000 +−1000 +0 +1000 +2000 +∆v[kms−1] +0 +0.4 +Sν [mJy] +B (C3) +H2O +[CII] +Figure 9. Cutouts and spectra of CO(7–6), CO(11–10), CO(16–15), and H2O line emission for galaxy B. The cutouts in each +panel show continuum emission (obtained by averaging over all line-free channels) and line emission (obtained by averaging over +all channels where the line is expected – see Section 2 for details), with contours starting at 2σ and increasing in steps of 3σ. +Apertures are shown as red circles, and used to obtain the spectra shown in the right panels. In each spectrum plot, we show +the [C ii] profiles from Hill et al. (2020), scaled to the expected frequency of the given line, and arbitrarily normalized. The +shaded regions show the integration ranges (set to be ±2σ about the [C ii] line – see Section 2) used to obtain line strengths. +The CO(7–6) line is blended with the [CI](2–1) line, and the expected central frequency (or for G, two central frequencies as the +[C ii] profile has two components) of the [CI](2–1) is marked in red. The [CI](2–1) line is fit by a Gaussian profile and subtracted, +and the original spectra are shown by the dashed lines. Similarly, the CO(16–15) line is blended with the OH doublet, and we +mark the mean frequency of each OH line in red (corresponsing to two frequencies for B and C, and four frequencies for G). +The OH doublet is fit by a scaled [C ii] profile (see Section 2) and subtracted, and the original spectra are shown by the dashed +lines. +In particular, SPT0550−53 shows an extended radio morphology/jet, well resolved in all three ATCA frequencies, +which is more naturally explained by a lower redshift radio-loud galaxy. Further, the optical spectrum of the lens +SPT0550−53 shows AGN emission lines. Neither SPT0125−50 nor SPT0202−62 show AGN signatures in their optical +spectra. +SPT0125−50 is curious as the ATCA 2 GHz flux density is very close to that expected from the FIR- +radio relation, however ASKAP reveals a 3 mJy source well centered on SPT0125−50, implying an incredibly steep +α = −2.24. Thus any K-correction to lower rest frame frequencies than that probed by 2.2 GHz observations quickly +places SPT0125−50 significantly above the FIR-radio correlation. +These results also beg the question of whether the radio emission in other lensed SPT-SMGs might be contaminated +from the often massive lens galaxy (Rotermund 2020). While in some examples, especially in SPT0538−50, the radio +emission is directly identified as coming from the ALMA-detected lensed SMG components, in others the Einstein +radius of the lensed source (Spilker et al. 2016) is too small to be detected offset from the lens galaxy itself, even +at 9 GHz. The distribution in α constrained by the fits in figure 15 show a mean of −0.93 ± 0.14, offset steeper, +but still consistent, with the α = −0.8 found in samples of unlensed SMGs reported in section 3.2 (e.g., Thomson +et al. 2014). Several of the higher redshift sources in figure 3 do in fact show a marginal excess over that expected +from the FIR-radio correlation. This excess sometimes appears only due to the comparison shown at rest 1.4 GHz, +accentuating the K-correction from their steeper than average α we measure. Given the large uncertainties from the +often two-point α estimates, this may be inconsequential. Generally, optically identified AGN are relatively rare in +the distant red galaxies that are often found to lens these SMGs (Rotermund 2020), and the gravitationally boosted +radio signal associated with the high-SFR SMG is a more probable source of the strong radio emission we see in these +19 SPT-SMGs. Their radio emission is not obviously contaminated by their foreground lens. + +A radio-loud AGN in SPT2349 +23 +−1′′ +0′′ +1′′ +∆α +−1′′ +0′′ +1′′ +∆δ +153-GHz continuum +CO(7 − 6) +−2000 +−1000 +0 +1000 +2000 +∆v[kms−1] +0 +0.4 +0.8 +1.2 +1.6 +Sν [mJy] +C (C6) +[CI](2 − 1) +CO(7 − 6) +[CII] +−1′′ +0′′ +1′′ +∆α +−1′′ +0′′ +1′′ +∆δ +231-GHz continuum +CO(11 − 10) +−2000 +−1000 +0 +1000 +1000 +∆v[kms−1] +0 +0.4 +0.8 +Sν [mJy] +C (C6) +CO(11 − 10) +[CII] +−1′′ +0′′ +1′′ +∆α +−1′′ +0′′ +1′′ +∆δ +346-GHz continuum +CO(16 − 15) +−1000 +0 +1000 +2000 +∆v[kms−1] +0 +0.4 +0.8 +1.2 +Sν [mJy] +C (C6) +OH +CO(16 − 15) +[CII] +−1′′ +0′′ +1′′ +∆α +−1′′ +0′′ +1′′ +∆δ +141-GHz continuum +H2O +−2000 +−1000 +0 +1000 +2000 +∆v[kms−1] +0 +0.4 +Sν [mJy] +C (C6) +H2O +[CII] +Figure 10. Same as Fig. 9 but for galaxy C. +−1′′ +0′′ +1′′ +∆α +−1′′ +0′′ +1′′ +∆δ +153-GHz continuum +CO(7 − 6) +−2000 +−1000 +0 +1000 +2000 +∆v[kms−1] +0 +0.4 +0.8 +1.2 +1.6 +Sν [mJy] +G (C13) +CO(7 − 6) +[CII] +−1′′ +0′′ +1′′ +∆α +−1′′ +0′′ +1′′ +∆δ +231-GHz continuum +CO(11 − 10) +−2000 +−1000 +0 +1000 +2000 +∆v[kms−1] +0 +0.4 +0.8 +Sν [mJy] +G (C13) +CO(11 − 10) +[CII] +−1′′ +0′′ +1′′ +∆α +−1′′ +0′′ +1′′ +∆δ +346-GHz continuum +CO(16 − 15) +−1000 +0 +1000 +2000 +∆v[kms−1] +0 +0.4 +0.8 +1.2 +Sν [mJy] +G (C13) +OH +CO(16 − 15) +[CII] +−1′′ +0′′ +1′′ +∆α +−1′′ +0′′ +1′′ +∆δ +141-GHz continuum +H2O +−2000 +−1000 +0 +1000 +2000 +∆v[kms−1] +0 +0.4 +Sν [mJy] +G (C13) +H2O +[CII] +Figure 11. Same as Fig. 9 but for galaxy G. +REFERENCES +Babul, A., Sharma, P., & Reynolds, C. S. 2013, ApJ, 768, +11, doi: 10.1088/0004-637X/768/1/11 +Ballo, L., Heras, F. J. H., Barcons, X., & Carrera, F. J. +2012, A&A, 545, A66, doi: 10.1051/0004-6361/201117464 +Banerji, M., Chapman, S. 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We also include +unresolved LABOCA 850 µm flux densities (S850), ALMA redshifts, and best-fit spectral indices. +Name +tint +S2.2 +Beama +PAb +S† +5.5 +S† +9.0 +S†† +0.9 +S850 +z +αc +Comment††† +(hrs) +(µJy) +(′′×′′) +(deg) +(µJy) +(µJy) +(µJy) +(mJy) +SPT 0027−50 +0.58 +334±31 +8 × 4 +106 +<105 +<150 +1064±201 +48 +3.444 +−1.28±0.25 +n n +SPT 0103−45 +0.54 +229±29 +8 × 4 +16 +<111 +<159 +< 589 +125 +3.092 +– +n n +SPT 0109−47 +0.56 +1106±31 +9 × 5 +11 +824±36 +461±52 +1417±194 +109 +3.614 +−0.42±0.04 +yR yR +SPT 0125−47 +0.59 +586±35 +9 × 5 +112 +382±36 +193±49 +1835±201 +144 +2.515 +−0.82±0.10 +y mR +SPT 0125−50 +0.56 +365±29 +5 × 7 +110 +<102 +<156 +2792±194 +109 +3.959 +−2.25±0.12 +n n +SPT 0202−61 +0.68 +710±35 +9 × 5 +6 +225±40 +<153 +1943±179 +109 +5.018 +−1.16±0.09 +y n +SPT 0245−63 +0.71 +94±33 +9 × 4 +77 +<99 +<135 +< 598 +61 +5.626 +– +n n +SPT 0345−47 +0.68 +275±29 +8 × 6 +174 +<99 +<132 +701±194 +89 +4.296 +−1.03±0.38 +n n +SPT 0346−52 +0.72 +162±38 +5 × 9 +68 +<105 +<132 +< 603 +131 +5.656 +– +n n +SPT 0418−47 +0.70 +173±22 +8 × 5 +163 +<93 +<135 +430±198 +108 +4.224 +−0.99±0.67 +n n +SPT 0512−59 +0.56 +465±34 +7 × 5 +153 +<177 +<231 +1531±197 +75 +2.233 +−1.32±0.17 +n n +SPT 0529−54 +0.54 +260±50 +8 × 5 +153 +<120 +<180 +< 586 +118 +3.369 +– +n n +SPT 0532−50 +0.57 +489±39 +8 × 5 +154 +144±49 +<177 +1093±231 +118 +3.399 +−1.06±0.16 +m n +SPT 0538−50 +0.56 +581±36 +8 × 5 +157 +341±59 +168±58 +1490±237 +125 +2.786 +−0.81±0.13 +yR mR +SPT 0550−53 +0.55 +1288±48 +9 × 6 +169 +446±39 +270±56 +4060±187 +53 +3.128 +−1.22±0.05 +yR yR +SPT 0551−50 +0.56 +286±25 +8 × 6 +160 +159±45 +<171 +520±185 +74 +3.164 +−0.70±0.24 +m n +SPT 2031−51 +0.48 +269±31 +9 × 5 +51 +<123 +<186 +721±203 +65 +2.452 +−1.08±0.39 +n n +SPT 2134−50 +0.53 +334±47 +8 × 4 +33 +174±43 +<162 +804±196 +101 +2.780 +−0.90±0.44 +y n +SPT 2319−55 +0.54 +75±44 +8 × 4 +47 +<126 +<174 +< 600 +38 +5.293 +– +n n +SPT 2332−53 +0.56 +244±23 +9 × 5 +36 +146±41 +<162 +< 581 +57 +2.756 +−0.82±0.56 +m n +SPT 2349−56 +0.56 +215±27 +8 × 4 +30 +<120 +<162 +867±189 +106d +4.303 +−1.58±0.31 +n n +SPT 2353−50 +0.56 +24±53 +9 × 5 +30 +<138 +<159 +< 594 +41 +5.576 +– +n n +SPT 2357−51 +0.55 +131±19 +9 × 5 +31 +<108 +<156 +< 589 +53 +3.070 +– +n n +a The 2.2 GHz beam is quoted as x and y FHWM. The 5.5 GHz beam is typically 3.6′′ × 2.2′′. The 9.0 GHz beam is typically +2.2′′ × 1.2′′. +b The PA of the 2.2 GHz beam is the angle East of North. +c The radio spectral index α, defined as Sν ∝ να. +d In SPT2349−56 we have assumed source C with S850 µm = 4.7 mJy is the host of the AGN, although it could be B or G as +described in the text; here we still provide the unresolved LABOCA flux density. +† At 5.5 and 9.0 GHz, the 3σ upper limit is listed unless there is a detection at > 3σ at the ALMA position. +†† The 888 MHz measurements are from the ASKAP RACS survey, described in section 2.2. Sources with < 3σ positive signal +are listed at these limits. +††† Comments list whether 5.5 GHz and 9.0 GHz data show detections > 4σ (y), marginal detections (m) where < 4σ flux +density is measured at the ALMA position, or no detection (n). We indicate the four sources with resolved radio morphologies +(R), in SPT0538−50 clearly following the ALMA emission, although in SPT0109−47 and especially SPT0550−53, the resolved +emission appears to be an extended lobe or jet. + +28 +Chapman et al. +0:27:08 +07 +06 +05 +-50:07:05 +10 +15 +20 +25 +30 +RA +Dec +SPT0027 − 50 +ATCA2.2GHz +1:03:12 +11 +-45:38:40 +45 +50 +55 +39:00 +05 +RA +Dec +SPT0103 − 45 +ATCA2.2GHz +1:09:51 +50 +49 +-47:02:00 +05 +10 +15 +20 +25 +RA +Dec +SPT0109 − 47 +ATCA2.2GHz +1:25:08 +07 +06 +-47:23:45 +50 +55 +24:00 +05 +10 +RA +Dec +SPT0125 − 47 +ATCA2.2GHz +1:25:50 +49 +48 +47 +-50:38:10 +15 +20 +25 +30 +35 +RA +Dec +SPT0125 − 50 +ATCA2.2GHz +2:03:00 +00 +02:58 +57 +-61:21:00 +05 +10 +15 +20 +25 +RA +Dec +SPT0202 − 61 +ATCA2.2GHz +2:45:46 +45 +44 +43 +42 +-63:20:25 +30 +35 +40 +45 +50 +RA +Dec +SPT0245 − 63 +ATCA2.2GHz +3:45:12 +11 +10 +-47:25:25 +30 +35 +40 +45 +50 +RA +Dec +SPT0345 − 47 +ATCA2.2GHz +3:46:42 +41 +40 +-52:04:50 +55 +05:00 +05 +10 +15 +RA +Dec +SPT0346 − 52 +ATCA2.2GHz +4:18:41 +40 +39 +-47:51:40 +45 +50 +55 +52:00 +05 +RA +Dec +SPT0418 − 47 +ATCA2.2GHz +5:13:00 +12:58 +57 +-59:35:30 +35 +40 +45 +50 +55 +RA +Dec +SPT0512 − 59 +ATCA2.2GHz +5:29:04 +03 +02 +-54:36:30 +35 +40 +45 +50 +55 +RA +Dec +SPT0529 − 54 +ATCA2.2GHz +5:32:52 +51 +50 +-50:46:55 +47:00 +05 +10 +15 +20 +RA +Dec +SPT0532 − 50 +ATCA2.2GHz +5:38:18 +17 +16 +15 +-50:30:40 +45 +50 +55 +31:00 +05 +RA +Dec +SPT0538 − 50 +ATCA2.2GHz +5:50:02 +01 +00 +00 +-53:56:30 +35 +40 +45 +50 +55 +RA +Dec +SPT0550 − 53 +ATCA2.2GHz +5:51:41 +40 +39 +38 +-50:57:50 +55 +58:00 +05 +10 +15 +RA +Dec +SPT0551 − 50 +ATCA2.2GHz +20:31:00 +00 +30:58 +-51:12:15 +20 +25 +30 +35 +40 +RA +Dec +SPT2031 − 51 +ATCA2.2GHz +21:34:04 +03 +02 +-50:13:15 +20 +25 +30 +35 +40 +RA +Dec +SPT2134 − 50 +ATCA2.2GHz +23:19:23 +22 +21 +20 +-55:57:45 +50 +55 +58:00 +05 +10 +RA +Dec +SPT2319 − 55 +ATCA2.2GHz +23:32:28 +27 +26 +25 +-53:58:25 +30 +35 +40 +45 +50 +RA +Dec +SPT2332 − 53 +ATCA2.2GHz +23:49:44 +43 +42 +-56:38:15 +20 +25 +30 +35 +40 +RA +Dec +SPT2349 − 56 +ATCA2.2GHz +23:57:18 +17 +16 +-51:53:40 +45 +50 +55 +54:00 +05 +RA +Dec +SPT2357 − 51 +ATCA2.2GHz +23:53:40 +39 +38 +-50:09:55 +10:00 +05 +10 +15 +20 +RA +Dec +SPT2353 − 50 +ATCA2.2GHz +Figure 12. ATCA 2.2 GHz images (30′′ × 30′′) of the 23 SPT-SMGs observed, with ALMA 850 µm contours overlaid. + +A radio-loud AGN in SPT2349 +29 +1:09:51 +50 +49 +-47:02:00 +05 +10 +15 +20 +25 +RA +Dec +SPT0109 − 47 +ATCA5.5GHz +1:25:08 +07 +06 +-47:23:45 +50 +55 +24:00 +05 +10 +RA +Dec +SPT0125 − 47 +ATCA5.5GHz +2:03:00 +00 +02:58 +57 +-61:21:00 +05 +10 +15 +20 +25 +RA +Dec +SPT0202 − 61 +ATCA5.5GHz +5:32:52 +51 +50 +-50:46:55 +47:00 +05 +10 +15 +20 +RA +Dec +SPT0532 − 50 +ATCA5.5GHz +5:38:18 +17 +16 +15 +-50:30:40 +45 +50 +55 +31:00 +05 +RA +Dec +SPT0538 − 50 +ATCA5.5GHz +5:50:02 +01 +00 +00 +-53:56:30 +35 +40 +45 +50 +55 +RA +Dec +SPT0550 − 53 +ATCA5.5GHz +5:51:41 +40 +39 +38 +-50:57:50 +55 +58:00 +05 +10 +15 +RA +Dec +SPT0551 − 50 +ATCA5.5GHz +21:34:04 +03 +02 +-50:13:15 +20 +25 +30 +35 +40 +RA +Dec +SPT2134 − 50 +ATCA5.5GHz +23:32:28 +27 +26 +25 +-53:58:25 +30 +35 +40 +45 +50 +RA +Dec +SPT2332 − 53 +ATCA5.5GHz +Figure 13. ATCA 5.5 GHz images (30′′ × 30′′) of the nine detected SPT-SMGs, with ALMA 850 µm contours overlaid. +1:09:51 +50 +49 +-47:02:00 +05 +10 +15 +20 +25 +RA +Dec +SPT0109 − 47 +ATCA9.0GHz +1:25:08 +07 +06 +-47:23:45 +50 +55 +24:00 +05 +10 +RA +Dec +SPT0125 − 47 +ATCA9.0GHz +5:38:18 +17 +16 +15 +-50:30:40 +45 +50 +55 +31:00 +05 +RA +Dec +SPT0538 − 50 +ATCA9.0GHz +5:50:02 +01 +00 +00 +-53:56:30 +35 +40 +45 +50 +55 +RA +Dec +SPT0550 − 53 +ATCA9.0GHz +Figure 14. ATCA 9.0 GHz images (30′′ × 30′′) of the four detected SPT-SMGs, with ALMA 850 µm contours overlaid. + +30 +Chapman et al. +100 +101 +Frequency (GHz) +102 +103 +Flux (uJy) +SPT0027-50 + = -1.283 +/- 0.246 +100 +101 +Frequency (GHz) +103 +4 × 102 +6 × 102 +2 × 103 +Flux (uJy) +SPT0109-47 + = -0.425 +/- 0.042 +100 +101 +Frequency (GHz) +103 +Flux (uJy) +SPT0125-47 + = -0.825 +/- 0.099 +100 +101 +Frequency (GHz) +101 +102 +103 +104 +Flux (uJy) +SPT0125-50 + = -2.248 +/- 0.119 +100 +101 +Frequency (GHz) +102 +103 +Flux (uJy) +SPT0202-61 + = -1.155 +/- 0.089 +100 +101 +Frequency (GHz) +102 +103 +Flux (uJy) +SPT0345-47 + = -1.031 +/- 0.377 +100 +101 +Frequency (GHz) +101 +102 +103 +Flux (uJy) +SPT0418-47 + = -0.994 +/- 0.67 +100 +101 +Frequency (GHz) +102 +103 +Flux (uJy) +SPT0512-59 + = -1.32 +/- 0.167 +100 +101 +Frequency (GHz) +102 +103 +Flux (uJy) +SPT0532-50 + = -1.061 +/- 0.16 +100 +101 +Frequency (GHz) +102 +103 +Flux (uJy) +SPT0538-50 + = -0.814 +/- 0.131 +100 +101 +Frequency (GHz) +103 +104 +Flux (uJy) +SPT0550-53 + = -1.223 +/- 0.048 +100 +101 +Frequency (GHz) +102 +103 +Flux (uJy) +SPT0551-50 + = -0.704 +/- 0.238 +100 +101 +Frequency (GHz) +102 +103 +Flux (uJy) +SPT2031-51 + = -1.084 +/- 0.387 +100 +101 +Frequency (GHz) +102 +103 +104 +Flux (uJy) +SPT2134-50 + = -0.902 +/- 0.443 +100 +101 +Frequency (GHz) +101 +102 +103 +104 +Flux (uJy) +SPT2332-53 + = -0.825 +/- 0.567 +Figure 15. Radio spectral indices with errors constrained by MCMC modelling for gravitatinally lensed SPT-SMGs having +at least two detections between the ATCA and ASKAP followup. Grey shaded regions show the 1 and 2σ uncertainties on α +derived from the ATCA data. The brighter sources with steeper spectra are generally detected by the ASKAP RACS survey. +The fit for SPT2349−56 is shown in Figure 2. + diff --git a/gtAzT4oBgHgl3EQfavzP/content/tmp_files/load_file.txt b/gtAzT4oBgHgl3EQfavzP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f913e52c590cbf55df0a283340856a879793fd4e --- /dev/null +++ b/gtAzT4oBgHgl3EQfavzP/content/tmp_files/load_file.txt @@ -0,0 +1,2668 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf,len=2667 +page_content='Draft version January 5, 2023 Typeset using LATEX twocolumn style in AASTeX631 Brightest Cluster Galaxy Formation in the z=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3 Protocluster SPT 2349-56: Discovery of a Radio-Loud AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Scott C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Chapman,1, 2, 3, 4 Ryley Hill,3 Manuel Aravena,5 Melanie Archipley,6, 7 Arif Babul,8 James Burgoyne,9 Rebecca E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Canning,10 Carlos De Breuck,11 Anthony H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Gonzalez,12 Christopher C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Hayward,13 Seon Woo Kim,6 Matt Malkan,14 Dan P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Marrone,15 Vincent McIntyre,16 Eric Murphy,17 Emily Pass,18, 1 Ryan W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Perry,1 Kedar A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Phadke,6, 7 Douglas Rennehan,13 Cassie Reuter,6 Kaja M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Rotermund,19 Douglas Scott,3 Nick Seymour,16 Manuel Solimano,5 Justin Spilker,20 Anthony A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Stark,18 Nikolaus Sulzenauer,21 Nick Tothill,22 Joaquin D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Vieira,6, 7 David Vizgan,6 George Wang,3 and Axel Weiss21 1Department of Physics and Atmospheric Science, Dalhousie University, Halifax, NS, B3H 4R2, Canada 2NRC Herzberg Astronomy and Astrophysics, 5071 West Saanich Rd, Victoria, BC, V9E 2E7, Canada 3Department of Physics and Astronomy, University of British Columbia, Vancouver, BC, V6T1Z1, Canada 4Eureka Scientific Inc, Oakland, CA 94602, USA 5N´ucleo de Astronomia, Facultad de Ingenieria y Ciencias, Universidad Diego Portales, Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Ej´ercito 441, Santiago, Chile 6Department of Astronomy, University of Illinois, 1002 West Green St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=',' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' USA 20Department of Physics and Astronomy and George P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy, Texas A&M University, 4242 TAMU, College Station, TX 77843-4242, US 21Max-Planck-Institut f¨ur Radioastronomie, Auf dem Hugel 69, Bonn, D-53121, Germany 22School of Science, Western Sydney University, Locked Bag 1797, Penrith NSW 2751, Australia (Received XXX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Revised YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Accepted ZZZ) Submitted to ApJ ABSTRACT We have observed the z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3 protocluster SPT2349−56 with the Australia Telescope Compact Array (ATCA) with the aim of detecting radio-loud active galactic nuclei (AGN) amongst the ∼ 30 submil- limeter (submm) galaxies (SMGs) identified in the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We detect the central complex of submm sources at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2 GHz with a luminosity of L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2 = (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='42±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='56) × 1025 W Hz−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The Australian Square Kilometre Array Pathfinder (ASKAP) also detects the source at 888 MHz, constraining the radio spec- tral index to α = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3, consistent with ATCA non-detections at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5 and 9 GHz, and implying L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4, rest = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3) × 1026 W Hz−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' This radio luminosity is about 100 times higher than expected from star formation, assuming the usual far-infrared (FIR)-radio correlation, which is a clear indica- tion of an AGN driven by a forming brightest cluster galaxy (BCG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' None of the SMGs in SPT2349−56 show signs of AGN in any other diagnostics available to us (notably 12CO out to J = 16, OH163µm, Corresponding author: Scott C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Chapman scott.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='chapman@dal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='ca arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='01375v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='GA] 3 Jan 2023 2 Chapman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' [C ii]/IR, and optical spectra), highlighting the radio continuum as a powerful probe of obscured AGN in high-z protoclusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' No other significant radio detections are found amongst the cluster members, with stacking on either all members or just the ten most luminous members yielding non-detections consistent with the FIR-radio correlation for star-forming galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We compare these results to field samples of radio sources and SMGs, along with the 22 SPT-SMG gravitational lenses also observed in the ATCA program, as well as powerful radio galaxies at high redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Our results allow us to better understand the effects of this gas-rich, overdense environment on early supermassive black hole (SMBH) growth and cluster feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We estimate that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='7) × 1038 W of power are injected into the growing intra-cluster medium (ICM) by the radio-loud AGN, whose energy over 100 Myr is com- parable to the binding energy of the gas mass of the central halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The AGN power is also comparable to the instantaneous energy injection from supernova feedback from the 23 catalogued SMGs in the core region of 120 kpc projected radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The SPT2349−56 radio-loud AGN may be providing strong feedback on a nascent ICM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Keywords: Submillimeter astronomy (1647) — Galaxy evolution (594) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' INTRODUCTION Submillimeter (submm) galaxies (SMGs) are impor- tant sites of stellar mass build-up at cosmic noon and earlier (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=', Chapman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2003, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Smail et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2004), with star-formation rates (SFRs) as high as hun- dreds to thousands of solar masses per year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Several studies have also suggested that SMGs may be good tracers of dark matter halos at early cosmic time (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=', Blain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Dudzeviˇci¯ut˙e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Simulations conducted by Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2015) found that while many dark matter halos at z = 2–4 do not contain any SMGs, large and rare associations of five or more SMGs do trace massive overdensities of dark matter that have the potential of evolving into present- day massive clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Supporting this, in the recent past, several high-redshift protoclusters have been identified entirely through their submm emission (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=', Chapman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Daddi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Capak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Casey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Oteo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' G´omez- Guijarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' AGN and star-formation processes in galaxy evolution are clearly related (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Kormendy & Ho 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' En- hanced AGN activity relative to the field environment has been found in massive protoclusters at z = 2–3 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=', Pentericci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Lehmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Digby-North et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2010), which is likely related to the enhancement of star formation in galaxy protocluster members (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Elbaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Chapman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Brodwin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Casey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Gilli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The efficient quenching of star formation in clusters requires mechan- ical and radiative feedback, which is naturally generated by AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The resulting shocked hot gas detected as “cav- ities” in the cluster is constrained by resolved, extended X-ray emission (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Fabian 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The Clusters Around Radio-Loud AGN (CARLA) survey of around 400 high- redshift radio galaxies (HzRGs) from z = 1–3 (Wyleza- lek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2013) showed that in the majority of cases, the radio AGN is located near the center of the galaxy over- density as traced by their stellar mass (Spitzer-IRAC emission).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' This is strong evidence that radio galaxy feedback in a growing brighest cluster galaxy (BCG) is important for the evolution of massive galaxy clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Overdense regions at high redshift have likely not yet virialized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Their abundant reservoirs of cold gas and the ongoing mergers and galaxy encounters expected in the dense environments will trigger star formation and drive gas down to the supermassive black hole (SMBH) poten- tial well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' AGN require this nuclear accretion as a power source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' This is in contrast to low redshifts, where struc- tures are virialized, which prevents AGN from forming (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' van Breukelen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Studies of AGN in pro- toclusters has recently become a viable endeavor, with relatively deep Chandra and XMM-Newton observations at z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5–3 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=', Digby-North et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Travascio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Continuing to study the rich variety of protoclusters and extending these studies to earlier times can inform how host galaxies are af- fected by their SMBHs, as well as the connection to the surrounding environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' In the z = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='09 SSA22 pro- tocluster, 50% of the SMGs were found to host X-ray luminous AGN (Umehata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2019) – a clear excess over the 15% found for field SMGs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' At larger distances, an overdensity of ten SMGs found by the Hershel Space Telescope at z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='0 (Oteo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2018) has been studied by Chandra in the X-ray (Vito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2020) and in the radio (Oteo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2018), revealing no significant excess of AGN activity in the system over field SMGs (22% versus 15%, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The 2,500 deg2 survey conducted by the South Pole Telescope (SPT – Vieira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Everett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' A radio-loud AGN in SPT2349 3 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' ATCA-selected sources within a radius of 1 Mpc in projection (140′′) of SPT2349−56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' ID RA Dec Freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Flux (GHz) (µJy) ID1 – – 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='98 < 159†† ID1 – – 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='47 < 120†† ID1 23:49:42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='760 −56:38:25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='17 214±27 ID1† 23:49:42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='55 −56:38:19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='888 867±189 ID2 23:49:38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='838 −56:37:09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='63 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='17 547±36 ID2† 23:49:38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='750 −56:37:06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='888 1324±182 ID3 23:49:43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='692 −56:38:01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='82 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='17 135±26 † ASKAP measurement †† 3σ ATCA limit 2020) at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='0 mm, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='0 mm and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4 mm has uncovered a small population of nine millimeter sources ranging from z = 3–7, which are extremely luminous, yet ap- parently not gravitationally lensed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Spilker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Reuter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' A well characterized example of this is SPT2349−56, a proto- cluster system at z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='303 (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Ob- servations at 870 µm using the Large APEX BOlome- ter CAmera (LABOCA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Siringo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2009) on the At- acama Pathfinder Experiment (APEX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' G¨usten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2006) telescope (with a 19-arcsec beam size) first re- vealed an extended structure with two distinct lobes connected by a bridge with a combined flux density of S870 µm = (106 ± 8) mJy (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Follow-up observations with the Atacama Large Millimeter-submillimeter Array (ALMA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Woot- ten & Thompson 2009) measured the redshift of its brightest central source through 12CO lines (Strandet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2016), and then resolved the structure into over 30 submm-luminous sources (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Rotermund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2021), with a velocity disper- sion suggesting a central halo mass of around 1013 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' A VLT/MUSE observation reveals the presence of a Lyα blob (LAB), with a linear size of about 60 kpc, close to the core of SPT2349−56 (Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Apostolovski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' None of the other protocluster SMGs were detected as Lyα emitters (LAEs) in the MUSE data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Lyα halos are commonplace in most HzRG protoclusters (Venemans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Similar objects are often found in protoclusters identified through other means, for ex- ample optical galaxy overdensities (Overzier 2016), and indicate the presence of significant amounts of neutral gas in the assembling cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' This paper presents a search for radio detections of members of the SPT2349−56 cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Section 2 de- scribes the radio and (sub)millimeter observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Sec- Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' ALMA observing programs used for follow-up analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Details on additional ALMA Band-7 observations used in this paper can be found in (Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Here the frequency is the central frequency between the upper and lower sidebands, the continuum sensitivity is calculated at the center of the primary beam and averaged over the upper and lower sidebands, and the beam is the average circular synthesized beam FWHM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' ID Date Freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' σctm Beam Array (GHz) (µJy) (′′) 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='01543.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='T 03/20/16 148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='88 C36-2/3 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='00058.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='S 10/03/18 146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='8 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='28 C43-6 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='01313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='S 07/27/22 146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3 21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='27 C-6 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='01313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='S 09/01/22 231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='9 31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='47 C-4 tion 3 presents the results derived from the source ex- traction and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' In Section 4 we discuss the de- tected central radio source, the energy injected into a growing ICM, and the implications for radio-loud AGN in protoclusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We conclude in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Throughout our analysis, a Hubble constant of H0 = 70 km s−1 Mpc−1 and density parameters of ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='7 and Ωm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3 are assumed, resulting in a proper angular scale of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='88 kpc/′′ at z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' DATA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' ATCA observations SPT2349−56 was observed by the Australia Tele- scope Compact Array (ATCA) at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5, and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='0 GHz between 2012, January 23 to 27, as part of a pro- gram (C1563) to observe 23 SPT-SMGs (described in Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We used the Compact Array Broad- band Backend (CABB) configured in the 1M-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5k mode, which leads to a bandwidth of 2 GHz per correlator window with 1 MHz per channel of spectral resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The observations were performed in the most extended ATCA configuration, 6A, with six working 22 m anten- nas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The on source time was 34 min, which was typ- ical for all SPT-SMGs observed (see Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The data were edited, calibrated, and imaged using the Miriad package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Data affected by known radio fre- quency interference (RFI) or with bad visibility ranges were flagged accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We estimate an absolute cal- ibration uncertainty of 5% at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5 GHz, and 10% at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='0 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We inverted the visibilities using nat- ural weighting, leading to beam sizes of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='7′′ × 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2′′, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2′′ × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='1′′, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='0′′ × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3′′ at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5, and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='0 GHz, respectively, with associated RMS noise values of 27, 40, and 53 µJy beam−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Figure 1 displays the ATCA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2 GHz map surrounding SPT2349−56, re- 4 Chapman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 23:49:45 40 35 30 56:37:00 30 38:00 30 39:00 RA Dec SPT2349 − 56 ATCA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2GHz ALMA 350GHz K A B C G F L H J I N E D Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Background: ATCA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2 GHz imaging of the SPT2349−56 region, with gold contours highlighting the 106 mJy extended LABOCA source at 870µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The linear ATCA feature east of SPT2349−56 is from the synthesized beam structure of a bright 20 mJy source to the south (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' An ATCA radio source is identified near the LABOCA core, with an ASKAP source (white contours) overlapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The bright ATCA+ASKAP source to the northwest is identified with a Milky Way star (ID2 in Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Inset: A 20′′ × 20′′ zoom-in of ALMA 350 GHz continuum imaging (Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2022) with overlays of ATCA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2 GHz (cyan), ASKAP 888 MHz (white).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' ATCA contours start at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='7σ revealing the FWHM of the source (4′′ × 8′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' White contours (ASKAP) start at 3σ, and the FWHM of the source is 16′′ × 25′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' ALMA sources are named from Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2018) in order of their 850 µm flux density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The ATCA radio detection of the B-C-G complex of galaxies is evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' A radio-loud AGN in SPT2349 5 100 101 102 103 104 105 Wavelength (microns) 10 2 10 1 100 101 Flux Density (mJy) C B Arp220 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Spectral energy distribution showing the ATCA and ASKAP radio detections at rest wavelengths, and the flux densities of the brightest two of the three central SMGs (B – red circles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' C – black circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Also shown is the rest- frame optical photometry of ALMA source C (as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 1), which was modelled with a 3 × 1011 M⊙ stellar mass fit (Rotermund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Source B is undetected at these wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The Arp220 SED (dashed blue line) is normal- ized to the submm photometry, revealing that the SPT2349 BCG galaxy complex has significant excess in radio above the far-infrared (FIR)-radio correlation for star-forming galax- ies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5 GHz and 9 GHz limits are shown at 3σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The radio spectral index is constrained to α = − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='31 (fit- ted line), primarily by the ASKAP detection, and consistent with the upper limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The grey shadings show the 1 and 2σ uncertainties in the fit (Appendix C);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' an Arp220 α = − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='8 spectral index is ruled out at the 3σ level by the 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5 GHz non-detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' vealing a well-detected (8σ) source near the core of SPT2349−56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' No sources at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5 or 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='0 GHz are found in the vicinity of SPT2349−56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' ATCA sources surround- ing SPT2349−56 out to 1 Mpc in projection are listed in Table 1, and the wider-field ATCA map is shown in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The shortest baseline is 30 m and the images should be sensitive to emission on angular scales up to a few arcminutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' In principle, these data should not be miss- ing any flux on the scales covering both the ATCA and ASKAP (see below) sources, although the ATCA data will be less sensitive to lower surface brightness emis- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' However the short 34 min integration, with quite limited SNR, may still be missing some structure due to sparse uv coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Similar issues were discussed in an ATCA snapshot survey of distant HzRGs (De Breuck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2000) and are elaborated in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' ASKAP observations The Australian Square Kilometre Array Pathfinder (ASKAP) comprises 36 twelve-metre dishes located in the Inyarrimanha Ilgari Bundra1 at the CSIRO Murchi- son Radio-astronomy Observatory (MRO) in Western Australia, observing between 700 MHz and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='8 GHz, with an instantaneous bandwidth of up to 288 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' ASKAP is equipped with phased-array feeds (PAF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Hotan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' McConnell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2016), capable of simultaneously forming up to 36 independent beams, covering some 30 deg2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' SPT2349−56, along with all 22 of the lensed SPT- SMGs in the ATCA program, were observed by the Rapid ASKAP Continuum Survey (RACS, McConnell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2020), covering the sky south of +41 deg dec- lination at a central frequency of 887.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5 MHz, using 903 individual pointings with 15-minute observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The beam size at the location of SPT2349−56 is 24′′ × 13′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We retrieved the ASKAP image surrounding SPT2349−56 using the cutout server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' At the declination of SPT2349−56 the achieved RMS sensitivity is 189 µJy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The RMS is similar in the ASKAP images around the other 22 lensed SPT-SMGs, although the actual sensi- tivity depends on proximity to other nearby bright ra- dio sources (see Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The SPT2349−56 ATCA- detected source is not cataloged in the RACS, but we find a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='6σ peak approximately 5′′ from the ATCA source (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' ALMA observations Extensive ALMA properties of SPT2349−56 sources B, C, and G have already been published (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Rotermund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Here we present several new ALMA observations (Table 2), sup- porting our measurements of line emission in the context of searching for AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' ALMA Band-4 imaging (150 GHz) was obtained un- der three different programs in Cycles 3, 6, and 8, all targeting the brightest peak of the LABOCA source, and tuned to place CO(7–6) (νrest = 806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='652 GHz) and [C i](2–1) (νrest = 809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='34 GHz) in the upper sideband, and para-H2O(211–202) (νrest = 752.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='033 GHz) in the lower sideband.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The Cycle 3 program 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='01543.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='T (PI: K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Lacaille) was observed on March 20, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The array was in the C36-2/3 configuration with baselines ranging from 15 to 1 The name means ‘shared skies and stars’ in the local indigenous language, Wajarri Yamatji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 6 Chapman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 460 m, and provided a naturally-weighted synthesized beam size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='88′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Pallas and J2343−5626 were used to calibrate the amplitude and phase, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The Cycle 6 program (2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='00058.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' PI: S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Chapman) ob- servations were obtained on 2018, October 3rd in the C43-6 array configuration with baseline lengths of 15 to 2500 m, giving a naturally-weighted synthesized beam size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='28′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' J2056−4714 was used to calibrate the amplitude, while J2357−5311 was used to calibrate the phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Lastly, the Cycle 8 program (2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='01313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' PI: R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Canning) observations were obtained on 2022, July 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' These observations used the C-6 array configura- tion with baselines of 15 to 2500 m, giving a naturally- weighted synthesized beam size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='27′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' J2357−5311 was used to calibrate the amplitude, while J2336−5236 was used to calibrate the phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The Cycle 8 program (2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='01313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='S) also observed CO(11–10) (νrest = 1267.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='01 GHz) and continuum at about 230 GHz in Band 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' These observations, carried out on 2022, September 1, used the C-4 array configu- ration with baselines of 15 to 784 m, giving a naturally- weighted synthesized beam size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='47′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' J2357−5311 and J2258−2758 were used to calibrate the amplitude, while J2357−5311 and J2336−5236 were used to cali- brate the phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We also make use of previously-published Band 7 (345 GHz) ALMA Cycle 5 and 6 observations (Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The deep 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5′′-resolution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' synthe- sized beam) Cycle 5 data contain the CO(16–15) line (νrest = 1841.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='35 GHz) and an OH doublet;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' each of the doublets is actually composed of a triplet whose fre- quencies are about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='01 GHz separated, which is com- pletely unresolved by our spectral resolution, so we consider the OH line to be a doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The mean frequencies of the doublet are νrest = 1837.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='80 GHz and νrest = 1834.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='74 GHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' These lines are present in the up- per sideband, which was not previously analyzed or pub- lished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The high-resolution Cycle 6 data described by (Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2020) has a synthesized beam of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2′′ and is here used to further analyze kinematics through a moment analysis of the [C ii] line (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' All the data were calibrated using the standard observatory-supplied calibration script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Imaging was done using the CASA task tclean, using Briggs weighting with a robust parameter of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5, and in all cases channel widths were averaged down to a common 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='625 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The Cycle 6 and 8 observations covering the CO(7–6), [C i](2–1), and H2O lines were combined in uv space and then imaged together, while the Cycle 3 observation was imaged separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We chose this approach as the two data sets did not overlap entirely in frequency, which led to artefacts in the imaging step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The higher-resolution Cycle 6 and 8 data cubes were then convolved to match the resolution of the Cycle 3 data (about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='88′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The continuum was subtracted using the task imcontsub af- ter flagging all channels expected to contain line emis- sion based on previously-detected [C ii] lines given in Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' At each spatial pixel, imcontsub extracts a one-dimensional spectrum and calculates the average over all channels not flagged by the user, then subtracts this average and returns a continuum-subtracted data cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The same apertures used by Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2020) to ex- tract [C ii] line strengths and 350-GHz continuum flux densities were applied to sources B, C, and G in order to extract one-dimensional spectra for each line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The Cycle 3 and Cycles 6+8 CO(7–6), [C i](2–1), and H2O spectra were averaged to produce a final spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' De- tails on how line strengths and continuum flux densities (including our procedure for deblending lines) are given in Appendix B, and the spectra are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 9 – 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' All new continuum flux densities and line strengths are listed in Table 3, and the new continuum measure- ments are also shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' RESULTS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Identifying and characterizing radio sources We first searched for radio sources at the positions of known ALMA and optically-identified members of the SPT2349−56 protocluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' There is one strong ra- dio detection at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2 GHz (S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2 = 214 µJy) found near the SPT2349−56 core with ATCA (detected at 8σ), which corresponds to a less robust (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='6σ) detection with ASKAP at 888 MHz (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 1 and Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The ATCA source with a much smaller beam encompasses the bright central ALMA sources, named B, C, and G based on their rank-ordered 850 µm flux densities Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2 It is unclear from positional uncertainty and beam size whether the emission comes from all three galaxies or just a single source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Irrespective of this, the strong radio emission would be in excess from that expected from the far-infrared (FIR)-radio correlation (Helou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We analyse these issues in detail in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' There are no other significant (>3σ) ATCA or ASKAP detections of any known protocluster members (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The FIR-radio correlation for star-forming galaxies (Helou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Ivison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2010) would imply S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2 ≈ 12 µJy for a S850 = 5 mJy source at z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The ten brightest SPT2349−56 SMGs (excluding B, C, 2 These three sources are named C3, C6, and C13 in Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2020) based on their rank-ordered [C ii] line strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' A radio-loud AGN in SPT2349 7 1 2 3 4 5 Redshift 1031 1032 1033 1034 1035 1036 L(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4, rest) (erg s 1 Hz 1) ALESS14 HDF850.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='1 SPT2349 8C 1435 TN J1338 MRC1138 1 2 3 4 5 Redshift 102 103 104 L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4 \\ L350 SPT2349 ALESS14 ALESS66 AGN SMG Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Left: Redshift vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4 GHz radio power using the GOODS-N sample (Barger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2017) and radio-excess candidates from the ALESS sample (Thomson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The detection threshold for the GOODS-N radio sample is shown with the blue curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Red circles show sources detected above the 3σ level at 850 µm, while black circles show sources not detected at this level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' SPT2349−56 has about 10 times more radio power than any radio source found in GOODS-N, although it has more than 500 times lower radio power than other well-studied radio-loud galaxies that were used to identify high redshift protoclusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Right: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4 GHz luminosity over 350 GHz luminosity vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' redshift for the submm sources with spectroscopic redshifts in GOODS-N (red circles), and lower limits on radio sources undetected in the submm (red triangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Also shown is the ATCA survey of lensed SPT-SMGs described in Appendix C (purple squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The blue dashed line region shows where the submillimeter luminosities and radio luminosities produce consistent estimates of SFRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' None of the GOODS-N SMGs show any excess radio emission over the FIR-radio relation, while two of the ALESS sources do have a clear excess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Some of the higher redshift SPT-SMGs show a marginal radio excess, discussed in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' SPT2349−56 is about 100 times higher than the median relation at rest 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The HzRGs lie about 500 times higher with their measured S850 = 6 − 12 mJy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' and G) span 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='8–15 mJy, with an average of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='7 mJy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Thus even the brightest SMGs would only be expected to be at the 1σ level in our ATCA map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' A radio stack- ing analysis on these remaining ten brightest SMGs finds (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='0±10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='0) µJy, which is completely consistent with the average 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2 GHz emission expected from the FIR-radio correlation, ⟨S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2⟩ = 12 µJy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Stacking on all 40 known cluster members yields −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='0±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='8 µJy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We then consider if there might be other radio sources in SPT2349−56 that could be cluster members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We searched for robustly-detected radio sources in the sur- roundings of SPT2349−56 out to 1 Mpc in projection (140′′ in radius) from the core, roughly the region stud- ied with ALMA by Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We find two ATCA sources above 5σ, ID2 and ID3 in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' ID2 is identi- fied to a bright star, and is also detected by ASKAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' ID3 has a clear optical counterpart, which does not have properties (especially non-detections in the g-band) of optical sources likely to be near z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3 (Rotermund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We thus focus on the properties of the central ID1 radio source, starting with the positional uncertainty, ∆α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' From Condon (1997), we can derive the synthe- sized beam positional uncertainty for the ATCA and ASKAP detections, assuming that the beam is a single 2D Gaussian with an RMS ‘width’ σ = FWHM/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='354 in each coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' In the limit where centroiding uncertainty dominates over systematic astrometry er- rors and for uncorrelated Gaussian noise, we have ∆α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='6 (SNR)−1 FWHM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' For both the ATCA and ASKAP sources in SPT2349−56, we have confirmed that the source size and position angle is indistinguish- able from other brighter, unresolved sources in the field, in agreement with the synthesized beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We conclude that the SPT2349−56 radio source is unresolved with our current data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' For the ATCA source (ID1) detected at SNR=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='9 and a beam size of 4′′ × 8′′ (PA = 27 deg east of north), the positional uncertainty is therefore 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3′′ × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='6′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' For the ASKAP source detected with SNR of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='6 and a beam size of 24′′ × 13′′ (PA = 89 deg east of north) the po- sitional uncertainty is therefore 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='0′′ × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='7′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' There is a 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='1′′ roughly northern offset between the ATCA and ASKAP sources, which is consistent at the joint 2σ level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 8 Chapman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The ASKAP centroid is most consistent with the ALMA source A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Comparison of our wider field ATCA map and the ASKAP RACS map reveals that the major- ity of the sources show excellent astrometric alignment, but we also identify a few other ATCA sources with ASKAP counterparts with several arcsecond offsets (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' In two cases, there is a robust association of the ATCA position to other cataloged objects (from 2MASS), suggesting the offset to the ASKAP position is likely due to measurement error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' For ID1, the more robust ATCA position and association to the B, C, and G galaxies in SPT2349−56 is the most likely interpreta- tion, with the ASKAP source being assumed to be en- tirely related to the ATCA source for the purposes of de- riving a radio spectral index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The 5′′ offset is not entirely unexpected, but may be significant enough to require a physical interpretation rather than just measurement er- ror (Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' It could for instance be related to a radio core-jet morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' However, as noted, the 30m minimum baselines of ATCA would not resolve out flux on scales smaller than several arcmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' While it’s not clear why the sources are offset, it appears more likely to be instrumental than physical based on the analysis in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Physical interpretation of ID1 We first constrain the radio spectral index to esti- mate and compare luminosities between sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The radio source ID1 has a steep spectrum with an index of α = − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='28, constrained by the ASKAP 888 MHz detection, and the non-detections at 5 and 9 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The uncertainty can be estimated by propagation of errors on the two frequencies as follows: ∆α = � SNR−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2 + SNR−2 888 ln(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='89) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (1) In Appendix C, we describe a MCMC method to as- sess the uncertainty for any number of spectral measure- ments, and show this distribution in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The spec- trum is too steep to be consistent with synchrotron ra- diation due to shock acceleration of cosmic ray electrons from supernovae (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=', star formation), where Thomson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2014) recently constrained α = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='79±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='06 specif- ically for high-z SMGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The steep SPT2349−56 spec- trum seems to demand an AGN interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The radio luminosity can then be assessed by assum- ing it is associated with one of the central SPT2349−56 galaxies at z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' With a specific luminosity of L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2 = (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3) × 1025 W Hz−1, it is far larger than ex- pected from star formation through the FIR-radio cor- relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' For reference, the FIR-radio correlation for star-forming galaxies (Ivison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2010) would im- ply L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4 × 1024 W Hz−1 for a similar S850 = 5 mJy source at z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Adopting the measured spectral index above, the radio excess increases to greater than a factor 100 at a rest-frame of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4 GHz, with L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4, rest = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3) × 1026 W Hz−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' This strong radio excess suggests the presence of an AGN (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Guidetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' however, the radio emission is still distinctly less luminous than powerful radio galaxies, like those re- siding in other structures studied at these redshifts, by a few orders of magnitude (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' MRC 1138, for in- stance, is almost 1000 times more powerful in radio, and it is also hosted by the obvious BCG of the protocluster (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Hatch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We then compare SPT2349−56 to radio sources from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 3, the redshift versus radio power is shown using the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3 deg2 GOODS-N VLA sample (Barger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2017), which is highly complete in spec- troscopic redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We compute the rest-frame radio lu- minosity using the equation L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4 = � 4πd2 L S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4/1029� (1 + z)−(α+1) erg s−1 Hz−1, (2) where dL is the luminosity distance (in cm) and S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4 is the flux density in units of µJy observed at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' This equation assumes Sν ∝ να, and we adopt a radio spectral index of α = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='8 (Ibar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2010) for the GOODS-N sources, and the measured α for SPT2349−56 and the literature HzRG sources (in fact all very close to −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Shown for comparison are several well-studied HzRGs that were used as beacons to uncover massive galaxy overdensities: MRC 1138 (Large et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Seymour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' TN J1338 (De Breuck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 1999);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' and 8C 1435 (Lacy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' SPT2349−56 has around 10 times more radio power than any radio source found in GOODS-N, but it has less than 500 times the radio power of these HzRGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Figure 3 also directly assesses the departure of SPT2349−56 from the radio-FIR correlation by plot- ting the luminosity ratio of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4 GHz to 350 GHz versus redshift for all GOODS-N submm sources with spec- troscopic redshifts (Barger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Huber in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' All of the submm sources in GOODS-N are radio-detected, even at z = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2, and the submm luminos- ity and radio luminosity produce consistent estimates of the SFRs for all sources – there is no sign of AGN from their radio emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' A similar analysis of the sub- set of gravitationally lensed SPT-SMGs also observed from ATCA in this program (Appendix C) suggests the majority (87%) also follow this relation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' however, there are three very significant outliers in this sample which is most likely attributed to an AGN contribution from the foreground lensing galaxy (discussed further in Ap- pendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' SPT2349−56 is an outlier by a factor of A radio-loud AGN in SPT2349 9 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Moment maps of B, C, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Left: 850-µm continuum (red contours) from high-resolution (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3′′) combined Cycle 5 and Cycle 6 ALMA data (Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2020), shown overlaid over 3-orbit HST F160W imaging (Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Mid-left: [C ii] moment-0 maps from Cycle 6 high-resolution ALMA data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Mid-right: [C ii] moment-1 maps (in velocity units) from Cycle 6 ALMA data, with the zero velocity centered at the peak of each galaxy’s [C ii] emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' All three sources show a clear velocity gradient (listed in Table 4, along with dynamical mass comparisons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Right: [C ii] moment-2 maps (in velocity units) are shown from lower-resolution data to increase the SNR, revealing centrally concentrated dispersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' In all panels, the synthesized beam FWHM is shown in the bottom-left corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' about 100 from this envelope (assuming the radio emis- sion is coming exclusively from ALMA source C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The HzRGs shown in the left panel of figure 3 have com- parable S850 = 6 − 12 mJy to other SMGs shown (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=', Dannerbauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' De Breuck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 1999), and would remain about 500 times above SPT2349−56 in the radio/submm ratio plot in the right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' By con- trast, the GOODS-N radio sources without submm de- tection rise significantly above this envelope, into the AGN regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Thomson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2014) have used deep JVLA (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4 GHz) and GMRT (610 MHz) to study the 76 ALMA-identified SMGs in the CDFS field (the ALESS survey – e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=', Simpson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' They find four SMGs whose radio- FIR values are > 2σ above the sample median, which they classify as potential AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The most robust of these (ALESS 066.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='1) is a strong X-ray source with an inverted radio spectrum (α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Of the remaining three, one (ALESS 014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='1) has a flat radio spectrum (α > −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='1) and an obviously high radio luminosity, while the other two (ALESS 094.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='1 and ALESS 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='1) have spectral in- dex limits consistent with star formation (α ∼ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We show these four SMGs in figure 3, where it is clear that none are comparable to SPT2349−56 in radio lumi- nosity or departure from the FIR-radio relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' In fact, two of the four are not at all unusual in their properties relative to the other samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Radio emission provides an extinction-free probe of AGN (which even X-ray cannot claim, since practical sensitivity limits preclude the detection of the most ob- scured, Compton-thick AGN with NH > 1024 cm−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Traditionally radio AGN are divided into two sub- sets (Padovani 2017): (i) radio-loud AGN L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4 > 1024 W Hz−1, which exhibit steep spectrum radio jets and lobes on kpc scales (Yun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 1999);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' and (ii) radio- quiet AGN, with flat-spectrum, lower luminosity radio emission, typically contained within a compact, several pc, core (Blundell & Kuncic 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' SPT2349−56 is solidly a radio-loud AGN, whereas most of the other candidate AGN found in the surveys described above (GOODS-N and ALESS) cannot clearly be defined as such.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Resolved properties of the ‘BCG’ sources Given the radio detection in SPT2349−56, it is of interest to assess the properties of the B, C, and G ALMA sources, and to compare them to other proto- cluster members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' As noted, these are three very submm- luminous sources in the core region (S850 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='7 mJy for B, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='7 mJy for C, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3 mJy for G), with only source A being brighter, although two even more luminous sources are present in the northern extension (sources N1 and N2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The most distinguishing features of this trio (beyond their flux-ordered source names serendipitously spelling out ‘BCG’) are their locations near the center-of-mass of the cluster core, and their immediate environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' They are very close neighbours (they lie within an arc- second of each other), and are likely to be interacting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Further, there is a notable arc seen in [C ii] surrounding the three galaxies (Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Sulzanauer, in prep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Source C does distinguish itself with an anoma- lously narrow [C ii] and CO(4–3) line width for its lumi- nosity (Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Rotermund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2021) iden- tified C as a significant outlier from the SPT2349−56 galaxy sample in its Mdyn/Mgas ratio inferred from the narrow CO(4–3) line width and large luminosity, similar to many high-z QSOs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=', Narayanan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Wal- ter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2019), where selection effects favoring face-on orientation offer viable explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' It is also noteworthy that source C has by far the largest stellar mass of any cluster member (> 1011 M⊙ Roter- mund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' It is associated with a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='0 300 200 D 200 1" 1" 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='7 150 9 100 [Jy kms- [km [kms 0" 0 S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4 100 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='1 300 50 0" 1" _1" 0" 1" 1" 0" 1" Aa Aa Aa10 Chapman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Continuum and line properties of B, C, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' S147 and S231 are the continuum flux densities at 147 and 231 GHz, respectively, while the other columns provide various line strengths (the line is indicated by the subscript).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The OH doublet arises from blended hyper-fine triplets centered at 1835 and 1838 GHz, and the H2O line is the para-211–202 line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' ID RA,Dec S147 S231 FCO(16−15) FCO(11−10) FCO(7−6) FH2O FOH F[CI](2−1) µJy µJy Jy km s−1 Jy km s−1 Jy km s−1 Jy km s−1 Jy km s−1 Jy km s−1 B 23:49:42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='79, -56:38:24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='0 589±15 3322±156 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='12±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='26±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='73±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='22±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='93±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='46±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='03 C 23:49:42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='84, -56:38:25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='1 336±11 1810±118 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='10±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='17±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='61±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='15±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='83±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='35±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='03 G 23:49:42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='74, -56:38:25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='1 181±23 136±9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='10±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='12±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='18±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='02±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='22±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='10±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='02 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Physical properties of B, C, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' SFRH2O is the SFR estimated using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 5, S850/FH2O is the ratio of 850 µm continuum flux density (from Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2020) to H2O line strength, and SFRLIR/SFRH2O is the ratio of the FIR-derived SFR (from Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2020) to the H2O-derived SFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Vp−p is the peak-to-peak velocity from moment-1 maps, while FWHMcen is the central velocity dispersion (multiplied by 2 √ 2 ln 2) from moment-2 maps (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 4), and FWHMint is the width of the [Cii] line after fitting a single Gaussian to the lines shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 9–11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Mdyn, disk is the dynamical mass derived using Vp−p and a disk model (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 3), while Mdyn, cen and Mdyn, int are dynamical masses derived using the velocity dispersion measurements and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' ID SFRH2O S850/FH2O SFRLIR/SFRH2O Vp−p Mdyn, disk FWHMcen Mdyn, cen FWHMint Mdyn, int M⊙ yr−1 10−3 km−1 s km s−1 1010 M⊙ km s−1 1010 M⊙ km s−1 1010 M⊙ B 1100±410 31±3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='8+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4 600±50 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2 540±20 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='8 612±10 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5 C 750±280 31±2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='8+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4 240±50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='8 280±20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4 358±5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2 G 80±60 65±23 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='8 690±50 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='1±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5 520±20 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='8 901±54 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='8±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='8 bright and very compact HST F160W source (Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2022), as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 1 and 4, and it has been sug- gested to be the seed of a growing BCG galaxy in this ongoing mega-merger (Rennehan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' [C ii] kinematics We consider here a more detailed analysis of the kine- matic properties of the B, C, and G galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Using high- resolution Cycle 6 [C ii] data (Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2020), which has a synthesized beam of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2′′, we construct moment 0, 1, and 2 maps of the B, C, and G sources and ana- lyze the resolved velocity and dispersion fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We use the CASA task immoments, focusing on channels between ±3σ of the best-fit [C ii] line, and masking pixels < 4 times the RMS per channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Since second moments are particularly sensitive to noise (being a squared term), we use uv-combined Cycle 5 and 6 data cubes (described in Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2020) to calculate the moment 2 maps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' for reference, the resolution of the combined data is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The results (moments 0, 1, and 2) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' All three sources show a clear velocity gradient and re- solved, centrally-concentrated dispersion, characteristic of rotationally-supported disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' From these velocity gra- dients and velocity dispersion maps we extract peak-to- peak velocities, Vp−p, and central velocity dispersions, FWHMcen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We draw a line along the semi-major axis of each galaxy, then from the moment 1 map calculate the velocity difference between the two ends, and from the moment 2 map extract the velocity dispersion at the midpoint of the line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We find that moving the position angle of the line by ±10 deg and moving the midpoint of the line by 5 pixels results in a peak-to-peak velocity change of ±50 km s−1 and a central velocity dispersion change of ±10 km s−1 (±20 km s−1 in FWHM), so we quote these as our uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The results are given in Table 4, multiplied by a factor of 2 √ 2 ln 2 to estimate a FWHM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We use these peak-to-peak velocities and central dis- persions to estimate masses assuming a disk model, with the enclosed dynamical mass given by Mdyn, disk[M⊙] = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='35 × 105 [Vp−p/⟨sin(i)⟩]2 R, (3) where Vp−p is the peak-to-peak velocity in km s−1, R is the radius in kpc, and i is the inclination an- gle of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We adopt a mean inclination suit- able for a collection of randomly oriented disks of ⟨sin(i)⟩ = π / 4 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='79 (see Law et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2009), and we use the half-light radii from Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2022), estimated by fitting S´ersic profiles to the high-resolution ALMA [C ii] moment 0 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The results are given in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' A radio-loud AGN in SPT2349 11 The dynamical masses were derived previously (Roter- mund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2021) from the unresolved velocity disper- sions, using the width of the integrated [C ii] lines shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 9–10, with an assumption about the structure of the source based on the virial theorem, using the re- lation Mdyn[M⊙] = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='81 × 105 FWHM2R, (4) where FWHM is a one-dimensional velocity dispersion (multiplied by a factor of 2 √ 2 ln 2) in km s−1, and R is the radius of the virialised structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' First, we use the resolved central velocity dispersion, FWHMcen, adopt- ing the [C ii] size measurements from Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2022) and the central resolved velocity dispersions from the moment 2 maps (Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Next we use the width of the integrated [C ii] line, FWHMint, obtained by fitting a single Gaussian model to the [C ii] spectra shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 9–11 and given in Table 4, again using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 4 and the same [C ii] size measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The resulting dy- namical masses are provided in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Considered in the context of a disk model, source C does show a similar dynamical mass comparing both its central and integrated velocity dispersion (Table 4, and Rotermund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' however, it still appears to have substantially lower mass (six times lower) than B from any kinematics analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Inclination is reasonably con- strained, since the aspect ratio of these galaxies is re- solved by ALMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' While it remains an uncertainty in any mass modelling, the aspect ratios of B and C are similar at ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='8 (major to minor axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Sources B and G have similarly large inferred disk masses (18 × 1011 M⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' However, the distinct double- horned profile of source G (Appendix B) is direct evi- dence for a rotating disk or bar-like structure at high inclination (explaining the broad velocity profile), while the profile for B is possibly due to a tidal torque in re- sponse to the interaction with C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Source G also has a higher aspect ratio (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3, major/minor axes) in moment-1 than B and C, suggesting the disk is seen closer to edge- on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Explicitly using this higher implied inclination in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 3 brings down the disk mass estimate by 25%, more consistent with the much lower gas mass of G compared with B and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' It is noteworthy that in projection at least, B is counter-rotating relative to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Several studies have pre- dicted that mergers configured with counter-rotating gas disks should lead to the most intense starbursts, and conditions for fueling the SMBHs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=', Mihos & Hern- quist 1994, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Di Matteo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Salom´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Submm line properties We then consider line diagnostics to elucidate which of the three might be most likely to host the radio-AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We first assess the [C ii]/FIR ratio, which has been shown to highlight AGN with a deficit compared with star-forming galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Stacey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' However at high luminosities, both AGN and SMGs (without ob- vious AGN) exhibit similar deficits in the ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2020) have shown that all three of B, C, and G are ‘deficit sources’ in [C ii]/FIR, inhabiting similar re- gions in the [C ii]/FIR-to-FIR plot as many luminous AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' However, this work also showed that all 12 of the most luminous SMGs in SPT2349−56 have comparable [C ii]/FIR ratios, and none of these are obviously AGN from any available diagnostics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' One possibility to consider is that the FIR estimates are being affected by an AGN in one of B, C, or G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Since the shortest wavelength measured by ALMA is 160µm in the rest frame, the peak of the SED is not sampled, and there is little constraint on whether the dust might be substantially hotter than the Td ≈40 K estimated in Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' To test this we make use of the para-H2O(211–202) lines observed in the ALMA Band 4 dataset (Table 3 and Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 9–11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' H2O is strongly coupled to the FIR radiation field whether it is being produced by star-formation or AGN (Omont et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Jarugula et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2021) compiled a sample of low- and high-z submm galaxies with para-H2O(211– 202) measurements (including two sources, SPT0346−52 and SPT0311−58, from the same parent sample as SPT2349−56), and found that a simple single-parameter scaling relation described the correlation between LH20 and SFR (derived from FIR) of the form SFR [M⊙ yr−1] = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='75) × 10−5LH2O [L⊙].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (5) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' A simple test is to first take the ratio of 850µm contin- uum flux density to H2O line strength, where measure- ment errors are mostly small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' These values are given in Table 4, where we have used S850 values from Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Using the same modified blackbody SED as in Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2020) to model the continuum flux density emission, a dust temperature of 40 K at a redshift of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3 means that S850 = 1 mJy corresponds to 115 M⊙ yr−1, and so Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 5 implies S850/FH2O = (42 × 10−3) km−1 s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' B and C sit significantly below this value, implying they might have higher FIR than currently estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' In con- trast, G is significantly above the relation, which could imply cooler dust and lower FIR than previously esti- mated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' This would make G less of a deficit source in [C ii]/FIR, and less likely to be considered an AGN by this criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 12 Chapman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 5 10 15 12CO Jup 106 107 108 Lline (L ) Mrk231 M82 o|vo o o |v |v|v|v | | G C B 5 10 15 12CO Jup 10 1 100 Normalized Lline (L ) Mrk231 M82 o|v o o o |v |v |v |v | | G C B Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' SLED for B, C, and G in the CO Jupper = 2, 4, 7, 11, and 16 transitions (new data for the Jupper = 7, 11, 16 lines are shown in Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The ATCA detection of CO(2–1) (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2018) is shown as an upper limit as it is an unresolved measurement over the core region including at least B, C, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Open circles illustrate the J = 2 division if the luminosities scale from J = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We compare to the LFIR = 3 × 1012 L⊙ AGN-dominated galaxy Mrk231 (van der Werf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2010), and the LFIR = 3 × 1010 L⊙ starburst M82 (Kamenetzky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2012), here normalized to Mrk231 at CO(7–6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' In the right panel the SPT2349−56 galaxies are also normalized to the CO(7–6) luminosity of Mrk231 for comparison of the excitation curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Source G is undetected in the Jupper = 11 and 16 lines, while B and C (shown as 2σ upper limits) are marginally detected in CO(16–15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 5, we can also estimate SFRs directly for B, C, and G using our measured para-H2O(211–202) line strengths (Table 4), and compare these with the SFRs from the FIR (taken from Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' These re- lations show the same behaviour as our ratio of mea- surements above, that B and C both may have higher FIR (from hotter dust) than estimated from our current ALMA data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We note that systematic errors in convert- ing to these physical quantities are large (as listed in 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We next consider the CO spectral line energy distribu- tion (SLED), which can distinguish AGN with high ex- citation lines driven by X-ray dominated regions (XDRs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' van der Werf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Here we present higher- J CO transitions (Section 2) than have previously been published in Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' All of B, C, and G are well-detected in CO(7–6) observations, while B and C are detected in CO(11–10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Remarkably B and C may be marginally detected in CO(16–15) due to the sensi- tivity of the deep Band-7 ALMA data presented in Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2020), although strictly they are upper limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' All the new line channel maps and one-dimensional spectra are shown in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' SLEDs are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 5 from the available J =2, 4, 7, 11, and 16 transitions, and compared to AGN and starburst templates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The ATCA detection of CO(2-–1) (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2018) is shown as an upper limit as it is an unresolved measurement over the core region including at least B, C, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' However for illustration, we show the division into B, C, and G of the integrated J = 2 luminosity assuming they scale with the J = 4 fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' None of B, C, or G appear to have high excitation SLEDs similar to AGN like Mrk231 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' van der Werf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2010), and are all similar to or less excited than M82 at high-J (Kamenetzky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2012), with the caveat that G has only upper limits beyond J = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Source B has the highest excitation SLED confirmed of the three, lying near the M82 SLED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Source B also has a stronger cool/warm gas component similar to Mrk231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' However, all three are reasonably characterized with a combination of cool and warm star-forming Photo Dis- sociation Regions (PDR) components, and without sig- nificant XDR contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Detailed SLED modelling of SPT2349−56 sources will appear in a future contri- bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Finally, the 163µm OH doublet in B, C, and G can be compared to Runco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2020), who studied 178 local galaxies in six of the 14 OH transitions in the FIR range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' They found the highest frequency OH163µm (detected in 25 galaxies) is the only OH doublet which is always in A radio-loud AGN in SPT2349 13 emission, with most transitions often appearing in ab- sorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Runco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2020) presented the correlations of the equivalent width, EW(OH), with various galaxy properties and line ratios, finding EW(163µm) is not well established as a direct AGN indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' For example, while galaxies with lower X-ray luminosities exclusively have low EW(OH), the full range of EW is seen for the highest X-ray luminosities (Runco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' How- ever, a strong correlation is found for EW(OH) with the ratio of AGN activity to SFR, suggesting this is a bet- ter predictor of EW(OH) than the total AGN power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' In figure 6, we compare the equivalent width, EW(163µm), to local starbursts, LINERs and Seyfert galaxies from Runco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The EW(163µm)=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='076µm and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='095µm measured for B and C respectively are amongst the highest found locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (G is similarly high, but is only marginally detected in OH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Their EW(163µm) are more similar to values in local Seyfert galaxies than starburst galaxies, the latter having EW(163µm)∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='02– 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='05µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' It is not yet clear at z > 4 what is “normal” for EW(163µm), since the line has only before been de- tected locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' However, these results may provide ini- tial evidence that B and C do in fact present as AGN through some submm-wave diagnostics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Optical and near-infrared properties Finally, we summarize optical and near-infrared spec- tra taken with the Gemini and VLT observatories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' A VLT XSHOOTER spectrum (λobs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='35–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4 µm) was obtained which targeted B and C (Rotermund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2021), covering redshifted Lyα through [OII]3727.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' No lines were detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' A VLT-MUSE spectral cube was used to extract one-dimensional spectra at the locations of each of B, C, and G (Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Apostolovski in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' ), but no lines are detected in any of these galaxies (nor any of the SPT2349−56 SMGs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Non-detections are not particularly surprising given the faintness of the galaxies at 6450 ˚A, the wavelength of redshifted Lyα, where C has S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='63 µm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='061 µJy, while B and G are undetected to S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='63 µm < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='01 µJy (Rotermund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The diffi- culty of spectroscopy in the near-infrared at 19,768 ˚A in the vicinity of the redshifted [OII]3727 means these limits on line equivalent widths are also not particularly constraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Nonetheless, strong AGN often exhibit de- tectable high excitation lines in optically faint, obscured SMG hosts (Chapman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Chapman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Chapman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Danielson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2017), and it is surprising that this AGN in SPT2349−56 eludes all op- tical and near-infrared spectral detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Concluding remarks 1010 1011 1012 LIR (L ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='18 EW(OH163) ( m) B C G Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The equivalent width of the OH163µm doublet versus IR luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' B, C, and G (blue circles) have similar values, although the error in G is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We compare to all local galaxies detected in OH163µm from the compilation in Runco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Galaxies are colour coded by their classification as starbursts (lime), LINERs (green), Seyfert-1 (red), and Seyfert-2 and intermediate types (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' B, C, and G all appear well above local starbursts, although all three AGN types have a few examples this high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Thus while the radio observations do not have suffi- cient spatial resolution to uniquely identify one of the three galaxies as the AGN, the source properties them- selves suggest source C could be a likely host, consid- ering mainly its large stellar mass, along with narrow emission lines, and high EW(OH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Sources with sim- ilar radio luminosities in the local Universe are typi- cally found in massive hosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' However, the more FIR- luminous and much more dust-obscured source B might also be a possible host for the AGN, given the large dy- namical mass from kinematic modelling and the higher excitation SLED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Of course all three SMGs could have AGN components at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The fact that they are likely strongly interacting dispels the typical duty- cycle arguments that would disfavor this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' To make progress, we will need deeper radio data with better resolution, and sensitive infrared spectroscopic observations now possible with the James Webb Space Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' DISCUSSION 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Inferring AGN properties from radio power 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Jet power and energy input to the ICM 14 Chapman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Radio jets are thought to provide an important feed- back mode in galaxy clusters by preventing the cooling of hot (X-ray) gas surrounding central galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=', Mc- Namara & Nulsen (2012)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' This is named “jet-mode” feedback and is associated to radio sources characterised by radiatively-inefficient accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' However, radio jets can also drive massive gas outflows on galactic scales, another signature of AGN feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' A theoretical relation between radio luminosity and radio jet power was determined by Willott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (1999), and can be used to estimate the kinetic energy output of AGN (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=', Hardcastle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The jet power can be estimated by assuming that the mechanical power of the jet can be approximated as the energy of the detected radio cavity averaged over some timescale (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=', Bˆırzan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The X-ray-detectable “cavities” that result from AGN jet activity (O’Sullivan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2011) allow us to quantify the heating experienced by the intra-cluster medium (ICM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The energy contained in these cavities comes from the product of the pressure and volume (pV ) over the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' This is the work done by the jet to cre- ate the cavity, and the internal energy of the radio lobes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Under the assumption that the cavity is dominated by relativistic plasma, this becomes 4pV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Dividing the en- ergy of the cavity by the cavity age gives the power, Pcav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Thus the most direct inference we can make from the radio properties of SPT2349−56 adopts a relatively tight correlation observed between radio power and cav- ity power (Cavagnolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' O’Sullivan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Panessa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2015), where a fitted relation follows: log Pcav = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='07) log L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4 + (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='10), (6) yielding Pcav = (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='7) × 1038 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' This is strictly a lower limit to the jet power, and therefore energy injec- tion into the ICM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The true jet power depends on how the radio cavity is inflated (as described in Nusser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2006), with some energy from the jet being carried away by shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The relation of L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4 to Pcav is still affected by uncertainties due to the assumption that the cavity is dominated by relativistic plasma and the detectability of cavities within the sample used in O’Sullivan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2011), as discussed in their work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' This jet power is a sizeable amount of energy, given the potential well of the ∼1013 M⊙ SPT2349−56 halo (see below) constrained from the central velocity disper- sion and radial distribution of cluster members (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' This is also a significant addition to the already abundant energy injection from the 6600 M⊙ yr−1 of star formation being experienced by the core of SPT2349−56 from summing the SFRs of all member galaxies found in these works (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Rotermund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We take the instantaneous injection of energy at z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3 as ˙Ekin = 1 2 ˙Moutv2, (7) where ˙Mout is the total amount of gas ejected per unit time by galaxies and v is the outflow velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' While v is not measured in SPT2349−56 galaxies, the average outflow in high−z SMGs and other starforming galax- ies has been constrained with increasingly large samples (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=', Banerji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' F¨orster Schreiber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We adopt a typical 500 km s−1 wind speed for the SN- driven outflows in each SPT2349−56 galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' A mass outflow rate can then be found by converting SFRs into mass outflow rates ˙Mout by multiplying by a conser- vative mass loading factor η = ˙Mout/SFR = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' η could even be greater than one based on observational (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=', Newman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2012) and theoretical work (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=', Hop- kins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' However, the same amount of metals is found in stars and the ICM, which suggests equal- ity, ˙Mout ≈ SFR, (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=', Renzini & Andreon 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We therefore obtain ˙Ekin = (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='1) × 1038 W, where the uncertainty reflects both the range in SFR estimates and the range of likely wind velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' This energy injection is remarkably similar to that found from the radio-loud AGN above from equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Estimating the total mechanical energy injected by the radio jets requires an estimate of the radio source lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Brienza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2017) and Hardcastle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2019) have suggested that remnant sources fade rapidly, with most of the observed remnant radio galaxies being relatively young, with ages between 50 to 100 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' With τ = 100 Myr for SPT2349−56, we find Emech = Pcav × τ = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2) × 1054 J, assuming only the uncertainty in the Pcav scaling relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Binding energy of the halo gas We then turn to estimating the binding energy of the gas in the SPT2349−56 halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Giodini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2010) demonstrated that the mechanical energy from jets is comparable to the binding energy (Ebinding) in galaxy groups, while it is lower by a factor of 102–103 in clus- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Since the SPT2349−56 halo mass is comparable to a large group today, and the entire protocluster is expected to form a massive cluster by z = 0, it is thus of interest to investigate how Ebinding compares to our estimate of Emech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We define the binding energy as the total potential energy needed to push the ICM gas within R500 (the radius where the mean dark matter halo density drops to 500 times the critical density) beyond R200 (the radius where the mean dark matter halo density drops to 200 A radio-loud AGN in SPT2349 15 times the critical density, which we assume to be equal to the virial radius).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2022) estimated M200 (the mass contained within R200) to be (9 ± 5) × 1012 M⊙, corresponding to R200 = (120 ± 70) kpc at z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3, and R500 can be computed if one assumes a density profile for the dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Following Giodini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2010), the binding energy is computed as Ebinding = � Mgas,500 0 [φ(r) − φ(R200)] dMgas = 4 π � R500 0 φ(r) ρgas(r) r2 dr, (8) where the constant term φ(R200) is small compared to the potential within R500 and can be ignored, and ρgas is the gas mass density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Assuming the gas mass density follows the dark matter density but scaled by a single gas-mass fraction param- eter, fgas, we can adopt an NFW dark matter profile to write the binding energy as (see Giodini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2010 for details) Ebinding = fgas4 π ρcrit δc A r3 s � c500 0 ln(1 + x) (1 + x)2 dx, (9) where x = r/rs, with rs being the characteristic ra- dius related to the halo concentration parameter by c = R200/rs, δc is a numerical factor that depends only on the halo concentration parameter c, A scales with M200 and also depends on c, and ρcrit is the criti- cal density at the redshift of interest (here 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We compute the concentration parameter using the mass- dependent relation of Macci`o et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2007);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' they find a linear trend between log c and log Mvir (which we assume is equal to M200), and we find c = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='9, corresponding to rs = 15 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We note that c = 5 is typically adopted for massive clusters > 1014 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' With the concentra- tion parameter known, we calculate R500 = 80 kpc and c500 = R500/rs = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We cannot estimate the halo gas mass directly in SPT2349−56, beyond summing the measured cold gas masses in individual galaxies from the core region and inferring additional cool and warm gas components in the halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Summing the H2 gas masses from the 23 SMGs within the cluster core from Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2020) yields Mgas,cool = 3 × 1011 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The unseen gas compo- nents in the halo are more uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' A trend observed in groups and clusters is an increase of the fraction of hot gas with total system mass (Connor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2014), approximately following fgas ∝ M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='1−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2, where 1013 M⊙ groups typically have fgas of around 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The LX-M relation has been shown to remain approx- imately self-similar out to z = 2 (Mantz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2018), in- cluding X-ray-detected clusters at z = 2 (Gobat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' However, for low-mass systems the gas mass frac- tions may evolve with redshift (Connor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Re- gardless, this in itself does not constrain the ICM gas fraction, which requires more detailed X-ray properties than LX to be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Based on the above, we will assume for SPT2349−56 a gas mass of 10% of the halo mass, or 9 × 1011 M⊙, which nominally requires that Mgas,hot = 6 × 1011 M⊙, unless there are substantial cold flows feeding the submm galaxies (Dekel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We estimate Ebinding = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='7 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4) × 1054 J, where the uncertainty has been propagated from M200 and the uncertainty in the M200-c scaling relation using a Markov chain Monte Carlo (MCMC) approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The radio feedback alone therefore conceivably pro- vides all of the energy required to unbind the total gas in the cluster core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The stellar feedback has a compara- ble energy input, and could also be unbinding the cluster gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' However, the total energy is a minimum condition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' the energy must also couple efficiently to the ICM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' An energetic jet may not couple to the bulk of the ICM gas (Babul et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Yang & Reynolds 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Cielo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Any hot ICM established at z > 4 may not be in hy- drostatic equilibrium since cold inflows likely dominate the flow of gas in protocluster halos (Dekel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The infalling gas only increases the energy required to inflate a bubble in the nascent ICM, acting as an addi- tive term to Ebinding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' While L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4 is fixed, the work done on an inflowing medium will be higher than for an ambi- ent static medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Therefore the Pcav (∝ 4pV ) to L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4 relationship might not hold when inflows dominate the halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Yajima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2022) and Trebitsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2021) have begun to explore some of these issues in hydrody- namical simulations of protoclusters, aiming to better understand AGN feedback and the impact of massive starburst galaxies in forming clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We leave more detailed calculations to future work (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Rennehan, in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Inferred X-ray luminosity and accretion rate A correlation also exists between radio power and X- ray luminosity (LX) for radio-loud AGN (Ballo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2012), although there is substantial scatter in this rela- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' While the correlation appears to be similar over a large range (nine orders of magnitude) in X-ray lumi- nosity, there is a range of over 100 in LX for a given radio luminosity in well populated areas of the correla- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The relation plotted in Ballo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2012) is char- acterized at 5 GHz rest frame, which we measure almost directly (through the ASKAP detection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Using the cor- 16 Chapman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' relation, we find that L5 = 7 × 1025 W Hz−1 in the radio corresponds to LX = 1038 W (where the X-ray luminos- ity is between 2 and 10 keV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We conclude that the X-ray emission from the central AGN in SPT2349−56 can be easily detected by XMM-Newton or Chandra under the full range of possible LX = 1037−39 W suggested by this correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Finally, taking source C as the most likely host, we can infer the SMBH mass from the stellar mass that has been well characterized for C (Rotermund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' For M∗ = 4 × 1011 M⊙, the SMBH mass is 7 × 108 M⊙ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' From this, we can infer the range of Eddington luminosities with respect to the range in X-ray luminosity constrained by the radio power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' In other words, how close to the maximal rate of accretion is the SPT2349−56 AGN if its SMBH is close to that implied by the stellar mass of source C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' In partic- ular, following Ballo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2012), our measurements of L5 GHz/MBH constrain LX/LEdd to the range of roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='005 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='05, based on their distribution shown (their figure 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Directly measuring the X-ray properties of SPT2349−56 will allow substantial progress in charac- terizing the system and its environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Implications of the steep spectrum For an optically-thin synchrotron source, the spec- trum will steepen in spectral index from low to high frequencies by ∆α = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5 if the source lifetime is greater than the timescale for energy-loss from the radiating electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' This leads to a concave spectral shape with a characteristic bend frequency, νb (Kellermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Thus the age of the electron population within radio jets contributes to the steepness of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Three effects will then decrease νb as the source red- shift increases (Krolik & Chen 1991): (1) for a fixed bend frequency ν∗ in the rest frame, the observed bend νb = ν∗ / (1+z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2) losses due to inverse Compton scat- tering off the microwave background rise with redshift as (1 + z)4, so that for a fixed time electrons spend in the radiating region, the lowest energy electron that can cool has a frequency (or energy), which decreases with increasing redshift;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' and (3) flux-limited samples result in a selection effect that favors low ν∗ at high-z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Sources must have higher emissivity at higher redshift to be in- cluded in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' They also must have stronger implied magnetic fields, and therefore more rapid syn- chrotron losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' A combination of these effects has been used to explain the observed trend that higher-redshift radio galaxies have steeper spectral indices (Carilli & Walter 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' van Breukelen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The ultra-steep spectral indices of HzRGs (up to the α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='6 we find in SPT2349−56) is a main selection criterion for identifying these powerful radio sources in the distant Universe (De Breuck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Broderick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' All three HzRGs shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2 in fact have α very close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' It is of note that SPT2349−56 would have been discovered by these HzRG surveys over one to two decades ago had the radio source been 10–100 times more radio luminous, and even cursory submm followup would then have revealed the extended S850 = 110 mJy source that belies its nature as a submm-luminous protocluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' A steep spectrum generally argues for self-absorbed synchrotron, and a lack of electron injection (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=', Rad- cliffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Thus the steep α in SPT2349−56 could represent a dying radio source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' In this case, the ATCA flux should be extended over the same area as the ASKAP data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Thus if there is no physical offset between ATCA and ASKAP, and this is just a mea- surement uncertainty, SPT2349−56 could be a young and completely unresolved compact radio source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' On the other hand, this might be a “contained” or “frus- trated” radio source inside a dense medium, sometimes referred to as a compact steep spectrum source, or CSS (Padovani 2017), but an issue with this interpretation is that the luminosity of the source is low relative to these typical GHz-peaked sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' If self-absorbed syn- chrotron is contributing to the steep spectrum, the ob- servational constraints would mean that the break fre- quency is well below about 5 GHz in the rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' In principle this break frequency can provide a constraint on the age of the radio source, but since we do not con- strain this break with the current data, we do not pursue this further here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' However if the radio emission is due to a CSS then it would have to be older than 500 Myr to have a break frequency below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='9 GHz (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='8 GHz rest) (Padovani 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Connection to the LAB The powering sources of Ly-α blobs (LABs) have of- ten been identified broadly with the photoionizing emis- sion from a close ionizing source (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=', a QSO, Geach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Overzier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2013), shocks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=', Taniguchi & Shioya 2000), or “cooling radiation” during gravita- tional collapse of the gas (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=', Haiman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The SPT2349−56 LAB (shown in figure 7) was originally hypothesized to be heated by some combination of the three ALMA sources that reside near or within it (Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Apostolovski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' However, given that the LAB center is only 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5′′ (31 kpc in projection) offset from ALMA source C, it could instead be heated by the radio- loud AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The LAB is centered on the weak SMG, N, which was originally identified through its [C ii] emis- sion (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' N is a luminous infrared galaxy A radio-loud AGN in SPT2349 17 23:49:43 42 56:38:15 20 25 30 35 RA Dec M N I B C G Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Does the radio-loud AGN power the LAB?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The background shows HST F160W imaging with ATCA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2 GHz contours (cyan) and the Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Apostolovski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=') MUSE Lyα contours (lime).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The center of the LAB lies 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5′′ (31 kpc) from the radio source centroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' ALMA 850µm contours are shown (coral), but sources M and N are too weak to see in this representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The LAB is centered near the SMG, N, which was originally identified through its [C ii] emission and undetected in continuum (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' N has S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='1mm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='18 mJy and an LIR = 4 × 1011 L⊙, with an implied SFR=35 M⊙ yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (LIRG) with S850 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='27±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='04 mJy, LFIR = 4 × 1011 L⊙, and a substantial M ∗ = 3 × 1010 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' It is a plausi- ble, but somewhat unlikely power source for the lumi- nous LAB (whose total luminosity is 3 × 1042 erg s−1, or 3 × 1035 W);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' source N fails to provide the necessary UV ionizing photons by at least a factor of ten, scaling from its meager R-band flux density of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='37 µJy (similar to the analysis in Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Apostolovski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We can directly estimate the AGN X-ray emission expected for powering the Ly-α blob following Overzier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2013), assuming that the fraction of ionizing photons that will cascade to Lyα is 68% (case B recombination).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' This number likely exceeds the actual amount of ionizing ra- diation available due to the absorption by dust by a fac- tor of around 10, which we account for here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We then assume a radio-quiet QSO spectrum given by Richards et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The predicted observed frame (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2–12 keV) X-ray luminosity would be 2 × 1037 W, which is compa- rable to the low end of the expected range of LX from the SPT2349−56 radio source, as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The radio AGN may therefore be at least as plausible a heat- ing source as N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Regarding the Lyα blob being spatially offset from the AGN position, we note that in the radio source B3 J2330 at z = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='1 (Matsuda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2009), the peak of the Lyα emission was also found to be similarly offset from the HzRG itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Even in the z = 4 Distant Red Core (DRC) LAB, there is a roughly 3′′ (21 kpc) offset from the X-ray-emitting AGN that is proposed as the LAB’s power source (Vito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' However, these are rare cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Venemans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2007) showed that gen- erally the AGN is very near the center of the Lyα halo, which grants some geometrical credence to the idea that the Lyα halo is ionized by the central AGN’s photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' In SPT2349−56, this is harder to argue, but the Lyα could be completely absorbed by the copious amounts of dust in the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The SPT pre-selection (as with the Herschel selection of the DRC) may favor finding sources with such offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' AGN fractions in protoclusters As described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='1, with 27 µJy RMS at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2 GHz, we are sensitive to moderately-luminous and heavily-obscured z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3 AGN among the 30 sub- millimeter galaxies identified in the SPT2349−56 struc- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' They need to lie approximately 5 times above the radio-FIR relation to be significantly (5σ) detected by ATCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' In GOODS-N (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 3), there are seven radio sources (all lacking submm detection) that satisfy this threshold, all of which lie at at redshifts less than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Another seven such radio sources lie 2–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5 times above the relation, extending to a redshift of about 4, which would not be detected by our observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The fact that all submm-detected sources in GOODS-N, and 74 of 76 SMGs in ALESS, are consistent with the radio- FIR relation does signify that radio-loud AGN are not common amongst the submm-luminous population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' No significant radio emission is found from any other (non- SMG) cluster members or candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' With our cur- rent radio depth, the radio-AGN content among SMGs in this protocluster is constrained to be less than 10% (three of 30 members), and most likely 3% (assuming C is the host of the ATCA radio source).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' However, the radio-loud AGN are only about 10% of the total AGN population in the field (Barger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Radcliffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The X-ray AGN fraction remains unconstrained, and given that many X-ray AGN are not radio emit- ters (Barger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2007), our AGN fraction estimates in SPT2349−56 are lower limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' In the z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='0 DRC protocluster (Oteo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2018), a central galaxy is radio-undetected, but is a Compton- thick X-ray AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Only one of the three X-ray-identified AGN is detected in the radio – DRC6 (S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5 = 128 µJy, S9 = 120 µJy), indicating a flat-spectrum source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' In this 18 Chapman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' case, the radio-AGN in the DRC lies towards the edge of the projected distribution of SMGs (offset from the core of the cluster).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Thus without X-ray data, we cannot tell if the total AGN fraction of SPT2349−56 is differ- ent from that in the DRC (23%, Vito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' As another example, in the core of the z = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='09 SSA22 pro- tocluster, the SMGs have a 50% X-ray AGN fraction, with four of eight SMGs detected by Chandra (Umehata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2019), significantly larger than the DRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Radio sources and cluster evolution Given that SPT2349−56 is conceivably the most mas- sive and active halo we know of at z > 4, an open ques- tion concerns the feedback or radio mode that this AGN is operating in, and how it is shaping the early core evo- lution of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' With the current data, having only the two photometric points characterizing the ra- dio emission, and not even localizing it uniquely to one galaxy, we cannot definitively address these issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Most radio-loud AGN appear to be hosted in recent or ongo- ing mergers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=', Ramos Almeida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Chiaberge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' In this light it may not be too surprising to find a radio-loud AGN in the core of SPT2349−56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Given that the radio luminosity of SPT2349−56 is mod- est for an HzRG, we may be seeing a radio-loud AGN fueled via radiatively-inefficient flows with low accretion rates (Best & Heckman 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' In this picture, the gas supplying the radio galaxy is frequently associated with hot X-ray halos surrounding massive galaxies, groups and clusters, as part of a radio-AGN feedback loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' This contrasts with more luminous radio sources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' TN J1338) thought to be fuelled at higher rates through radiatively efficient standard accretion disks by cold gas (Best & Heckman 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' These more luminous radio sources are hypothesized to have fuel brought in through mergers and interactions, which are in fact abundant in SPT2349−56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The debate thus remains open as to whether we are seeing a decaying radio source, or a radio source quickly building in luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' By better speci- fying the radio emission and its origin, we could learn about the build-up and state of the ICM that may al- ready be present at z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' CONCLUSIONS We have presented ATCA radio observations of SPT2349−56, a starbursting and gas-rich protoclus- ter, consisting of over 30 SMGs at z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We placed SPT2349−56 in context with µJy radio sources in the GOODS-N and ALESS fields, and with the other 22 gravitationally-lensed SPT SMGs also observed with ATCA in our program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We also studied in detail the cen- tral galaxies identified by ALMA in SPT2349−56 near this strong radio detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' – We detected a single source at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2 GHz in SPT2349−56, spatially coincident with the central three luminous members of the protocluster, denoted B, C, and G in Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' While the ATCA radio centroid lies close to source C, which has the largest stellar mass in the protocluster, we cannot rule out that the radio emission is coming from B or G, or even a combination of the galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Under any of the possibilities above, the 214 µJy flux density at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2 GHz translates to more than 20 times the radio luminosity expected from the FIR-radio correla- tion defined by star-forming galaxies, and suggests that an AGN is driving the radio emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The radio source has a steep spectrum, with an in- dex of α = − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='31, constrained by the ASKAP 888MHz detection, and the non-detections at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5 and 9 GHz, consistent with an AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' No other clear signs of AGN activity have yet been detected in this protocluster using any other diagnostics available to us (CO SLEDs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' EW(OH163µm), [C ii]/FIR ratios;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' optical spectra), highlighting the radio contin- uum as a powerful probe of obscured AGN in high-z protoclusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The three SMGs likely associated to the radio source have amongst the highest gas and dynamical mass of the protocluster members (Rotermund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' More- over, high resolution ALMA imaging resolves this sys- tem into multiple interacting, star-forming clumps, with a surrounding arc of [C ii] emission (Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Sulzanauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' This is consistent with the idea that the availability of large amounts of gas and galaxy interactions, both of which are enhanced in gas- rich overdensities at high redshift, can trigger fast and obscured SMBH accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' – No significant radio emission (nor any other robust AGN signature) is found from any other cluster member, constraining the radio-loud AGN content among SMGs in this protocluster to no more than 10% (three of 30 members), and likely just 3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' A radio stacking analy- sis on the remaining ten brightest SPT2349−56 SMGs finds (11±10) µJy, which is consistent with the average 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2 GHz emission from star formation via the FIR-radio correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We thus find no evidence that nuclear accre- tion powering radio emission exists below our detection threshold in other SMG members of SPT2349−56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' How- ever, radio-loud AGN represent only 10% of all AGN, and X-ray observations and JWST infrared spectroscopy would be the next key steps to constrain AGN in this system and compare to AGN fractions found in other protoclusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The SPT2349−56 radio-loud AGN has a luminos- ity density of L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4 × 1025 W Hz−1, extrapolating to A radio-loud AGN in SPT2349 19 L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4, rest = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3) × 1026 W Hz−1 with the measured α = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='6, which is still over two orders of magnitude less luminous than the powerful radio galaxies normally studied at these redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Many such HzRGs have rich protocluster environments, however it remains unclear if the opposite is true, that all massive z > 4 protoclusters have a central radio galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The fact that the radio AGN is detected in the hy- pothesized central seed of a growing BCG galaxy with significant stellar mass already in place makes this dis- covery an important new ingredient in understanding the formation and evolution of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The radio luminosity was used to infer a radio jet power of Pcav = (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='7) × 1038 W, sufficiently large as to provide a dominant feedback on the cooling gas in the 1013 M⊙ halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The radio luminosity also suggests a strong X-ray source with LX = 1038 W (integrated be- tween 2 and 10 keV), easily detectable by Chandra or XMM-Newton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' SPT2349−56 therefore has a high lumi- nosity AGN, even if in the form of a highly obscured quasar, and JWST will be a powerful tool to uncover its properties through high ionization infrared emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' ACKNOWLEDGEMENTS The Australia Telescope Compact Array is part of the Australia Telescope National Facility (https://ror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='org/05qajvd42), which is funded by the Australian Government for operation as a National Facility managed by CSIRO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The Australian SKA Pathfinder is part of the Australia Telescope National Facility (https://ror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='org/05qajvd42) which is managed by CSIRO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Operation of ASKAP is funded by the Australian Government with support from the Na- tional Collaborative Research Infrastructure Strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' ASKAP uses the resources of the Pawsey Supercom- puting Centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Establishment of ASKAP, the Murchi- son Radio-astronomy Observatory and the Pawsey Su- percomputing Center are initiatives of the Australian Government, with support from the Government of Western Australia and the Science and Industry En- dowment Fund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We acknowledge the Wajarri Yamatji people as the traditional owners of the Observatory site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The National Radio Astronomy Observatory is a facility of the National Science Foundation oper- ated under cooperative agreement by Associated Uni- versities, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' This paper makes use of the follow- ing ALMA data: ADS/JAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='ALMA#2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='01543.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='T, ADS/JAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='ALMA#2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='00058.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='S, and ADS/JAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='ALMA#2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='01010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' ALMA is a part- nership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Repub- lic of Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The Joint ALMA Observatory is op- erated by ESO, AUI/NRAO and NAOJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=', and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' gratefully acknowledge support for this re- search from NSERC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Manuel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' acknowledges support from FONDECYT grant 1211951, CONICYT + PCI + INSTITUTO MAX PLANCK DE ASTRONOMIA MPG190030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' acknowledge support from CONICYT + PCI + REDES 190194 and ANID BASAL project FB210003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=', Melanie A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' are supported by the Center for AstroPhysical Surveys at the National Center for Supercomputing Applications as an Illinois Survey Science Graduate Fellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' APPENDIX A In this appendix, we further assess the offsets between source centroids in the 888 MHz ASKAP image and the ATCA 2 GHz image that were discussed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='1 (Fig 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We measured peak fluxes in both images for all sources within a 13′ radius of SPT2349−56, measured their centroids, and calculated radial offsets for each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The offsets appear random in orientation, with the mean x and y offset being close to zero (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3′′, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 8 we plot the radial offsets versus ASKAP flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' SPT2349−56 shows the largest offset, which could indicate that its origin may be physical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' It has a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='6σ deviation from the median (excluding SPT2349−56) offset of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Even restricting the analysis to those sources with comparable flux densitites and SNRs (S888 < 2 mJy) only increases the median offset to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='6′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' APPENDIX B In this appendix and Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 9 – 11, we show the CO J = 7, 11, and 16 lines for the central B, C, and G sources, whose line strengths are plotted in the SLED diagram (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We also show the H2O lines that are used to compare with the FIR luminosity estimates from Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' In order to measure line strengths, the bright and well-detected [C ii] lines provided in Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2020) were used as a template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' These [C ii] lines were fit by single and double Gaussian profiles, and we selected the integration range 20 Chapman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 100 101 102 S(888MHz) (mJy) 1 2 3 4 5 Offset (arcsec) SPT2349 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Assessing the offsets between ASKAP and ATCA sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Top: The 888 MHz ASKAP image surrounding SPT2349−56 (green LABOCA contours, showing ALMA sources as green dots) with ATCA 2 GHz contours overlaid (26′ × 19′ field shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Bottom: The radial offsets between the centroids for sources common in both images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' SPT2349−56 shows the largest offset, which might therefore require a physical interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' It has a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='6σ deviation from the median (excluding SPT2349−56) offset of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4′′ offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 宣 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='A radio-loud AGN in SPT2349 21 by scaling the [C ii] profile to the rest frequency of the line of interest and then summing channels between −2σ and 2σ (where σ is the standard deviation of the best-fitting linewidth), or for cases where two Gaussians were a better fit, from −2σL to +2σR, where σL and σR are from the left and right Gaussian fits, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The CO(7–6) line is blended with the [C i](2–1) line, and the CO(16–15) line is blended with the OH doublet, so these had to be fit and subtracted before integrating over the CO lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' For the former case, where both CO(7–6) and [C i](2–1) are both well-detected in B and C, we simultaneously fit single Gaussian profiles at the locations of the two lines, then subtract the best-fit [C i](2–1) model from the spectrum, and sum over the relevant channels as described above (then vice-versa to obtain [C i](2–1) line strengths).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' For source G, we do not see any strong line features around the expected [C i](2–1) frequency, so we simply sum over the CO(7–6) and [C i](2–1) channels in the raw spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The CO(16–15) line is not well-detected for any sources but the OH doublet is, so we fit a Gaussian to these OH lines and subtract the models before summing over the CO(16–15) channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' In the fit we force the amplitude of each doublet component to be equal, and we fix the width of each doublet component to be equal to the width of the [C ii] line (described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' see Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Since the profile for G is two Gaussians, we include an additional OH doublet component of equal amplitude and fixed frequency separation/width to match the [C ii] profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' This leaves two free parameters in all fits: the frequency of the first doublet, and the amplitude of the all the components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 9 we can see that for source B the two OH doublet components are blended with each other due to the large FWHM of the system, and for G the four components blend into three peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Lastly, Band 4 and Band 6 continuum flux densities were estimated by averaging over all line-free channels in the original (non continuum-subtracted) data cubes (again using the [C ii] line as a template).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We combined channels from the lower and upper sideband of these observations, meaning they are at observed frequencies of 147 and 231 GHz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Band 7 continuum flux densities (around the CO(16–15) and OH lines) are already provided in Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' APPENDIX C Here we describe the ATCA observations of the full sample of 23 SPT-SMGs observed in the survey program, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' These SPT-SMGs were drawn from the complete sample of 81 sources (Reuter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2020), selecting those that had the best redshift constraints at the time of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' All but three of the 23 SPT-SMGs are detected at > 4σ significance at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2 GHz flux densities are measured at peak pixels (Table 5), as in all cases the sources are unresolved in the 8′′ × 5′′ beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The restored beam sizes and position angles are also listed in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' ALMA 850 µm overlays are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 12 (data from Spilker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Reuter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' LABOCA 850 µm fluxes and ALMA-derived redshifts from Reuter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2020) are also listed in Table 5 for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Most sources are not detected or only marginally detected at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5GHz (nine detections at > 3σ) and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='0 GHz (four detections at > 3σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' For those sources detected at these higher frequencies with ATCA, we measure flux densities from peak pixels when the source is unresolved, or as aperture measurements when the source is resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We show the nine sources detected at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5 GHz in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 13, the four sources detected at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='0 GHz in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We have also searched for detections in the ASKAP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='888 MHz RACS survey described in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2, listing their flux densities in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We find 15 of the 23 sources are significantly detected by ASKAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We derive radio spectral indices directly for all sources with at least two radio detections, and list these together with flux densities in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The data was fit according to a linear function using a Markov Chain Monte Carlo algorithm (MCMC) implemented by the emcee package (Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' This MCMC package samples the posterior probability function, and is used to determine the error contours shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2, as well as the uncertainties on α in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We summarize their radio spectral indices in in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 15 and Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The lensed SMGs are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 3, where we estimate their rest 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4 GHz luminosities directly using the measured α, or with α = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='8 if only detected at a single radio frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' In general these SPT-SMGs follow the same FIR-radio correlation as the other field samples shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' However, three sources are highly significant outliers from the FIR-radio correlation: SPT0125−50 at z =3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='96;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' SPT0202−61 at z = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='02;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' and SPT0550−53 at z = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Given how rare such strong outliers are in the field SMG samples (only one of 76 SMGs shows anywhere near this level of radio excess in the ALESS SMG sample – Thomson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2014), we propose that the lensing galaxy rather than SMG may be the more likely radio-AGN in these three cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' These radio excess sources exhibit steeper radio indices than typical star-forming galaxies, comparable to or exceeding SPT2349−56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Without knowing if the lens or source redshift is correct, we cannot reasonably apply the radio K-correction to estimate the rest 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4 GHz luminosity, and therefore we do not include these three in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 22 Chapman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' −1′′ 0′′ 1′′ ∆α −1′′ 0′′ 1′′ ∆δ 153-GHz continuum CO(7 − 6) −2000 −1000 0 1000 2000 ∆v[kms−1] 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='6 Sν [mJy] B (C3) [CI](2 − 1) CO(7 − 6) [CII] −1′′ 0′′ 1′′ ∆α −1′′ 0′′ 1′′ ∆δ 231-GHz continuum CO(11 − 10) −2000 −1000 0 1000 2000 ∆v[kms−1] 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='8 Sν [mJy] B (C3) CO(11 − 10) [CII] −1′′ 0′′ 1′′ ∆α −1′′ 0′′ 1′′ ∆δ 346-GHz continuum CO(16 − 15) −1000 0 1000 2000 ∆v[kms−1] 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2 Sν [mJy] B (C3) OH CO(16 − 15) [CII] −1′′ 0′′ 1′′ ∆α −1′′ 0′′ 1′′ ∆δ 141-GHz continuum H2O −2000 −1000 0 1000 2000 ∆v[kms−1] 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4 Sν [mJy] B (C3) H2O [CII] Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Cutouts and spectra of CO(7–6), CO(11–10), CO(16–15), and H2O line emission for galaxy B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The cutouts in each panel show continuum emission (obtained by averaging over all line-free channels) and line emission (obtained by averaging over all channels where the line is expected – see Section 2 for details), with contours starting at 2σ and increasing in steps of 3σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Apertures are shown as red circles, and used to obtain the spectra shown in the right panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' In each spectrum plot, we show the [C ii] profiles from Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' (2020), scaled to the expected frequency of the given line, and arbitrarily normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The shaded regions show the integration ranges (set to be ±2σ about the [C ii] line – see Section 2) used to obtain line strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The CO(7–6) line is blended with the [CI](2–1) line, and the expected central frequency (or for G, two central frequencies as the [C ii] profile has two components) of the [CI](2–1) is marked in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The [CI](2–1) line is fit by a Gaussian profile and subtracted, and the original spectra are shown by the dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Similarly, the CO(16–15) line is blended with the OH doublet, and we mark the mean frequency of each OH line in red (corresponsing to two frequencies for B and C, and four frequencies for G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The OH doublet is fit by a scaled [C ii] profile (see Section 2) and subtracted, and the original spectra are shown by the dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' In particular, SPT0550−53 shows an extended radio morphology/jet, well resolved in all three ATCA frequencies, which is more naturally explained by a lower redshift radio-loud galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Further, the optical spectrum of the lens SPT0550−53 shows AGN emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Neither SPT0125−50 nor SPT0202−62 show AGN signatures in their optical spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' SPT0125−50 is curious as the ATCA 2 GHz flux density is very close to that expected from the FIR- radio relation, however ASKAP reveals a 3 mJy source well centered on SPT0125−50, implying an incredibly steep α = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Thus any K-correction to lower rest frame frequencies than that probed by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2 GHz observations quickly places SPT0125−50 significantly above the FIR-radio correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' These results also beg the question of whether the radio emission in other lensed SPT-SMGs might be contaminated from the often massive lens galaxy (Rotermund 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' While in some examples, especially in SPT0538−50, the radio emission is directly identified as coming from the ALMA-detected lensed SMG components, in others the Einstein radius of the lensed source (Spilker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2016) is too small to be detected offset from the lens galaxy itself, even at 9 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The distribution in α constrained by the fits in figure 15 show a mean of −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='14, offset steeper, but still consistent, with the α = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='8 found in samples of unlensed SMGs reported in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=', Thomson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Several of the higher redshift sources in figure 3 do in fact show a marginal excess over that expected from the FIR-radio correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' This excess sometimes appears only due to the comparison shown at rest 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4 GHz, accentuating the K-correction from their steeper than average α we measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Given the large uncertainties from the often two-point α estimates, this may be inconsequential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Generally, optically identified AGN are relatively rare in the distant red galaxies that are often found to lens these SMGs (Rotermund 2020), and the gravitationally boosted radio signal associated with the high-SFR SMG is a more probable source of the strong radio emission we see in these 19 SPT-SMGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Their radio emission is not obviously contaminated by their foreground lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' A radio-loud AGN in SPT2349 23 −1′′ 0′′ 1′′ ∆α −1′′ 0′′ 1′′ ∆δ 153-GHz continuum CO(7 − 6) −2000 −1000 0 1000 2000 ∆v[kms−1] 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='6 Sν [mJy] C (C6) [CI](2 − 1) CO(7 − 6) [CII] −1′′ 0′′ 1′′ ∆α −1′′ 0′′ 1′′ ∆δ 231-GHz continuum CO(11 − 10) −2000 −1000 0 1000 1000 ∆v[kms−1] 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='8 Sν [mJy] C (C6) CO(11 − 10) [CII] −1′′ 0′′ 1′′ ∆α −1′′ 0′′ 1′′ ∆δ 346-GHz continuum CO(16 − 15) −1000 0 1000 2000 ∆v[kms−1] 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2 Sν [mJy] C (C6) OH CO(16 − 15) [CII] −1′′ 0′′ 1′′ ∆α −1′′ 0′′ 1′′ ∆δ 141-GHz continuum H2O −2000 −1000 0 1000 2000 ∆v[kms−1] 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4 Sν [mJy] C (C6) H2O [CII] Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 9 but for galaxy C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' −1′′ 0′′ 1′′ ∆α −1′′ 0′′ 1′′ ∆δ 153-GHz continuum CO(7 − 6) −2000 −1000 0 1000 2000 ∆v[kms−1] 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='6 Sν [mJy] G (C13) CO(7 − 6) [CII] −1′′ 0′′ 1′′ ∆α −1′′ 0′′ 1′′ ∆δ 231-GHz continuum CO(11 − 10) −2000 −1000 0 1000 2000 ∆v[kms−1] 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='8 Sν [mJy] G (C13) CO(11 − 10) [CII] −1′′ 0′′ 1′′ ∆α −1′′ 0′′ 1′′ ∆δ 346-GHz continuum CO(16 − 15) −1000 0 1000 2000 ∆v[kms−1] 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2 Sν [mJy] G (C13) OH CO(16 − 15) [CII] −1′′ 0′′ 1′′ ∆α −1′′ 0′′ 1′′ ∆δ 141-GHz continuum H2O −2000 −1000 0 1000 2000 ∆v[kms−1] 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='4 Sν [mJy] G (C13) H2O [CII] Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 9 but for galaxy G.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5, and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='0 GHz, as well as ASKAP at 888 MHz, of 23 SPT SMGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We also include unresolved LABOCA 850 µm flux densities (S850), ALMA redshifts, and best-fit spectral indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Name tint S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2 Beama PAb S† 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5 S† 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='0 S†† 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='9 S850 z αc Comment††† (hrs) (µJy) (′′×′′) (deg) (µJy) (µJy) (µJy) (mJy) SPT 0027−50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='58 334±31 8 × 4 106 <105 <150 1064±201 48 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='444 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='28±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='25 n n SPT 0103−45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='54 229±29 8 × 4 16 <111 <159 < 589 125 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='092 – n n SPT 0109−47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='56 1106±31 9 × 5 11 824±36 461±52 1417±194 109 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='614 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='42±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='04 yR yR SPT 0125−47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='59 586±35 9 × 5 112 382±36 193±49 1835±201 144 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='515 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='82±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='10 y mR SPT 0125−50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='56 365±29 5 × 7 110 <102 <156 2792±194 109 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='959 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='25±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='12 n n SPT 0202−61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='68 710±35 9 × 5 6 225±40 <153 1943±179 109 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='018 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='16±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='09 y n SPT 0245−63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='71 94±33 9 × 4 77 <99 <135 < 598 61 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='626 – n n SPT 0345−47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='68 275±29 8 × 6 174 <99 <132 701±194 89 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='296 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='03±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='38 n n SPT 0346−52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='72 162±38 5 × 9 68 <105 <132 < 603 131 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='656 – n n SPT 0418−47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='70 173±22 8 × 5 163 <93 <135 430±198 108 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='224 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='99±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='67 n n SPT 0512−59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='56 465±34 7 × 5 153 <177 <231 1531±197 75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='233 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='32±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='17 n n SPT 0529−54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='54 260±50 8 × 5 153 <120 <180 < 586 118 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='369 – n n SPT 0532−50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='57 489±39 8 × 5 154 144±49 <177 1093±231 118 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='399 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='06±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='16 m n SPT 0538−50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='56 581±36 8 × 5 157 341±59 168±58 1490±237 125 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='786 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='81±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='13 yR mR SPT 0550−53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='55 1288±48 9 × 6 169 446±39 270±56 4060±187 53 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='128 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='22±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='05 yR yR SPT 0551−50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='56 286±25 8 × 6 160 159±45 <171 520±185 74 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='164 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='70±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='24 m n SPT 2031−51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='48 269±31 9 × 5 51 <123 <186 721±203 65 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='452 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='08±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='39 n n SPT 2134−50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='53 334±47 8 × 4 33 174±43 <162 804±196 101 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='780 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='90±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='44 y n SPT 2319−55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='54 75±44 8 × 4 47 <126 <174 < 600 38 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='293 – n n SPT 2332−53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='56 244±23 9 × 5 36 146±41 <162 < 581 57 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='756 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='82±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='56 m n SPT 2349−56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='56 215±27 8 × 4 30 <120 <162 867±189 106d 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='303 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='58±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='31 n n SPT 2353−50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='56 24±53 9 × 5 30 <138 <159 < 594 41 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='576 – n n SPT 2357−51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='55 131±19 9 × 5 31 <108 <156 < 589 53 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='070 – n n a The 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2 GHz beam is quoted as x and y FHWM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5 GHz beam is typically 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='6′′ × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='0 GHz beam is typically 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2′′ × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' b The PA of the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2 GHz beam is the angle East of North.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' c The radio spectral index α, defined as Sν ∝ να.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' d In SPT2349−56 we have assumed source C with S850 µm = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='7 mJy is the host of the AGN, although it could be B or G as described in the text;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' here we still provide the unresolved LABOCA flux density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' † At 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='0 GHz, the 3σ upper limit is listed unless there is a detection at > 3σ at the ALMA position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' †† The 888 MHz measurements are from the ASKAP RACS survey, described in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Sources with < 3σ positive signal are listed at these limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' ††† Comments list whether 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5 GHz and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='0 GHz data show detections > 4σ (y), marginal detections (m) where < 4σ flux density is measured at the ALMA position, or no detection (n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' We indicate the four sources with resolved radio morphologies (R), in SPT0538−50 clearly following the ALMA emission, although in SPT0109−47 and especially SPT0550−53, the resolved emission appears to be an extended lobe or jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 28 Chapman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 0:27:08 07 06 05 50:07:05 10 15 20 25 30 RA Dec SPT0027 − 50 ATCA2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2GHz 1:03:12 11 45:38:40 45 50 55 39:00 05 RA Dec SPT0103 − 45 ATCA2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2GHz 1:09:51 50 49 47:02:00 05 10 15 20 25 RA Dec SPT0109 − 47 ATCA2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2GHz 1:25:08 07 06 47:23:45 50 55 24:00 05 10 RA Dec SPT0125 − 47 ATCA2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2GHz 1:25:50 49 48 47 50:38:10 15 20 25 30 35 RA Dec SPT0125 − 50 ATCA2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2GHz 2:03:00 00 02:58 57 61:21:00 05 10 15 20 25 RA Dec SPT0202 − 61 ATCA2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2GHz 2:45:46 45 44 43 42 63:20:25 30 35 40 45 50 RA Dec SPT0245 − 63 ATCA2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2GHz 3:45:12 11 10 47:25:25 30 35 40 45 50 RA Dec SPT0345 − 47 ATCA2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2GHz 3:46:42 41 40 52:04:50 55 05:00 05 10 15 RA Dec SPT0346 − 52 ATCA2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2GHz 4:18:41 40 39 47:51:40 45 50 55 52:00 05 RA Dec SPT0418 − 47 ATCA2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2GHz 5:13:00 12:58 57 59:35:30 35 40 45 50 55 RA Dec SPT0512 − 59 ATCA2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2GHz 5:29:04 03 02 54:36:30 35 40 45 50 55 RA Dec SPT0529 − 54 ATCA2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2GHz 5:32:52 51 50 50:46:55 47:00 05 10 15 20 RA Dec SPT0532 − 50 ATCA2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2GHz 5:38:18 17 16 15 50:30:40 45 50 55 31:00 05 RA Dec SPT0538 − 50 ATCA2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2GHz 5:50:02 01 00 00 53:56:30 35 40 45 50 55 RA Dec SPT0550 − 53 ATCA2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2GHz 5:51:41 40 39 38 50:57:50 55 58:00 05 10 15 RA Dec SPT0551 − 50 ATCA2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2GHz 20:31:00 00 30:58 51:12:15 20 25 30 35 40 RA Dec SPT2031 − 51 ATCA2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2GHz 21:34:04 03 02 50:13:15 20 25 30 35 40 RA Dec SPT2134 − 50 ATCA2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2GHz 23:19:23 22 21 20 55:57:45 50 55 58:00 05 10 RA Dec SPT2319 − 55 ATCA2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2GHz 23:32:28 27 26 25 53:58:25 30 35 40 45 50 RA Dec SPT2332 − 53 ATCA2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2GHz 23:49:44 43 42 56:38:15 20 25 30 35 40 RA Dec SPT2349 − 56 ATCA2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2GHz 23:57:18 17 16 51:53:40 45 50 55 54:00 05 RA Dec SPT2357 − 51 ATCA2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2GHz 23:53:40 39 38 50:09:55 10:00 05 10 15 20 RA Dec SPT2353 − 50 ATCA2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2GHz Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' ATCA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='2 GHz images (30′′ × 30′′) of the 23 SPT-SMGs observed, with ALMA 850 µm contours overlaid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' A radio-loud AGN in SPT2349 29 1:09:51 50 49 47:02:00 05 10 15 20 25 RA Dec SPT0109 − 47 ATCA5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5GHz 1:25:08 07 06 47:23:45 50 55 24:00 05 10 RA Dec SPT0125 − 47 ATCA5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5GHz 2:03:00 00 02:58 57 61:21:00 05 10 15 20 25 RA Dec SPT0202 − 61 ATCA5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5GHz 5:32:52 51 50 50:46:55 47:00 05 10 15 20 RA Dec SPT0532 − 50 ATCA5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5GHz 5:38:18 17 16 15 50:30:40 45 50 55 31:00 05 RA Dec SPT0538 − 50 ATCA5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5GHz 5:50:02 01 00 00 53:56:30 35 40 45 50 55 RA Dec SPT0550 − 53 ATCA5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5GHz 5:51:41 40 39 38 50:57:50 55 58:00 05 10 15 RA Dec SPT0551 − 50 ATCA5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5GHz 21:34:04 03 02 50:13:15 20 25 30 35 40 RA Dec SPT2134 − 50 ATCA5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5GHz 23:32:28 27 26 25 53:58:25 30 35 40 45 50 RA Dec SPT2332 − 53 ATCA5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5GHz Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' ATCA 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='5 GHz images (30′′ × 30′′) of the nine detected SPT-SMGs, with ALMA 850 µm contours overlaid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 1:09:51 50 49 47:02:00 05 10 15 20 25 RA Dec SPT0109 − 47 ATCA9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='0GHz 1:25:08 07 06 47:23:45 50 55 24:00 05 10 RA Dec SPT0125 − 47 ATCA9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='0GHz 5:38:18 17 16 15 50:30:40 45 50 55 31:00 05 RA Dec SPT0538 − 50 ATCA9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='0GHz 5:50:02 01 00 00 53:56:30 35 40 45 50 55 RA Dec SPT0550 − 53 ATCA9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='0GHz Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' ATCA 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='0 GHz images (30′′ × 30′′) of the four detected SPT-SMGs, with ALMA 850 µm contours overlaid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 30 Chapman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' 100 101 Frequency (GHz) 102 103 Flux (uJy) SPT0027-50 = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='283 +/- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='246 100 101 Frequency (GHz) 103 4 × 102 6 × 102 2 × 103 Flux (uJy) SPT0109-47 = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='425 +/- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='042 100 101 Frequency (GHz) 103 Flux (uJy) SPT0125-47 = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='825 +/- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='099 100 101 Frequency (GHz) 101 102 103 104 Flux (uJy) SPT0125-50 = -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='248 +/- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='119 100 101 Frequency (GHz) 102 103 Flux (uJy) SPT0202-61 = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='155 +/- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='089 100 101 Frequency (GHz) 102 103 Flux (uJy) SPT0345-47 = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='031 +/- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='377 100 101 Frequency (GHz) 101 102 103 Flux (uJy) SPT0418-47 = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='994 +/- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='67 100 101 Frequency (GHz) 102 103 Flux (uJy) SPT0512-59 = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='32 +/- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='167 100 101 Frequency (GHz) 102 103 Flux (uJy) SPT0532-50 = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='061 +/- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='16 100 101 Frequency (GHz) 102 103 Flux (uJy) SPT0538-50 = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='814 +/- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='131 100 101 Frequency (GHz) 103 104 Flux (uJy) SPT0550-53 = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='223 +/- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='048 100 101 Frequency (GHz) 102 103 Flux (uJy) SPT0551-50 = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='704 +/- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='238 100 101 Frequency (GHz) 102 103 Flux (uJy) SPT2031-51 = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='084 +/- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='387 100 101 Frequency (GHz) 102 103 104 Flux (uJy) SPT2134-50 = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='902 +/- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='443 100 101 Frequency (GHz) 101 102 103 104 Flux (uJy) SPT2332-53 = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='825 +/- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content='567 Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Radio spectral indices with errors constrained by MCMC modelling for gravitatinally lensed SPT-SMGs having at least two detections between the ATCA and ASKAP followup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' Grey shaded regions show the 1 and 2σ uncertainties on α derived from the ATCA data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The brighter sources with steeper spectra are generally detected by the ASKAP RACS survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtAzT4oBgHgl3EQfavzP/content/2301.01375v1.pdf'} +page_content=' The fit for SPT2349−56 is shown in Figure 2.' metadata={'source': 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Yu1, Shun Liang1, Feng Li1, Yanpeng Zhang1, +Min Xiao2,3 and Zhaoyang Zhang1,‡ +1Key Laboratory for Physical Electronics and Devices of the Ministry of Education & Shaanxi Key Lab of Information +Photonic Technique, School of Electronic Science and Engineering, Faculty of Electronic and Information +Engineering, Xi’an Jiaotong University, Xi’an, 710049, China +2Department of Physics, University of Arkansas, Fayetteville, Arkansas, 72701, USA +3National Laboratory of Solid State Microstructures and School of Physics, Nanjing University, Nanjing 210093, +China + +Non-Hermitian systems with complex-valued energy spectra provide an extraordinary +platform for manipulating unconventional dynamics of light. Here, we demonstrate the +localization of light in an instantaneously reconfigurable non-Hermitian honeycomb +photonic lattice that is established in a coherently-prepared atomic system. One set of the +sublattices is optically modulated to introduce the absorptive difference between +neighboring lattice sites, where the Dirac points in reciprocal space are extended into +dispersionless local flat bands. When these local flat bands are broad enough due to larger +loss difference, the incident beam is effectively localized at one set of the lattices with +weaker absorption, namely, the commonly seen power exchange between adjacent +channels in photonic lattices is effectively prohibited. The current work unlocks a new +capability from non-Hermitian two-dimensional photonic lattices and provides an +alternative route for engineering tunable local flat bands in photonic structures. + +Loss is usually considered to be detrimental for wave manipulations as it leads to energy decay. +However, since the introduction of non-Hermitian physics within the context of quantum field theories +into optical and other systems, the combination of dispersion and loss in desired manners has inspired +considerable new phenomena, particularly, the unusual wave dynamical characteristics arising from the +complex-valued eigenvalue spectra [1]-7], such as non-Hermitian topological properties [8-10], +non-Hermitian skin effect [11,12], lasing and optical sensing [13,14], etc. Most of the counterintuitive +features in non-Hermitian optical settings [15-20] are closely related to the well-known exceptional +points [21,22], around which the complex eigenvalues or band structures undergo bifurcation and an +abrupt phase transition occurs. To date, the studies of non-Hermitian potentials have been extended to +high-dimensional framework, where exceptional rings emerge in momentum space, inside which local +flat bands become possible [18,19,23,24]. These pioneering works mainly focused on the exotic wave + +dynamics around exceptional rings by manipulating the degree of non-Hermiticity. +Recently, local flat bands inside exceptional rings are also theoretically proposed to govern the +behaviors of light in distinct manners, for example, chiroptical polarization response [25]. Expanding +the research interests from the vicinity of exceptional rings to the ring-surrounded areas will further +enrich the capabilities and applications of high-dimensional non-Hermitian systems. Nevertheless, the +experimental exploration on the underlying properties of the areas that surrounded by exceptional rings +is rarely reported. +One intrinsic property of the dispersionless flat band is to realize field localization. In general, the +localization of optical field in periodic photonic structures is achieved by taking advantage of energy +flat bands in the whole Brillouin zone, such as the localizations in one dimensional photonic chain [26], +photonic Lieb lattices [27,28], and moiré superlattices [29], among others. This is because the +non-dispersive flat band guarantees zero group velocity, therefore the wave package will propagate in a +dispersionless manner through media. In such a case, however, it is not necessary to have an entire flat +band for the field localization, as the Fourier components of the wave package usually cover part of the +momentum space, rather than the whole k space. +In this paper, we experimentally demonstrate the field localization in a non-Hermitian honeycomb +photonic lattice exhibiting local flat bands (around both K and K points) that are induced by the loss +difference between the two sublattice sets. The photonic lattice is constructed in a four-level N-type +85Rb atomic vapor cell with electromagnetically induced transparency (EIT) [30]. The instantaneous +reconfigurability of such photonic lattice supports the demonstration of the wave behaviors under +different parametric configurations. The honeycomb photonic lattice with the same complex refractive +index for all lattice sites is “written” by a coupling field with a hexagonal intensity distribution under +EIT in a three-level configuration [31-33]. The incident Gaussian probe beam is discretized into the +honeycomb geometry by the induced lattice, and the light in two neighboring channels (corresponding +to A and B sites) can experience power exchange due to evanescent coupling. With the introduction of +another one-dimensional (1D) periodic pump field to cover one set of sublattice [for example, the A +sublattice, see Figs. 1(a) and 1(c)] in the induced photonic structure, a four-level system is formed at the +covered regions. Namely, the susceptibilities of A and B sublattices are resulted from the interactions +between the probe beam and four- and three-level atomic configurations, respectively, leading to +different transmission characteristics for the light inside corresponding waveguide channels. When the +difference between the losses of A and B sites is large enough by changing the detuning of the 1D field, +there emerge the local flat band surfaces which can localize the fields at one set of sublattice, +suppressing the energy exchange between neighboring lattice sites. Such absorption-difference-induced +flat band surface exhibits the similar zero-dispersion property as commonly seen in flat band structures +such as Lieb lattices [27,28,34]. + + +FIG. 1. (a) Sketch of the experimental principle. (b) The four-level 85Rb atomic configuration. (c) The spatial +arrangements of the 1D periodic pump field E3 and the induced honeycomb lattice inside the medium. (d-e) The +observed output patterns of the probe field without and with A sublattice covered by the pump field, respectively. +The sketch of the experimental principle is shown in Fig. 1(a), where the honeycomb refractive index +distribution inside the atomic vapor cell is optically induced by a coupling field E2 with a hexagonal +intensity profile (by interfering three coupling beams) under EIT. The lattice sites with different colors +correspond to the A and B sublattices in the honeycomb photonic lattice. A Gaussian probe field E1 is +launched into the atomic vapor cell from the same side as E2 to excite one K valley, while the directions +of the three coupling beams identify three K points in the reciprocal space. A 1D periodic pump field +E3 by interfering two pump beams is injected to cover the A sublattice. More details about the +experimental settings are given in Supplemental Material [35]. +As given in Fig. 1(b), the probe field E1 together with the coupling field E2 drives a three-level +atomic configuration [1 (5S1/2, F2)3 (5P1/2)2 (5S1/2, F3)], in which an EIT window can be +effectively created to enhance the refractive index felt by the probe field. The pump field E3 drives the +transition of 14 (5P3/2) to excite an N-type four-level atomic configuration (only at A sites), which +can lead to different absorption for the two sets of sublattice. Detuning ∆� (� = 1, 2 and 3) in the +energy-level diagram represents the difference between the laser frequency and the frequency gap of the +two levels connected by Ei. The spatial beam arrangement of the pump field and honeycomb photonic +lattice is shown in Fig. 1(c). For the three-level EIT case without the 1D pump field, the resulted +refractive index is inversely related to the coupling-field intensity under certain range of frequency +detuning. With the two-photon detuning set as ∆�−∆�>0, the reconfigurable non-Hermitian photonic +lattice with honeycomb real and imaginary parts is effectively established [36]. The observed +transmitted probe pattern (at the output surface of the cell and captured by a charge coupled device +camera) under the condition of uniform susceptibility for both sublattices is shown in Fig. 1(d), which is +clearly a discrete pattern in the honeycomb geometry. With the pump field turned on to modulate the +imaginary part of A sublattice, the transmitted probe intensity [Fig. 1(e)] at A sites (marked by the blue + +(a) +(b)5P[3> +5P1/2E3 +E +E22 +5S1/2 +F=3XBdashed circles) is much weaker due to the larger imaginary part (representing stronger absorption). + +FIG. 2 (a)-(f) Detected field patterns by changing the pump detuning Δ� from 120 MHz to 20 MHz. The +two-photon detuning is fixed at Δ� − Δ�25 MHz; (g) The calculated imaginary parts of the susceptibility at A and B +sites from the density matrix method. The six symbols correspond to the values of detuning in (a-f). +One advantage of the photonic lattices based on atomic coherence is the easily accessible +tunability inherited from the multi-level atomic configuration. Here, we show that loss difference +between A and B sites of the constructed non-Hermitian honeycomb lattice can be reconfigured by +adjusting the laser parameters. Figures 2(a-f) demonstrate the evolution of the output intensity at A +(blue circles) and B (white circles) sublattices versus the detuning of the pump field (which only affects +the susceptibility of A sites) with other parameters fixed. With the pump detuning ∆� discretely tuned +from 120 MHz to 20 MHz, the experienced absorption in A sites becomes stronger, resulting in a +decrease of the intensity inside blue circles, while the intensity at uncovered B sites governed by a +three-level configuration exhibits no obvious change. These observed results clearly demonstrate the +increase of the loss difference between A and B sublattices with decreasing ∆�. +The interaction between the probe beam and the four-level atomic configuration can be described +by using the density matrix method with rotating-wave approximation [35, 37]. According to the +density matrix equations, the loss parameters ��,� = Im���,�� at the A and B sites are plotted in Fig. +2(g), with symbols denoting the detunings corresponding to Fig. 2(a-f). As one can see, when ∆� is +tuned from 120 MHz to 20 MHz , �� keeps constant while �� increases, resulting in the +enlargement of loss difference Δ� = �� − ��. The estimated loss difference values of Fig. 2(a-f) are + +A.-60N- +A.-20N1eA3-120MH1Z +M1n +△3-100MHZ +△3-80MHZt N.0DD +(eMax(np)1.6F1003�� = 0.49, 0.62, 0.88, 1.33, 2.27, and 4.59 (× 10��) , respectively. This result advocates the +instantaneous reconfigurability of the established non-Hermitian honeycomb lattice, which in turn +provides abundant space to govern the beam dynamics under various experimental conditions. + +FIG. 3. Band structures and eigenstates of the non-Hermitian honeycomb lattice system. (a, b) Bands of Re[�] +according to Eq. (2) for Δ� = 0 (a) and Δ� = 8.5 (b) with � = 12. �� and (�� + ��)/2 can be set arbitrarily as +they only shift the bands up or down. (c) Boundaries of the local flat band surfaces for different Δ�. The inset shows +the first Brillouin zone. (d, e) Calculated band structures (solid lines) for the synthesized honeycomb lattice with Δ� = +0 (d) and Δ� = 4.1 × 10�� (e), together with representative eigenstates (insets). The dashed lines are the +corresponding bands in Eq. (2) from the analytic tight-binding model for comparison. (f) Evolutions of the local flat +band around one K point by increasing loss difference ��. The inset shows the localized field pattern when the local +flat band (loss difference) is large enough for field localization. +Such a non-Hermitian honeycomb photonic lattice system can be described by the effective +tight-binding model with Hamiltonian [38,39]: + � = � +�� +� ∑ +�� �∙�� +� +� ∑ +��� �∙�� +� +�� +�, (1) +where ��,� = �� − ���,� are the normalized complex refractive indices at A and B sites, � is the +normalized coupling strength between neighboring sites, � = ���, ��� is the two-dimensional +wavenumber, and ��,�,� = (1,0), �− +� +� , √� +� � , �− +� +� , − √� +� � are the normalized coupling directions on the +� − � plane. The eigenvalues of this Hamiltonian, which characterize the propagation properties of the +probe beam inside the system, are found to be +��,� = �� − +� +� �(�� + ��) ± ��3 − � +�� +��� +� ++ 2 �cos +��� +� cos +√��� +� ++ cos�√3 ���� . (2) +When there is no loss difference (Δ� = �� − �� = 0), such eigenvalues can be visualized in the +band structure shown in Fig. 3(a). It is obvious that they degenerate at the Dirac points (K and K points, + +△y=2.76x10-60XheMRe(B +Re(B)A2=1.28△y=14.5Miax +iax +M +1.26△y=18.5M +K +1T +M +0.0 +ky +1.2 +1.0X +L +OL +1k. +0 +2 +3X10 +1.36 +d +· +△V=0real part of the eigenvalues) and locally show linear dispersion properties. However, when loss +difference is introduced(Δ� ≠ 0), the point degeneracy becomes circular-surface-like degeneracy and +the original linear dispersion evolves into local dispersionless flat band around the K and K points. This +can be seen in the band structure of Re[�] in Fig. 3(b) for Δ� = 8.5 and � = 12(normalized +parameters). The corresponding bands of Im[�] are shown in Fig. S2(a) in Supplemental Material [35]. +Such a local flat band surface can localize the incident light due to locally zero group velocity. +Moreover, in Fig. 3(c), we plot the boundaries of the local flat bands for different Δ�. It is clear that +with the increase of Δ�, the local flat band surfaces become broader, enabling better localization +performance. +The formation of the local flat band under non-zero loss difference can be understood from the +Dirac Hamiltonian approximation around the K and K points. By taking the Taylor series expansion of +Eq. (1), one can obtain +�� = ��� − ��� +������ +������� +�� − ��� +�, (3) +where �� = 3�/2, � is the amplitude of the wavenumber reference to K (K ) point, and � is the local +direction angle. The eigenvalues of this Dirac Hamiltonian are +��,� = �� − +� +� �(�� + ��) ± ����� − � +�� +���� +� +. (4) +Now, the degeneracy condition �� − (Δ�/2��)� = 0 is dependent on the amplitude of the local +wavenumber �, indicating that the degenerate point evolves to an exceptional ring [23,40]. In addition, +the size of the ring is linearly proportional to the loss difference Δ�. As the local flat band (real part of +the eigenvalues) is inside the exceptional ring, the size of the local flat band increases proportionally to +the magnitude of Δ�, as verified by Fig. 3(c). +We calculate the band structure of the synthesized photonic lattice with honeycomb susceptibility +distributions using the PDE module of COMSOL Multiphysics and the results are shown as the solid +lines in Fig. 3(d, e). One can see that the Dirac cone at K point for Δ� = 0 becomes local flat band +surfaces for Δ� = 4.6´10-�,agreeing with the analytic results (dashed lines) of Eq. (2) from the +tight-binding model, especially in the vicinity of K points. We note that the minor difference between +bandstructures close to Γ point based on the two methods is due to the simplicity of the symmetric +tight-binding model. To better demonstrate the formation of the local flat band surfaces, in Fig. 3(f), we +plot the bands around K point for four different Δ�. With the increase of the Δ� value, the local flat +band expands broader. Also, we show representative eigenstates in the inset of Fig. 3(d, e). It is worth +noting that, with the increase Δ�, the local flat bands not only emerge and expand, but also share the +similar eigenstates for each band. Particularly, for the eigenstates of the upper (lower) local flat band, +the field is localized at B (A) sites with low (high) loss. When a probe beam is obliquely incident into +the system to excite the K vicinity, the eigenstates of both the local flat bands will be excited. However, + +since the B lattice sites have lower loss than the A sites, the eigenstates of field concentrated at A sites +(lower band) will decay much faster and finally only the upper band survives, resulting in the field +localization at B sites. The inset of Fig. 3(f) shows the simulated results of the localized field pattern +when the probe beam propagates for a certain distance in the synthetic honeycomb photonic lattice with +relatively large loss difference. The probe beam will keep this localized profile when propagating inside +the lattice but experience the amplitude decay due to the loss. + +FIG. 4. Observed energy exchanges between sublattices A and B during the propagation of probe under different pump +parameters: (a) without the 1D pump field E3, (b) Δ� = 120 MHz, (c) Δ� = 40 MHz, and (d) Δ� = 20 MHz. The +squares and circles are the measured proportions of the transmitted intensities at A and B sites, respectively, while the +corresponding solid curves represent mathematic fitting based on the experimental measurements and provide a guide +to the eye. +In experiment, the field localization can then be verified by examining the energy exchange +between A and B sites. Under the condition of a large enough loss difference, the field can be localized +at B sites (three-level atomic region) and in principle there should be no energy transportation between +A and B sites. While the field fails to be localized under the condition of a relatively small loss +difference, the energy at A and B sites will couple to each other and resulting in energy fluctuation. In +the following, we show the energy exchange between two adjacent waveguides (corresponding to A and +B sublattices) at different Δ� in the experiment. One advantage of atomic media is that increasing the +atomic density (positively related to the temperature of the atomic vapor) is understood as extending the +effective propagating distance of the probe beam inside the photonic lattice [31,41]. With the pump +detuning adjusted to set different imaginary parts of the refractive index of the four-level atomic regions +(A sites), the transport dynamics of the probe beam are captured in Fig. 4 when the medium is heated +from 120 ℃ to 165 ℃. Actually, considering that the susceptibility is proportional to the atomic +density N for either the three- or four-level configuration [Eq. (S2) in the Supplemental Material [35], +higher temperature can cause larger imaginary part as well as larger loss difference Δ�, which will +result in weaker transmission but help to improve performance of the field localization. The relatively + +Fitting Curve for IB/(IA+IB)B +0.60.4120 +160 +120 +140 +1600.8 +(b) +(a) +Measured IA/(IA+IB) +Measured IB/(IA+IB) +0.6transmitted intensities �� and �� at A and B sites are measured by the software of the charge coupled +device camera and their proportions are represented by symbols in Fig. 4. +When the pump field is absent, all the sites on the honeycomb lattice possess the same +susceptibility controlled by a three-level configuration. In this case, the loss difference is zero +(Δ� = 0) and the commonly seen power exchange between neighboring waveguide channels is +observed in Fig. 4(a), where the initial intensities �� and �� are very close to each other. During the +propagation, one can see clearly twice energy exchanges indicated by the two points where �� ≈ ��. +With the small increase of Δ� by setting the pump detuning at ∆3 = 120 MHz [see Fig. 2(g)], as +shown in Fig. 4(b), the frequency of power exchange between A and B sites is reduced to only once at +~155℃, which means the energy exchange requires a longer propagating distance than the case without +the pump field. The slowing down of the power exchange indicates the formation of the local flat band +surface but still a limited area, which cannot fully localize the beam energy. With the pump detuning +decreased to ∆3 = 40 MHz [Fig. 4(c)], the difference between the imaginary parts is further enlarged +and a broader local flat band surface occurs. As a consequence, one can see that there is no power +exchange in the given propagation distance. For the very large loss difference shown in Fig. 4(d) with +∆� = 20 MHz, the ratio between the transmitted intensities �� and �� reaches ~1:4. Since the +generated local flat band surface have a large enough area and can confine the energy mainly into B +sublattices. Correspondingly, we simulate the beam propagation dynamics in the synthetic honeycomb +photonic lattice with the susceptibility distribution at the same order of magnitude as the experiment +[42]. The simulation results of the energy exchange depending on normalized propagating distance are +plotted in Fig. S3, which shows very similar energy evolution behaviors as the experimental results. +In conclusion, we experimentally demonstrate the field localization behavior by switching the +band structures from Dirac points with linear dispersion to dispersionless local flat band surfaces in a +reconfigurable non-Hermitian honeycomb photonic lattice that induced in a multi-level atomic vapor +cell. Such switching is derived from the induced loss difference between A and B sublattices, whose +susceptibility are controlled by four- and three-level atomic configurations, respectively. By easily +setting the laser parameters, the imaginary parts as well as the sizes of the formed local flat band +surface are effectively manipulated to govern the beam dynamics in individual waveguide channel. The +occurrence of the local flat band surface is verified by the vanishing of power exchange, indicating the +light is localized by the flat-band modes. Being different from the phenomena relying on the degree of +non-Hermiticity in previous works involving exceptional rings [19, 23-25], the localization of optical +fields requires only the proper manipulation of loss difference. Namely, the localization behavior arising +from loss difference is a result of the local flat band with shared eigenstates, and does not depend on the +exceptional points. 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Res. 2, 401 +(2022). + + diff --git a/ltE3T4oBgHgl3EQf6Avi/content/tmp_files/load_file.txt b/ltE3T4oBgHgl3EQf6Avi/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f1e47e244538d714666a24452682d93d46edd09e --- /dev/null +++ b/ltE3T4oBgHgl3EQf6Avi/content/tmp_files/load_file.txt @@ -0,0 +1,675 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf,len=674 +page_content='Loss-difference-induced localization in a non-Hermitian honeycomb photonic lattice Yuan Feng1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content='*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Zhenzhi Liu1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content='*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Fu Liu1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content='†,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Jiawei Yu1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Shun Liang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Feng Li1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Yanpeng Zhang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Min Xiao2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content='3 and Zhaoyang Zhang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content='‡ 1Key Laboratory for Physical Electronics and Devices of the Ministry of Education & Shaanxi Key Lab of Information Photonic Technique,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' School of Electronic Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Faculty of Electronic and Information Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Xi’an Jiaotong University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Xi’an,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' 710049,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' China 2Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' University of Arkansas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Fayetteville,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Arkansas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' 72701,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' USA 3National Laboratory of Solid State Microstructures and School of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Nanjing University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Nanjing 210093,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' China Non-Hermitian systems with complex-valued energy spectra provide an extraordinary platform for manipulating unconventional dynamics of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Here, we demonstrate the localization of light in an instantaneously reconfigurable non-Hermitian honeycomb photonic lattice that is established in a coherently-prepared atomic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' One set of the sublattices is optically modulated to introduce the absorptive difference between neighboring lattice sites, where the Dirac points in reciprocal space are extended into dispersionless local flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' When these local flat bands are broad enough due to larger loss difference, the incident beam is effectively localized at one set of the lattices with weaker absorption, namely, the commonly seen power exchange between adjacent channels in photonic lattices is effectively prohibited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' The current work unlocks a new capability from non-Hermitian two-dimensional photonic lattices and provides an alternative route for engineering tunable local flat bands in photonic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Loss is usually considered to be detrimental for wave manipulations as it leads to energy decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' However, since the introduction of non-Hermitian physics within the context of quantum field theories into optical and other systems, the combination of dispersion and loss in desired manners has inspired considerable new phenomena, particularly, the unusual wave dynamical characteristics arising from the complex-valued eigenvalue spectra [1]-7], such as non-Hermitian topological properties [8-10], non-Hermitian skin effect [11,12], lasing and optical sensing [13,14], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Most of the counterintuitive features in non-Hermitian optical settings [15-20] are closely related to the well-known exceptional points [21,22], around which the complex eigenvalues or band structures undergo bifurcation and an abrupt phase transition occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' To date, the studies of non-Hermitian potentials have been extended to high-dimensional framework, where exceptional rings emerge in momentum space, inside which local flat bands become possible [18,19,23,24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' These pioneering works mainly focused on the exotic wave dynamics around exceptional rings by manipulating the degree of non-Hermiticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Recently, local flat bands inside exceptional rings are also theoretically proposed to govern the behaviors of light in distinct manners, for example, chiroptical polarization response [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Expanding the research interests from the vicinity of exceptional rings to the ring-surrounded areas will further enrich the capabilities and applications of high-dimensional non-Hermitian systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Nevertheless, the experimental exploration on the underlying properties of the areas that surrounded by exceptional rings is rarely reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' One intrinsic property of the dispersionless flat band is to realize field localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' In general, the localization of optical field in periodic photonic structures is achieved by taking advantage of energy flat bands in the whole Brillouin zone, such as the localizations in one dimensional photonic chain [26], photonic Lieb lattices [27,28], and moiré superlattices [29], among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' This is because the non-dispersive flat band guarantees zero group velocity, therefore the wave package will propagate in a dispersionless manner through media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' In such a case, however, it is not necessary to have an entire flat band for the field localization, as the Fourier components of the wave package usually cover part of the momentum space, rather than the whole k space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' In this paper, we experimentally demonstrate the field localization in a non-Hermitian honeycomb photonic lattice exhibiting local flat bands (around both K and K\uf0a2 points) that are induced by the loss difference between the two sublattice sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' The photonic lattice is constructed in a four-level N-type 85Rb atomic vapor cell with electromagnetically induced transparency (EIT) [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' The instantaneous reconfigurability of such photonic lattice supports the demonstration of the wave behaviors under different parametric configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' The honeycomb photonic lattice with the same complex refractive index for all lattice sites is “written” by a coupling field with a hexagonal intensity distribution under EIT in a three-level configuration [31-33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' The incident Gaussian probe beam is discretized into the honeycomb geometry by the induced lattice, and the light in two neighboring channels (corresponding to A and B sites) can experience power exchange due to evanescent coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' With the introduction of another one-dimensional (1D) periodic pump field to cover one set of sublattice [for example, the A sublattice, see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' 1(a) and 1(c)] in the induced photonic structure, a four-level system is formed at the covered regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Namely, the susceptibilities of A and B sublattices are resulted from the interactions between the probe beam and four- and three-level atomic configurations, respectively, leading to different transmission characteristics for the light inside corresponding waveguide channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' When the difference between the losses of A and B sites is large enough by changing the detuning of the 1D field, there emerge the local flat band surfaces which can localize the fields at one set of sublattice, suppressing the energy exchange between neighboring lattice sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Such absorption-difference-induced flat band surface exhibits the similar zero-dispersion property as commonly seen in flat band structures such as Lieb lattices [27,28,34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' (a) Sketch of the experimental principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' (b) The four-level 85Rb atomic configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' (c) The spatial arrangements of the 1D periodic pump field E3 and the induced honeycomb lattice inside the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' (d-e) The observed output patterns of the probe field without and with A sublattice covered by the pump field, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' The sketch of the experimental principle is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' 1(a), where the honeycomb refractive index distribution inside the atomic vapor cell is optically induced by a coupling field E2 with a hexagonal intensity profile (by interfering three coupling beams) under EIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' The lattice sites with different colors correspond to the A and B sublattices in the honeycomb photonic lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' A Gaussian probe field E1 is launched into the atomic vapor cell from the same side as E2 to excite one K valley, while the directions of the three coupling beams identify three K\uf0a2 points in the reciprocal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' A 1D periodic pump field E3 by interfering two pump beams is injected to cover the A sublattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' More details about the experimental settings are given in Supplemental Material [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' As given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' 1(b), the probe field E1 together with the coupling field E2 drives a three-level atomic configuration [\uf07c1\uf0f1 (5S1/2, F\uf03d2)\uf0ae\uf07c3\uf0f1 (5P1/2)\uf0ae\uf07c2\uf0f1 (5S1/2, F\uf03d3)], in which an EIT window can be effectively created to enhance the refractive index felt by the probe field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' The pump field E3 drives the transition of \uf07c1\uf0f1\uf0ae\uf07c4\uf0f1 (5P3/2) to excite an N-type four-level atomic configuration (only at A sites), which can lead to different absorption for the two sets of sublattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Detuning ∆� (� = 1, 2 and 3) in the energy-level diagram represents the difference between the laser frequency and the frequency gap of the two levels connected by Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' The spatial beam arrangement of the pump field and honeycomb photonic lattice is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' For the three-level EIT case without the 1D pump field, the resulted refractive index is inversely related to the coupling-field intensity under certain range of frequency detuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' With the two-photon detuning set as ∆�−∆�>0, the reconfigurable non-Hermitian photonic lattice with honeycomb real and imaginary parts is effectively established [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' The observed transmitted probe pattern (at the output surface of the cell and captured by a charge coupled device camera) under the condition of uniform susceptibility for both sublattices is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' 1(d), which is clearly a discrete pattern in the honeycomb geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' With the pump field turned on to modulate the imaginary part of A sublattice, the transmitted probe intensity [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' 1(e)] at A sites (marked by the blue (a) (b)5P[3> 5P1/2E3 E E22 5S1/2 F=3XBdashed circles) is much weaker due to the larger imaginary part (representing stronger absorption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' 2 (a)-(f) Detected field patterns by changing the pump detuning Δ� from 120 MHz to 20 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' The two-photon detuning is fixed at Δ� − Δ�\uf03d25 MHz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' (g) The calculated imaginary parts of the susceptibility at A and B sites from the density matrix method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' The six symbols correspond to the values of detuning in (a-f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' One advantage of the photonic lattices based on atomic coherence is the easily accessible tunability inherited from the multi-level atomic configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Here, we show that loss difference between A and B sites of the constructed non-Hermitian honeycomb lattice can be reconfigured by adjusting the laser parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Figures 2(a-f) demonstrate the evolution of the output intensity at A (blue circles) and B (white circles) sublattices versus the detuning of the pump field (which only affects the susceptibility of A sites) with other parameters fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' With the pump detuning ∆� discretely tuned from 120 MHz to 20 MHz, the experienced absorption in A sites becomes stronger, resulting in a decrease of the intensity inside blue circles, while the intensity at uncovered B sites governed by a three-level configuration exhibits no obvious change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' These observed results clearly demonstrate the increase of the loss difference between A and B sublattices with decreasing ∆�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' The interaction between the probe beam and the four-level atomic configuration can be described by using the density matrix method with rotating-wave approximation [35, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' According to the density matrix equations, the loss parameters ��,� = Im���,�� at the A and B sites are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' 2(g), with symbols denoting the detunings corresponding to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' 2(a-f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' As one can see, when ∆� is tuned from 120 MHz to 20 MHz , �� keeps constant while �� increases, resulting in the enlargement of loss difference Δ� = �� − ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' The estimated loss difference values of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' 2(a-f) are A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content='-60N- A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content='-20N1eA3-120MH1Z M1n △3-100MHZ △3-80MHZt N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content='0DD (eMax(np)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content='6F1003�� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content='49, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content='62, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content='88, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content='33, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content='27, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content='59 (× 10��) , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' This result advocates the instantaneous reconfigurability of the established non-Hermitian honeycomb lattice, which in turn provides abundant space to govern the beam dynamics under various experimental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Band structures and eigenstates of the non-Hermitian honeycomb lattice system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' (a, b) Bands of Re[�] according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' (2) for Δ� = 0 (a) and Δ� = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content='5 (b) with � = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' �� and (�� + ��)/2 can be set arbitrarily as they only shift the bands up or down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' (c) Boundaries of the local flat band surfaces for different Δ�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' The inset shows the first Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' (d, e) Calculated band structures (solid lines) for the synthesized honeycomb lattice with Δ� = 0 (d) and Δ� = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content='1 × 10�� (e), together with representative eigenstates (insets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' The dashed lines are the corresponding bands in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' (2) from the analytic tight-binding model for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' (f) Evolutions of the local flat band around one K point by increasing loss difference ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' The inset shows the localized field pattern when the local flat band (loss difference) is large enough for field localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Such a non-Hermitian honeycomb photonic lattice system can be described by the effective tight-binding model with Hamiltonian [38,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content='39]: � = � �� � ∑ �� �∙�� � � ∑ ��� �∙�� � �� �,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' (1) where ��,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content='� = �� − ���,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content='� are the normalized complex refractive indices at A and B sites,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' � is the normalized coupling strength between neighboring sites,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' � = ���,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' ��� is the two-dimensional wavenumber,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' and ��,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content='�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content='� = (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content='0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' �− � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' √� � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' �− � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' − √� � � are the normalized coupling directions on the � − � plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' The eigenvalues of this Hamiltonian, which characterize the propagation properties of the probe beam inside the system, are found to be ��,� = �� − � � �(�� + ��) ± ��3 − � �� ��� � + 2 �cos ��� � cos √��� � + cos�√3 ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' (2) When there is no loss difference (Δ� = �� − �� = 0), such eigenvalues can be visualized in the band structure shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' It is obvious that they degenerate at the Dirac points (K and K\uf0a2 points, △y=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content='76x10-60XheMRe(B Re(B)A2=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content='28△y=14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content='5Miax iax M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content='26△y=18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content='5M K 1T M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content='0 ky 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content='0X L OL 1k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' 0 2 3X10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content='36 d △V=0real part of the eigenvalues) and locally show linear dispersion properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' However, when loss difference is introduced(Δ� ≠ 0), the point degeneracy becomes circular-surface-like degeneracy and the original linear dispersion evolves into local dispersionless flat band around the K and K\uf0a2 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' This can be seen in the band structure of Re[�] in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' 3(b) for Δ� = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content='5 and � = 12(normalized parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' The corresponding bands of Im[�] are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' S2(a) in Supplemental Material [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Such a local flat band surface can localize the incident light due to locally zero group velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Moreover, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' 3(c), we plot the boundaries of the local flat bands for different Δ�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' It is clear that with the increase of Δ�, the local flat band surfaces become broader, enabling better localization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' The formation of the local flat band under non-zero loss difference can be understood from the Dirac Hamiltonian approximation around the K and K\uf0a2 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' By taking the Taylor series expansion of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' (1), one can obtain �� = ��� − ��� ������ ������� �� − ��� �, (3) where �� = 3�/2, � is the amplitude of the wavenumber reference to K (K\uf0a2 ) point, and � is the local direction angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' The eigenvalues of this Dirac Hamiltonian are ��,� = �� − � � �(�� + ��) ± ����� − � �� ���� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' (4) Now, the degeneracy condition �� − (Δ�/2��)� = 0 is dependent on the amplitude of the local wavenumber �, indicating that the degenerate point evolves to an exceptional ring [23,40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' In addition, the size of the ring is linearly proportional to the loss difference Δ�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' As the local flat band (real part of the eigenvalues) is inside the exceptional ring, the size of the local flat band increases proportionally to the magnitude of Δ�, as verified by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' 3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' We calculate the band structure of the synthesized photonic lattice with honeycomb susceptibility distributions using the PDE module of COMSOL Multiphysics and the results are shown as the solid lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' 3(d, e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' One can see that the Dirac cone at K point for Δ� = 0 becomes local flat band surfaces for Δ� = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content='6´10-�,agreeing with the analytic results (dashed lines) of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' (2) from the tight-binding model, especially in the vicinity of K points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' We note that the minor difference between bandstructures close to Γ point based on the two methods is due to the simplicity of the symmetric tight-binding model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' To better demonstrate the formation of the local flat band surfaces, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' 3(f), we plot the bands around K point for four different Δ�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' With the increase of the Δ� value, the local flat band expands broader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Also, we show representative eigenstates in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' 3(d, e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' It is worth noting that, with the increase Δ�, the local flat bands not only emerge and expand, but also share the similar eigenstates for each band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Particularly, for the eigenstates of the upper (lower) local flat band, the field is localized at B (A) sites with low (high) loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' When a probe beam is obliquely incident into the system to excite the K vicinity, the eigenstates of both the local flat bands will be excited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' However, since the B lattice sites have lower loss than the A sites, the eigenstates of field concentrated at A sites (lower band) will decay much faster and finally only the upper band survives, resulting in the field localization at B sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' The inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' 3(f) shows the simulated results of the localized field pattern when the probe beam propagates for a certain distance in the synthetic honeycomb photonic lattice with relatively large loss difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' The probe beam will keep this localized profile when propagating inside the lattice but experience the amplitude decay due to the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Observed energy exchanges between sublattices A and B during the propagation of probe under different pump parameters: (a) without the 1D pump field E3, (b) Δ� = 120 MHz, (c) Δ� = 40 MHz, and (d) Δ� = 20 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' The squares and circles are the measured proportions of the transmitted intensities at A and B sites, respectively, while the corresponding solid curves represent mathematic fitting based on the experimental measurements and provide a guide to the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' In experiment, the field localization can then be verified by examining the energy exchange between A and B sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Under the condition of a large enough loss difference, the field can be localized at B sites (three-level atomic region) and in principle there should be no energy transportation between A and B sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' While the field fails to be localized under the condition of a relatively small loss difference, the energy at A and B sites will couple to each other and resulting in energy fluctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' In the following, we show the energy exchange between two adjacent waveguides (corresponding to A and B sublattices) at different Δ� in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' One advantage of atomic media is that increasing the atomic density (positively related to the temperature of the atomic vapor) is understood as extending the effective propagating distance of the probe beam inside the photonic lattice [31,41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' With the pump detuning adjusted to set different imaginary parts of the refractive index of the four-level atomic regions (A sites), the transport dynamics of the probe beam are captured in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' 4 when the medium is heated from 120 ℃ to 165 ℃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Actually, considering that the susceptibility is proportional to the atomic density N for either the three- or four-level configuration [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' (S2) in the Supplemental Material [35], higher temperature can cause larger imaginary part as well as larger loss difference Δ�, which will result in weaker transmission but help to improve performance of the field localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' The relatively Fitting Curve for IB/(IA+IB)B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content='4120 160 120 140 1600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content='8 (b) (a) Measured IA/(IA+IB) Measured IB/(IA+IB) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content='6transmitted intensities �� and �� at A and B sites are measured by the software of the charge coupled device camera and their proportions are represented by symbols in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' When the pump field is absent, all the sites on the honeycomb lattice possess the same susceptibility controlled by a three-level configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' In this case, the loss difference is zero (Δ� = 0) and the commonly seen power exchange between neighboring waveguide channels is observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' 4(a), where the initial intensities �� and �� are very close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' During the propagation, one can see clearly twice energy exchanges indicated by the two points where �� ≈ ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' With the small increase of Δ� by setting the pump detuning at ∆3 = 120 MHz [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' 2(g)], as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' 4(b), the frequency of power exchange between A and B sites is reduced to only once at ~155℃, which means the energy exchange requires a longer propagating distance than the case without the pump field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' The slowing down of the power exchange indicates the formation of the local flat band surface but still a limited area, which cannot fully localize the beam energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' With the pump detuning decreased to ∆3 = 40 MHz [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' 4(c)], the difference between the imaginary parts is further enlarged and a broader local flat band surface occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' As a consequence, one can see that there is no power exchange in the given propagation distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' For the very large loss difference shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' 4(d) with ∆� = 20 MHz, the ratio between the transmitted intensities �� and �� reaches ~1:4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Since the generated local flat band surface have a large enough area and can confine the energy mainly into B sublattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Correspondingly, we simulate the beam propagation dynamics in the synthetic honeycomb photonic lattice with the susceptibility distribution at the same order of magnitude as the experiment [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' The simulation results of the energy exchange depending on normalized propagating distance are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' S3, which shows very similar energy evolution behaviors as the experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' In conclusion, we experimentally demonstrate the field localization behavior by switching the band structures from Dirac points with linear dispersion to dispersionless local flat band surfaces in a reconfigurable non-Hermitian honeycomb photonic lattice that induced in a multi-level atomic vapor cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Such switching is derived from the induced loss difference between A and B sublattices, whose susceptibility are controlled by four- and three-level atomic configurations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' By easily setting the laser parameters, the imaginary parts as well as the sizes of the formed local flat band surface are effectively manipulated to govern the beam dynamics in individual waveguide channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' The occurrence of the local flat band surface is verified by the vanishing of power exchange, indicating the light is localized by the flat-band modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Being different from the phenomena relying on the degree of non-Hermiticity in previous works involving exceptional rings [19, 23-25], the localization of optical fields requires only the proper manipulation of loss difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Namely, the localization behavior arising from loss difference is a result of the local flat band with shared eigenstates, and does not depend on the exceptional points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Our work not only uncovers a new property of non-Hermitian honeycomb photonic systems but also opens the door for the experimental exploration on the capabilities of rings-surrounded flat bands in high-dimensional non-Hermitian systems, promisingly extending to quantum, mechanical, and electrical non-Hermitian systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' This work was supported by National Key R&D Program of China (2018YFA0307500), National Natural Science Foundation of China (62022066, 12074306, 11804267), and the Key Scientific and Technological Innovation Team of Shaanxi Province (2021TD-56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' †fu.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Feng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Liang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Zhang, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Xiao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Experimental demonstration of optical Bloch oscillation in electromagnetically induced photonic lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Fundam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} +page_content=' 2, 401 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE3T4oBgHgl3EQf6Avi/content/2301.04787v1.pdf'} diff --git a/mNFST4oBgHgl3EQfKDg1/content/tmp_files/2301.13735v1.pdf.txt b/mNFST4oBgHgl3EQfKDg1/content/tmp_files/2301.13735v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5a77b61dddd78f11a362caf84fa9aabc7de6f962 --- /dev/null +++ b/mNFST4oBgHgl3EQfKDg1/content/tmp_files/2301.13735v1.pdf.txt @@ -0,0 +1,2815 @@ +Flipper games for monadically stable graph classes* +Jakub Gajarský† +Nikolas Mählmann‡ +Rose McCarty§ +Pierre Ohlmann† +Michał Pilipczuk† +Wojciech Przybyszewski† +Sebastian Siebertz‡ +Marek Sokołowski† +Szymon Toru´nczyk† +Abstract +A class of graphs C is monadically stable if for any unary expansion � +C of C , one cannot +interpret, in first-order logic, arbitrarily long linear orders in graphs from � +C . It is known that +nowhere dense graph classes are monadically stable; these encompass most of the studied +concepts of sparsity in graphs, including classes of graphs that exclude a fixed topological +minor. On the other hand, monadic stability is a property expressed in purely model-theoretic +terms and hence it is also suited for capturing structure in dense graphs. +For several years, it has been suspected that one can construct a structure theory for +monadically stable graph classes that mirrors the theory of nowhere dense graph classes in +the dense setting. In this work we provide a next step in this direction by giving a character- +ization of monadic stability through the Flipper game: a game on a graph played by Flipper, +who in each round can complement the edge relation between any pair of vertex subsets, and +Connector, who in each round is forced to localize the game to a ball of bounded radius. This +is an analog of the Splitter game, which characterizes nowhere dense classes of graphs (Grohe, +Kreutzer, and Siebertz, J. ACM ’17). +We give two different proofs of our main result. The first proof is based on tools borrowed +from model theory, and it exposes an additional property of monadically stable graph classes +that is close in spirit to definability of types. Also, as a byproduct, we give an alternative +proof of the recent result of Braunfeld and Laskowski (arXiv 2209.05120) that monadic stabil- +ity for graph classes coincides with existential monadic stability. The second proof relies on +the recently introduced notion of flip-wideness (Dreier, Mählmann, Siebertz, and Toru´nczyk, +arXiv 2206.13765) and provides an efficient algorithm to compute Flipper’s moves in a win- +ning strategy. +Acknowledgements. +We thank Patrice Ossona de Mendez for his valuable contributions to +this paper. +*This paper is part of projects that have received funding from the European Research Council (ERC) (grant agree- +ment No 948057 – BOBR) and from the German Research Foundation (DFG) with grant agreement No 444419611. +†University of Warsaw, Poland +‡University of Bremen, Germany +§Princeton University, USA. Supported by European Research Council (ERC) grant No. 714704 – CUTACOMBS +and National Science Foundation (NSF) grant No. DMS-2202961. +arXiv:2301.13735v1 [cs.LO] 31 Jan 2023 + +erc +European Research Council +Established by the European CommissionContents +1 +Introduction +3 +I +Prelude +8 +2 +Preliminaries +9 +2.1 +Model theory +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +9 +2.2 +Flips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +11 +2.3 +Flip-wideness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +13 +2.4 +Flip-connectivity +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +13 +3 +Variants of the Flipper game +15 +3.1 +Relating the game variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +16 +4 +Relations to flip-wideness and existential monadic stability +19 +4.1 +From Pseudo-Flipper game to flip-wideness . . . . . . . . . . . . . . . . . . . . . . +19 +4.2 +From Flipper game to existential monadic stability . . . . . . . . . . . . . . . . . . +20 +II +Model-theoretic proof +23 +5 +Additional model-theoretic preliminaries +24 +6 +Pattern-free classes +27 +7 +Finite separators in pattern-free stable models +32 +7.1 +Case of finitely many types +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +32 +7.2 +Balls have finitely many types over a model . . . . . . . . . . . . . . . . . . . . . . +35 +7.3 +Inductive proof +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +36 +8 +Separator wins in monadically stable classes +36 +III +Algorithmic Flipper game +38 +9 +Outline +38 +10 Predictable flip-wideness +40 +10.1 Classifiers +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +41 +10.2 Proof of the result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +44 +11 Winning strategy +51 +11.1 Strategies and runtimes +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +51 +11.2 Finalizing the argument +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +53 +2 + +1 +Introduction +Monadic stability is a notion of logical tameness for classes of structures. Introduced by Baldwin +and Shelah [BS85] in the context of model theory1, it has recently attracted attention in the field +of structural graph theory. We recall the logical definition below. One of the main contributions +of this paper is to provide a purely combinatorial characterization of monadically stable classes +of graphs. Our characterization is effective, and can be employed in algorithmic applications, +as we explain later. +In this paper we focus on (undirected, simple) graphs, rather than arbitrary structures. A +graph is modelled as a relational structure with one symmetric binary relation signifying adja- +cency. By a class of graphs we mean any set of graphs. For a class of graphs C , a unary expansion +of C is any class � +C of structures such that each �G ∈ � +C is obtained from some graph in G ∈ C +by adding some unary predicates. Thus, the elements of � +C can be regarded as vertex-colored +graphs from C . A class of graphs C is called monadically stable if one cannot interpret, using +a fixed formula ϕ( ¯x, ¯y) of first-order logic, arbitrarily long linear orders in any unary expan- +sion � +C of C . More precisely, for every unary expansion � +C and formula ϕ( ¯x, ¯y) with | ¯x| = | ¯y| +(over the signature of � +C ) there is a bound ℓ such that there is no structure �G ∈ +� +C and tuples +¯a1, . . . , ¯aℓ ∈ V( �G) ¯x such that �G |= ϕ(¯ai, ¯aj) if and only if i ⩽ j. More generally, C is monadically +dependent (or monadically NIP) if one cannot interpret, using a fixed formula ϕ( ¯x, ¯y) of first-order +logic, all finite graphs in any unary expansion of C . Thus, from the model-theoretic perspective, +the intuition is that being monadically dependent is being non-trivially constrained: for any +fixed interpretation, one cannot interpret arbitrarily complicated structures in vertex-colored +graphs from the considered class. On the other hand, graphs from monadically stable classes +are “orderless”, in the sense that one cannot totally order any large part of them using a fixed +first-order formula. +Baldwin and Shelah proved that in the definitions, one can alternatively rely on formu- +las ϕ(x, y) with just a pair of free variables, instead of a pair of tuples of variables [BS85, +Lemma 8.1.3, Theorem 8.1.8]. Moreover, they proved that monadically stable theories are tree +decomposable [BS85, Theorem 4.2.17], providing a structure theorem for such theories, although +one of a very infinitary nature. A more explicit, combinatorial structure theorem for monad- +ically stable and monadically dependent is desirable for obtaining algorithmic results for the +considered classes, as we discuss later. +On the other hand, Braunfeld and Laskowski [BL22] very recently proved that for hereditary +classes of structures C that are not monadically stable or monadically dependent, the required +obstructions (total orders or arbitrary graphs) can be exhibited by an existential formula ϕ( ¯x, ¯y) +in the signature of C , without any additional unary predicates. Among other things, this shows +that for hereditary classes of structures, the notions of monadic stability coincides with the +more well-known notion of stability, and similarly, monadic dependence coincides with depen- +dence (NIP). Furthermore, since the formulas are existential, this result can be seen as a rather +explicit, combinatorial non-structure theorem for hereditary classes that are not monadically +stable (resp. monadically dependent). Still, they do not provide explicit structural results for +classes that are monadically stable or monadically dependent. +Explicit, combinatorial and algorithmic structural results for monadically dependent and +monadically stable classes are not only desired, but also expected to exist, based on the known +1Formally, Baldwin and Shelah [BS85], as well as Braunfeld and Laskowski [BL22], study monadically dependent +and monadically stable theories, rather than classes of structures. Some of their results transfer to the more general +setting of monadically dependent/stable classes of structures. +3 + +examples of such classes that have been studied in graph theory and computer science. As +observed by Adler and Adler [AA14] based on the work of Podewski and Ziegler [PZ78], all +nowhere dense graph classes are monadically stable. A class C is nowhere dense if for every fixed +r ∈ N, one cannot find r-subdivisions2 of arbitrarily large cliques as subgraphs of graphs in C . +In particular, every class excluding a fixed topological minor (so also the class of planar graphs, +or the class of subcubic graphs) is monadically stable. In fact, it follows from the results of Adler +and Adler [AA14] and of Dvoˇrák [Dvo18] that monadic stability and monadic dependence are +both equivalent to nowhere denseness if one assumes that we work with a class of sparse graphs +(formally, with a class of graphs that excludes a fixed biclique as a subgraph). However, since +they are defined in only model-theoretic terms, monadic stability and monadic dependence are +not bound to sparsity; they can be used to understand and quantify structure in dense graphs +as well. +The pinnacle of the theory of nowhere dense graph classes is the result of Grohe, Kreutzer, +and Siebertz [GKS17] that the model-checking problem for first-order logic is fixed-parameter +tractable on any nowhere dense class of graphs. +Theorem 1.1 ([GKS17]) For every nowhere dense graph class C , first-order sentence ϕ, and ε > 0, there +exists an algorithm that given an n-vertex graph G ∈ C decides whether G |= ϕ in time OC ,ϕ,ε(n1+ε). +Monadically dependent classes include all monadically stable classes, in particular all nowhere +dense classes, but also for instance all classes of bounded twin-width [BKTW20]. An analogous +result, with 1 + ε replaced by 3, holds for all classes C of ordered graphs3 of bounded twin- +width [BGOdM+22]. +In light of the discussion above, monadic stability and monadic dependence seem to be +well-behaved generalizations of nowhere denseness that are defined in purely model-theoretic +terms; hence these concepts may be even better suited for treating the model-checking problem +for first-order logic. This motivated the following conjecture [war16], which has been a subject +of intensive study over the last few years4. +Conjecture 1.1 Let C be a monadically dependent graph class. There exists a constant c ∈ N depending +only on C and, for every first-order sentence ϕ, an algorithm that, given a n-vertex graph G ∈ C , decides +whether G |= ϕ in time OC ,ϕ(nc). +Conjecture 1.1 is not even resolved for monadically stable classes. To approach this conjec- +ture, it is imperative to obtain explicit, combinatorial structure theorems for monadically stable +and in monadically dependent graph classes, with a particular focus on finding analogs of the +tools used in the proof of Theorem 1.1. Our work contributes in this direction. We provide cer- +tain tree-like decompositions for graphs in monadically stable graph classes, which can be most +intuitively explained in terms of games. On the one hand, our decompositions generalize a sim- +ilar result for nowhere dense classes, recalled below. On the other hand, they are remininiscent +of the tree decomposability property proved by Baldwin and Shelah, but are more explicit and +finitary in nature. +Splitter game. +The cornerstone of the proof of Theorem 1.1 is a game-theoretic characteriza- +tion of nowhere denseness, through the Splitter game. This game has a fixed radius parameter +2The r-subdivision of a graph H is the graph obtained from H by replacing every edge with a path of length r + 1. +3Ordered graphs are graphs equipped with a total order. +4To the best of our knowledge the conjecture was first explicitly discussed during the open problem session of the +Algorithms, Logic and Structure Workshop in Warwick, in 2016, see [war16]. +4 + +r ∈ N and is played on a graph G between two players, Splitter and Connector, who make moves +in rounds alternately. In each round, Splitter first chooses any vertex u and removes it from the +graph. Next, Connector has to select any other vertex v, and the game gets restricted to the +subgraph induced by the ball of radius r with center at v. The game ends with Splitter’s victory +when there are no vertices left in the graph. +Theorem 1.2 ([GKS17]) A class C of graphs is nowhere dense if and only if for every r ∈ N there exists +k ∈ N such that for every G ∈ C , Splitter can win the radius-r Splitter game on G within k rounds. +Very roughly speaking, Theorem 1.2 shows that any graph from a nowhere dense class can +be hierarchically decomposed into smaller and smaller parts so that the decomposition has +height bounded by a constant k depending only on the class and the locality parameter r. This +decomposition is used in the algorithm of Theorem 1.1 to guide model-checking. +Flipper game. +In this work we introduce an analog of the Splitter game for monadically stable +graph classes: the Flipper game. Similarly to before, the game is played on a graph G and there +is a fixed radius parameter r ∈ N. There are two players, Flipper and Connector, which make +moves in rounds alternately. In her move, Flipper selects any pair of vertex subsets A, B (possi- +bly non-disjoint) and applies the flip between A and B: inverts the adjacency between any pair +(a, b) of vertices with a ∈ A and b ∈ B. Connector’s moves are exactly as in the Splitter game; +in every round, he needs to select a ball of radius r, and the game is restricted to the subgraph +induced by this ball. The game is won by Flipper once there is only one vertex left. +We remark that the Flipper game is a radius-constrained variant of the natural game for +graph parameter SC-depth, which is functionally equivalent to shrubdepth, in the same way that +the Splitter game is a radius-constrained variant of the natural game for treedepth. SC-depth +and shrubdepth were introduced and studied by Ganian et al. in [GHN+12, GHN+19]. +Our main result is the following analog of Theorem 1.2 for monadically stable classes. +Theorem 1.3 A class C of graphs is monadically stable if and only if for every r ∈ N there exists k ∈ N +such that for every graph G ∈ C , Flipper can win the radius-r Flipper game on G within k rounds. +Let us compare Theorem 1.3 with another recent characterization of monadic stability, pro- +posed by Gajarský and Kreutzer, and proved by Dreier, Mählmann, Siebertz, and Toru´nczyk +[DMST22], through the notion of flip-wideness. This notion is an analog of uniform quasi-wideness, +introduced by Dawar [Daw10]. Without going into technical details, a class of graphs C is uni- +formly quasi-wide if for any graph G ∈ C and any large enough set of vertices A in G, one can +find many vertices in A that are pairwise far from each other after the removal of a constant +number of vertices from G. As proved by Nešetˇril and Ossona de Mendez [NO11], a class of +graphs is uniformly quasi-wide if and only if it is nowhere dense. Flip-wideness is an analog +of uniform quasi-wideness obtained similarly to the Flipper game: by replacing the concept of +deleting a vertex with applying a flip; see Definition 1 for a formal definition. The fact that +monadic stability is equivalent to flip-wideness (as proved in [DMST22]) and to the existence of +a short winning strategy in the Flipper game (as proved in this paper) suggests the following: +the structural theory of monadically stable graph classes mirrors that of nowhere dense graph +classes, where the flip operation is the analog of the operation of removing a vertex. +We give two very different proofs of Theorem 1.3. The first proof is based on elementary +model-theoretic techniques, and it provides new insight into the properties of monadically sta- +ble graph classes. As a side effect, it gives a new (though non-algorithmic) proof of the main +result of [DMST22]: equivalence of monadic stability and flip-wideness. On the other hand, the +5 + +second proof relies on the combinatorial techniques developed in [DMST22]. It has the advan- +tage of being effective, and provides an efficient algorithm for computing Flipper’s moves in a +winning strategy. +Model-theoretic proof. +The following statement lists properties equivalent to monadic stabil- +ity uncovered in our model-theoretic proof. Notions not defined so far will be explained later. +Theorem 1.4 Let C be a class of graphs. Then the following conditions are equivalent: +1. C is monadically stable. +2. C has a stable edge relation and is monadically dependent with respect to existential formulas +ϕ(x, y) with two free variables. +3. C has a stable edge relation and is pattern-free. +4. For every r ∈ N every model G of the theory of C , every elementary extension H of G, and every +vertex v ∈ V(H) − V(G), there is a finite set S ⊆ V(G) that r-separates v from G. +5. For every r ∈ N there is k ∈ N such that Separator wins the Separation Game of radius r on +every G ∈ C in at most k rounds. +6. For every r ∈ N there is k ∈ N such that Flipper wins the Flipper game of radius r on every +G ∈ C in at most k rounds. +7. C is flip-wide. +(1) +mon. stable +(2) +stable edge relation ++ ex. mon. dependent +(3) +stable edge relation ++ pattern-free +(4) +separable +(5) +Separator wins +(6) +Flipper wins +(7) +flip-wide +trivial +Sec. 6 +Sec. 7 +Sec. 8 +Sec. 3.1 +Sec. 3.1 + 4.1 +Sec. 4.2 +[DMST22] +FIGURE 1: The implications that constitute Theorem 1.4, together with the sections in which they are proved. +The implications that constitute Theorem 1.4 are illustrated in Fig. 1. Note that Theorem 1.3 +is the equivalence (1)↔(6). Let us give a brief overview of the presented conditions. +Conditions (1) and (2), respectively, are monadic stability and a weak form of existential +monadic stability. Recall that Baldwin and Shelah proved that it is sufficient to consider formu- +las ϕ(x, y) with two free variables in the definition of monadic stability (instead of formulas +ϕ( ¯x, ¯y)), whereas Braunfeld and Laskowski proved that it is sufficient to consider existential +formulas ϕ( ¯x, ¯y) that do not involve additional unary predicates. The condition (2) lies some- +where in between: it implies that it is sufficient to consider existential formulas ϕ(x, y) with +two variables, possibly involving additional unary predicates. In particular, it implies the re- +sult of Baldwin and Shelah (in the case of graph classes) and is incomparable with the result of +Braunfeld and Laskowski. Our proof uses different techniques. +6 + +Condition (3) is a rather explicit combinatorial condition. Roughly, a class C is pattern-free if +it does not encode, using a quantifier-free formula ϕ(x, y), the class of r-subdivided cliques, for +any fixed r ⩾ 1. See Definition 2 for details. +Condition (4) is phrased in the language of model theory and serves a key role in our proof. +It resembles a fundamental property called “definability of types”, and in essence it says the +following: whenever working with a model G of the theory of C , every element of any elemen- +tary extension of G can be robustly “controlled” by a finite subset of G. We believe that the +new notion of r-separation used here is of independent interest. It refers to non-existence of +short paths after applying some flips governed by S. The implication (3)→(4) is the core part of +our proof. +Conditions (5) and (6) assert the existence of a short winning strategy in the Flipper game +and its technical variant, the Separation game. In essence, in the Separation game we restrict the +moves of Flipper (now called Separator) by requiring that the flips are definable using a bounded +number of vertices. Also, we measure distances somewhat differently, so that intuitively we +take into account all possible flips that Separator could have made. The implication (4)→(5) +is proved by proposing a strategy for Separator and using compactness combined with (4) to +argue that it leads to a victory within a bounded number of rounds. +Finally, condition (7) is the notion of flip-wideness, whose equivalence with monadic sta- +bility was proved by Dreier et al. [DMST22]. We prove the implication (5)→(7) by (essentially) +providing a strategy for Connector in the Separator game when the class is not flip-wide. Then +we rely on the implication (7)→(1) from [DMST22] to close the circle of implications; this proves +the equivalence of (1)-(7) with the exception of (6). We remark that (7)→(1) is the easy implica- +tion of [DMST22], hence our reasoning can also serve as an alternative proof of the flip-wideness +characterization given in [DMST22]. +To put the Flipper game into the picture, we separately prove the implications (5)→(6)→(2). +The implication (5)→(6) relies on a conceptually easy, but technically not-so-trivial translation of +the strategies. In the implication (6)→(2) we use obstructions to existential monadic stability to +give a strategy for Connector in the Flipper game that enables her to endure for arbitrarily long. +Algorithmic proof. +We also give a purely combinatorial proof of (the forward implication of) +Theorem 1.3, which in particular provides a way to efficiently compute Flipper’s moves in a +winning strategy. Formally, we show the following. +Theorem 1.5 Let C be a monadically stable class of graphs. Then for every radius r ∈ N there exist +k ∈ N and a Flipper strategy flip⋆ such that the following holds: +– When playing according to flip⋆ in the Flipper game of radius r on any graph G ∈ C , Flipper +wins within at most k rounds. +– The moves of flip⋆ on an n-vertex graph G ∈ C can be computed in time OC ,r(n2). +The main idea behind the proof of Theorem 1.5 is to rely on the result of Dreier et al. that +monadically stable graph classes are flip-wide [DMST22]. Using the combinatorial tools devel- +oped in [DMST22], we strengthen this property: we prove that the set of flips F whose applica- +tion uncovers a large scattered set Y (a set of vertices that are pairwise far from each other) can be +selected in a somewhat canonical way, so that knowing any 5-tuple of vertices in Y is enough to +uniquely determine F. We can then use such strengthened flip-wideness to provide a winning +strategy for Flipper; this roughly resembles the Splitter’s strategy used by Grohe et al. in their +proof of Theorem 1.2, which in turn relies on uniform quasi-wideness. +7 + +Theorem 1.5, the algorithmic version of Theorem 1.3, is the key to any algorithmic applica- +tions of the Flipper game. In particular, it was very recently already used by Dreier, Mählmann, +and Siebertz [DMS] to approach the first-order model checking problem on monadically stable +graph classes and even solve it on structurally nowhere dense classes, an important subclass +of monadically stable classes. On a high level, the proof of [DMS] follows the approach of +Grohe et al. [GKS17] on nowhere dense classes. Essentially, by Gaifman’s Locality Theorem +the model checking problem reduces to computing which formulas up to a certain quantifier +rank q are true in the local r-neighborhoods of the input graph. Here, the numbers q and r de- +pend only on the input formula. The set of formulas of quantifier rank q that are true in a local +neighborhood is called the local q-type of the neighborhood. The computation of local q-types +is done recursively in a recursion guided by the Splitter game, which hence terminates after a +bounded number of steps. A naive branching into all local neighborhoods, however, leads to +a too high running time. This issue is solved by grouping close by elements into clusters and +computing the local types of all neighborhoods that are grouped in one cluster in one recursive +call. Whenever the clusters can be collected into a sparse neighborhood cover, which is the case +in nowhere dense graph classes, this leads to an efficient model checking algorithm. Dreier et +al. [DMS] showed that the Splitter game in the above approach can be replaced by the Flipper +game and present an efficient model checking algorithm on all monadically stable classes that +admit sparse (weak) neighborhood covers. Their result does not fully solve the model check- +ing problem on monadically stable classes as the question whether sparse weak neighborhood +covers for monadically stable classes exist remains an open problem. Dreier et al. only showed +the existence of such covers for structurally nowhere dense classes. We refer to [DMS] for the +details. +Organization. +The paper is split into three parts. +Part I is devoted to introducing the basic notions and relations between them. In particular, +we prove the implications (5)→(6)→(2) of and (5)→(7) of Theorem 1.4. +In Part II we present the model-theoretic proof of Theorem 1.3. More precisely, we prove the +implications (2)→(3)→(4)→(5) of Theorem 1.4. Together with the implication (5)→(7) proved in +Part I, the (easy) implication (7)→(1) proved in [DMST22], and the trivial implication (1)→(2), +this closes the cycle of implications between the conditions (1), (2), (3), (4), (5), (7). And with the +implications (5)→(6)→(2) proved in Part I, this completes the proof of Theorem 1.4. +In Part III we prove Theorem 1.5. +Part I +Prelude +In this part, we introduce the basic notions of interest: monadic stability, variants of the Flipper +game, flip-wideness, and prove some relations between them. +In Section 2 we establish notation and common definitions, and present basic model-theoretic +background. Also, we discuss the notions of flips, of flip-connectivity, and of flip-wideness. +In Section 3 we define three variants of the Flipper game, and prove results relating some of +them to the others. In particular, we define the Separation game, which is a variant of the Flip- +per game in which Flipper’s moves are required to be definable, so Flipper is more constrained. +8 + +On the other hand, the balls are measured in different ways in the two games, so it is not imme- +diately clear how the two games compare. However, the implication (5)→(6) of Theorem 1.4 is +relatively easy, and is proved in Section 3.1. +In Section 4 we relate variants of the Flipper game to other notions around monadic stability: +flip-wideness and existential monadic stability. In particular, we derive flip-wideness from a +strategy of Separator in the Separation game (this is the implication (5)→(7) of Theorem 1.3). +Also we derive existential monadic stability from a strategy of Flipper in the Flipper game (this +is the implication (6)→(2)). +2 +Preliminaries +All graphs in this paper are simple and loopless but not necessarily finite. For a vertex v of a +graph G, we write N(v) for the (open) neighborhood of v in G; so N(v) := {u ∈ V(G) | uv ∈ +E(G)}. For a set of vertices X ⊆ V(G), we write G[X] for the subgraph of G induced by X, +and G − X for the subgraph of G induced by V(G) − X. For two sets X, Y ⊆ V(G) the bipartite +graph semi-induced by X and Y in G, denoted G[X, Y], is the bipartite graph with parts X and Y, +and edges uv for u ∈ X, v ∈ Y with uv ∈ E(G). By |G| we denote the cardinality of the vertex +set of G. +For vertices a, b ∈ V(G), an (a, b)-path is a path with ends a and b. Similarly, for sets A, B ⊆ +V(G), an (A, B)-path is a path where one end is in A and the other end is in B. We write △ for +the symmetric difference of two sets. Similarly, for two graphs G and G′ on the same vertex-set, +we write G△G′ for the graph with vertex-set V(G) = V(G′) and edge-set E(G)△E(G′). +2.1 +Model theory +We work with first-order logic over a fixed signature Σ that consists of (possibly infinitely many) +constant symbols and of relation symbols. A model is a Σ-structure, and is typically denoted +M, N, etc. We usually do not distinguish between a model and its domain, when writing, for +instance, m ∈ M or f : M → X, or X ⊆ M. A graph G is viewed as a model over the signature +consisting of one binary relation denoted E, indicating adjacency between vertices. +If ¯x is a finite set of variables, then we write ϕ( ¯x) to denote a first-order formula ϕ with +free variables contained in ¯x. We may also write ϕ( ¯x1, . . . , ¯xk) to denote a formula whose free +variables are contained in ¯x1 ∪ . . . ∪ ¯xk. We will write x instead of {x} in case of a singleton +set of variables, e.g. ϕ(x, y) will always refer to a formula with two free variables x and y. We +sometimes write ϕ( ¯x; ¯y) to distinguish a partition of the set of free variables of ϕ into two parts, +¯x and ¯y; this partition plays an implicit role in some definitions. +If U is a set and ¯x is a set of variables, then U ¯x denotes the set of all valuations ¯a: ¯x → U of ¯x +in U. Such a valuation is also called an ¯x-tuple. For a formula ϕ( ¯x) and an ¯x-tuple ¯m ∈ M ¯x, we +write M |= ϕ( ¯m), or M, ¯m |= ϕ( ¯x), if the valuation ¯m satisfies the formula ϕ( ¯x) in M. +A quantifier-free formula is a formula that does not use quantifiers. An existential formula is a +formula of the form ∃y1 . . . ∃yl.α, where α is quantifier-free. +We will use the notion of an atomic type only in the context of a finite relational signature Σ. +In this case, an atomic type with variables ¯x is a formula α( ¯x) that is a conjunction of clauses of +the form R(x1, . . . , xk) or ¬R(x1, . . . , xk), where R ∈ Σ ∪ {=} is a relation symbol of arity k and +x1, . . . , xk are variables from ¯x, such that each possible such clause or its negation occurs as a +9 + +conjunct in α. The atomic type of a tuple ¯a ∈ M ¯x is the unique atomic type α( ¯x) that is satisfied +by ¯a in M. +Stability and dependence. +A formula ϕ( ¯x; ¯y) is stable in a class C of structures if there exists +k ∈ N such that for every M ∈ C , there are no sequences ¯a1, . . . ¯ak ∈ M ¯x and ¯b1, . . . , ¯bk ∈ M ¯y +such that +M |= ϕ(¯ai; ¯bj) +⇐⇒ +i < j, +for 1 ⩽ i, j ⩽ k. +We say that a class C of graphs has a stable edge relation if the formula E(x; y) is stable in C . +Equivalently, C excludes some ladder as a semi-induced subgraph, where a ladder (often called +also half-graph) of order k is the graph with vertices a1, . . . , ak, b1, . . . , bk and edges aibj for all +1 ⩽ i < j ⩽ k; see Fig. 2. Note that replacing < by ⩽ in the above definitions does not change +them. +a1 +b1 +a2 +b2 +a3 +b3 +a4 +b4 +a5 +b5 +a6 +b6 +FIGURE 2: A ladder (half-graph) of order 6. +A formula ϕ( ¯x; ¯y) is dependent, or NIP (standing for “not the independence property”) in a +class C if there exists k ∈ N such that for every M ∈ C , there are no tuples ¯a1, . . . , ¯ak ∈ M ¯x and +¯bJ ∈ M ¯y for J ⊆ {1, . . . , k} such that +M |= ϕ(¯ai; ¯bJ) +⇐⇒ +i ∈ J, +for 1 ⩽ i ⩽ k and J ⊆ {1, . . . , k}. +Observe that a formula which is stable is also dependent. A class C is stable (resp. dependent) if +every formula ϕ( ¯x; ¯y) is stable (resp. dependent) in C . +Let Σ be a signature and let �Σ be a signature extending Σ by (possibly infinitely many) +unary relation symbols and constant symbols. A �Σ-structure � +M is a lift of a Σ-structure M if M +is obtained from � +M by forgetting the symbols from �Σ − Σ. A class of �Σ-structures � +C is a unary +expansion of a class of Σ-structures C if every structure � +M ∈ � +C is a lift of some structure M ∈ C . +A class C of structures is monadically stable if every unary expansion � +C of C is stable. Similarly, +C is monadically dependent (or monadically NIP) if every unary expansion � +C of C is dependent. +A class C is simply existentially monadically dependent (resp. simply existentially monadically sta- +ble) if every existential formula ϕ(x, y) (with two free variables) is dependent (resp. stable), in +every unary expansion � +C of C . A single structure M is monadically stable (resp. monadically +dependent) if the class {M} is. Note that a class which is monadically stable (resp. monadically +dependent) is stable (resp. dependent). Also, a class which is simply existentially monadically +stable is simply existentially monadically dependent, and has a stable edge relation. +We remark that even though monadic dependence and monadic stability are defined in +terms of formulas ϕ( ¯x, ¯y), where ¯x and ¯y are tuples of free variables, rather than single variables, +by the results of Baldwin and Shelah [BS85, Lemma 8.1.3, Theorem 8.1.8], it is enough to con- +sider formulas ϕ(x, y) with two free variables instead. Our definitions of existential monadic +stability and existential monadic dependence involves existential formulas ϕ(x, y) with two +10 + +free variables only. As follows from Theorem 1.4, for graph classes monadic stability is equiva- +lent to existential monadic stability, thus strengthening the result of Baldwin and Shelah, when +restricted to graph classes, by additionally showing that it suffices to consider existential for- +mulas. +2.2 +Flips +For the purpose of the following definition, it is convenient to assume that all considered graphs +have a domain contained in some fixed, universe Ω. +Atomic flips. +An atomic flip is an operation F specified by a pair (A, B) of (possibly intersect- +ing) subsets of Ω, which complements the adjacency relation between the sets A and B in a +given graph G. Formally, for a graph G, the graph obtained from G by applying the atomic +flip F is the graph denoted G ⊕ F with vertex set V(G), where, for distinct vertices u, v in V(G), +uv ∈ E(G ⊕ F) +⇐⇒ +� +uv /∈ E(G), +if (u, v) ∈ (A × B) ∪ (B × A); +uv ∈ E(G), +otherwise. +In the remainder of this section, it will be convenient to identify an atomic flip F with the pair +(A, B) of sets defining it and to write G ⊕ (A, B) instead of G ⊕ F. Note that, in our definition, +we do not require A and B to be subsets of V(G). Instead, we will allow A and B to be any +subset of Ω. In particular, G ⊕ (A, B) = G ⊕ (A ∩ V(G), B ∩ V(G)). This is useful as we often +work with induced subgraphs. As an example, for every graph G and vertex v ∈ V(G), the +atomic flip Fv = ({v}, N(v)) isolates the vertex v in G, meaning that v is an isolated vertex of +G ⊕ Fv. We denote by Flip(G) = {(A, B) | A, B ⊆ V(G)} the set of all the atomic flips defined +by subsets of vertices of G. +Observation 2.1 For every graph G and atomic flips F1, F2 we have G ⊕ F1 ⊕ F2 = G ⊕ F2 ⊕ F1. +Furthermore, when considering a sequence of atomic flips, we may restrict to the case where +no atomic flip appears more than once in the sequence since performing twice the same atomic +flip twice leaves the graph unchanged, thanks to the following. +Observation 2.2 For every graph G and atomic flip F we have G ⊕ F ⊕ F = G. +This justifies the following definition. +Sets of flips. +A set of flips {F1, . . . , Fk} defines an operation F that, given a graph G, results in +the graph +G ⊕ F := G ⊕ F1 ⊕ · · · ⊕ Fk. +Note that the order in which we carry out the atomic flips does not matter, according to Ob- +servation 2.1 and that it would be useless to consider mutlisets, according Observation 2.2. +Abusing terminology, we will often just say that the operation F is a set of flips, and write +F = {F1, . . . , Fk}. +Observation 2.3 If F and F′ are sets of flips, and F △ F′ is the symmetric difference of F and F′, then +G ⊕ F ⊕ F′ = G ⊕ (F △ F′). +The next observation is elementary but crucial. +11 + +Observation 2.4 Let G be a graph, let X be a subset of the vertices of G, and let F = (A, B) be an atomic +flip. Suppose two vertices u, v have the same neighborhood on X in G and have the same membership in +A and B (i.e. u ∈ A ⇐⇒ v ∈ A and u ∈ B ⇐⇒ v ∈ B). Then u and v have the same neighborhood +on X in G ⊕ F. +Finally, flips commute with vertex deletions. As a consequence, we have +Observation 2.5 For every graph G, every subset A of V(G), and every set of flips F, we have G[A] ⊕ +F = (G ⊕ F)[A]. +Remark 2.1. Specifying an atomic flip by a pair of subsets of Ω will be convenient for carrying out +our proofs. We note that we could also define an atomic flip by a single set A ⊆ Ω producing the +graph G ⊕ (A, A) that is obtained by complementing all adjacencies within A. Up to a constant +factor, both definitions yield the same expressive power: every atomic flip (A, B) is equivalent +to the set of flips defined by {(A ∪ B, A ∪ B), (A, A), (B, B)}. +Flip terminology. +For conciseness, we will use the term flip to refer to an atomic flip. Therefore, +a flip may be identified with a pair (A, B) of vertex sets. +F-flips. +Let F be a family of subsets of Ω. Then an F-flip is a set of flips of the form {F1, . . . , Fk}, +where each flip Fi is a pair (A, B) with A, B ∈ F. Note that there are at most 2|F|2 different F- +flips. In our context, the family F will usually be a partition of the set V(G) ⊆ Ω of vertices of +some graph G. An F-flip of a graph G, where F is a family of subsets of V(G), is a graph G′ +obtained from G after applying an F-flip. Whenever we speak about an F-flip, it will be always +clear from the context whether we mean a graph or the family of flips used to obtain it. +S-classes. +Let G be a graph, and let S ⊆ V(G) be a finite set of vertices. Consider the equiva- +lence relation ∼S on V(G), in which two vertices a, b are equivalent if either a, b ∈ S and a = b, +or a, b /∈ S and N(a) ∩ S = N(b) ∩ S. An S-class is an equivalence class of ∼S. In other words, it +is a set of vertices either of the form {s} for some s ∈ S, or of the form +{v ∈ V(G) − S | N(v) ∩ S = T} +for some T ⊆ S. The S-class of a vertex v ∈ V(G) is the unique S-class which contains v. Hence, +V(G) is partitioned into S-classes, and the number of S-classes is at most |S| + 2|S|. +S-flips. +An S-flip of a graph is an F-flip G′ of G, where F is the partition of V(G) into S-classes. +Note that there are 22O(|S|) many S-flips of a given graph G. +Note that a direct consequence of Observation 2.4 is that S-flips preserve S-classes. More- +over, S-flips satisfy the following transitivity property. +Lemma 2.6 If G is a graph, S and T are finite subsets of V(G), G′ is an S-flip of G, and G′′ is a T-flip +of G′, then G′′ is an (S ∪ T)-flip of G. +Proof. We first consider the case where S = T. By Observation 2.4, the S-classes of G′ are the +same as the S-classes of G. Thus, G′′ is an S-flip of G. +Assume S ̸= T. The partition into (S ∪ T)-classes is a common refinement of the partitions +into S-classes and into T-classes. It follows that G′ is an (S ∪ T)-flip of G and G′′ an (S ∪ T)-flip +of G′. Thus, the statement follows from the case where S = T. +12 + +Finally, the S-flips behave well with the extraction of an induced subgraph. +Lemma 2.7 Let G be a graph and S ⊆ X ⊆ V(G) be sets with S finite. If G′ is an S-flip of G, then +G′[X] is an S-flip of G[X]. +Proof. Let F be the set of flips (between S-classes of G) used to obtain G′ from G, so that G′ = +G ⊕ F. By Observation 2.5 we have G′[X] = G[X] ⊕ F′, where F′ is obtained from F by replacing +each S-class C used in any flip by C ∩ X. Now just note that the S-classes of G[X] are exactly the +sets of the form C ∩ X, where C is an S-class of G. Therefore, F′ applies flips between S-classes +in G[X], so G′[X] is an S-flip of G[X]. +2.3 +Flip-wideness +The following notion of flip-wideness was introduced in [DMST22]. Given a graph G and a set +of vertices A ⊆ V(G) we call A distance-r independent if all vertices in A are pairwise at distance +greater than r in G. +Definition 1 (Flip-wideness). A class of graphs C is flip-wide if for every r ∈ N there exists a +function Nr : N → N and a constant sr ∈ N such that for all m ∈ N, G ∈ C , and A ⊆ V(G) +with |A| ⩾ Nr(m), there exists a set F of flips with |F| ⩽ sr and B ⊆ A with |B| ⩾ m such that B +is distance-r independent in G ⊕ F. +Flip wideness is known to be equivalent to monadic stability, with a polynomial time algo- +rithmic version. +Theorem 2.8 ([DMST22]) Let C be a class of graphs. Then, the following are equivalent: +1. C is monadically stable; +2. C is flip-wide; +3. for every r ∈ N there exists a function Nr : N → N and a constant sr ∈ N such that for all +m ∈ N, G ∈ C with n vertices, and A ⊆ V(G) with |A| ⩾ Nr(m), we can compute in time +fC (r) · n3 (for some function fC ) a set F of flips with |F| ⩽ sR and a set B ⊆ A with |B| ⩾ m +such that B is distance-r independent in G ⊕ F. +2.4 +Flip-connectivity +Let G be a graph and P a partition of V(G). We will mostly be interested in the case when P +is the partition of V(G) into S-classes, for some finite S ⊆ V(G). In this case, in the notation +below we will write S instead of P. +The following notion is central in our proof of Theorem 1.4. We say that vertices a and b of +G are r-separated over P, denoted by5 +a +r +|⌣ +P +b, +if there exists a P-flip H of G such that distH(a, b) > r. We set +Br +P(a) := {b | b is not r-separated from a over P}. +5The symbol |⌣ denotes forking independence in stable theories. Its use here is justified by the relationship of +r-separation and forking independence in monadically stable theories, see next footnote. +13 + +We define a more general notion below, where instead of single vertices a and b we may have +sets of vertices. We also justify the use of the notation Br +P(a), by observing that this is indeed a +ball in some metric. +For any r ∈ N and any graph G, finite partition P of V(G), and sets A, B ∈ V(G), we write +A +r +|⌣ +P +B +if there exists a P-flip H of G such that H has no (A, B)-path of length at most r. If A |⌣ +r +P B we +say that A and B are r-separated over P. Note that when A ∩ B ̸= ∅, A and B are not r-separated +over any partition P. Note also that in the case when P is the partition into S-classes, we can +assume that every vertex in S is isolated in H, because this can be achieved through an S-flip. +If A consists of a single vertex a and B of a single vertex b, then we write a |⌣ +r +P b for A |⌣ +r +P B. +We use similar notation for a |⌣ +r +P B and A |⌣ +r +P b. Note that A |⌣ +r +P B is a stronger condition than +a |⌣ +r +P b for all a ∈ A and b ∈ B, since we require that the same set P and the same P-flip H +are used for all a ∈ A and b ∈ B. We write ̸ |⌣ +r +P to denote the negation of the relation |⌣ +r +P. If +A ̸ |⌣ +r +P B we say that A and B are r-connected over P. +For two vertices u, v, denote by distP(u, v) the smallest number r ∈ N such that u and v +are r-connected over P, or +∞ if no such number exists. This can be equivalently described as +follows. For u, v ∈ V(G), a P-flip-path from u to v in G is a collection consisting of one (u, v)- +path in H for each P-flip H of G. Note that a P-flip-path might not exist. The length of a P-flip +path is the supremum of the lengths of its paths. Then, equivalently, distP(u, v) is the infimum +of the lengths of all P-flip-paths (where if there is no P-flip-path then distP(u, v) = +∞). +Now we prove that distP(·, ·) is a metric, where we allow metrics to take on the value of ++∞. Thus, in particular, the relation distP(·, ·) < +∞ is an equivalence relation on V(G): the +set of all pairs (u, v) ∈ V(G)2 such that distP(u, v) < +∞ is transitive, reflexive, and symmetric. +Lemma 2.9 For any graph G and finite partition P of V(G), distP(·, ·) is a metric on V(G). +Proof. First of all, notice that for any different u, v ∈ V(G) it holds that u |⌣ +1 +P v, because we have +distH(u, v) > 1 either for H = G or for H being the complement of G, which is a P-flip. So in +general distP(a, b) = 0 if and only if a = b. From the definition we also have that distP(·, ·) +is symmetric. Finally, a P-flip-path from a to b can be naturally composed with a P-flip-path +from b to c in a path-by-path manner, yielding a P-flip-path from a to c. This proves the triangle +inequality. +Now that we know distP(·, ·) is a metric, it is sensible to define an analog of the “ball of +radius r around a vertex”. This is exactly the notion Br +P(v) defined earlier, as +Br +P(v) = {w ∈ V(G) | distP(v, w) ⩽ r}. +In case P is the partition into S-classes, for some finite S ⊆ V(G), we write A |⌣ +r +S B and Br +S(v) +to denote A |⌣ +r +P B and Br +P(v), respectively 6. This notion behaves well under taking subsets of +6We comment on the relationship between r-separation and forking independence in monadically stable graphs, +thus justifying the use of the symbol |⌣. As a subset of authors will show in subsequent work, for a monadically stable +graph G, its elementary superstructure H, and two vertices a, b ∈ V(H), a and b are forking independent over H if +and only if for every r ∈ N there is a finite S ⊆ G such that a |⌣ +r +S b. This is related to a result of Ivanov, characterising +forking independence in nowhere dense (or superflat) graphs [Iva93]. +14 + +S as follows. +Lemma 2.10 For any r ∈ N ∪ {+∞}, graph G, and finite sets T ⊆ S ⊆ V(G), if a is r-separated from +b over T, then a is also r-separated from b over S. +Proof. Since a |⌣ +r +T b, there exists a T-flip H of G such that H contains no (a, b)-path of length at +most r. As T ⊆ S, H is also an S-flip. Hence a |⌣ +r +S b. +3 +Variants of the Flipper game +We define three variants of the Flipper game. +Flipper game. +Fix a radius r. The Flipper game of radius r is played by two players, Flipper +and Connector, on a graph G as follows. At the beginning, set G0 := G. In the ith round, for +i > 0, the game proceeds as follows. +– If |Gi−1| = 1 then Flipper wins. +– Connector chooses a vertex v in Gi−1 and we set Gloc +i−1 to be the subgraph of Gi−1 induced +by the ball Br(v) of radius r around v in Gi−1. +– Flipper chooses an atomic flip F and applies it to produce Gi, i.e. Gi = Gloc +i−1 ⊕ F. +A k-round play of the Flipper game can be represented as a sequence of graphs G0, Gloc +0 , G1, +Gloc +1 , G2, . . . , Gloc +k−1, Gk, where G0 = G and Gloc +i +is the subgraph of Gi induced by the ball of radius +r around some v ∈ V(G) and Gi+1 is a (X, Y)-flip of Gloc +i +for some X, Y ⊆ V(Gloc +i +). Note that +V(Gi) ⊇ V(Gj) whenever i < j. +One can consider an extended variant of the Flipper game where Flipper in the ith move +applies a set F of flips to Gloc +i−1 to obtain Gi, where |F| ⩽ g(i) for some function g: N → N. This +does not change the game significantly – if Flipper wins this extended game in m rounds, then +Flipper wins the standard Flipper game in ∑m +i=1 g(i) rounds. +Pseudo-Flipper game. +We now introduce a variant of the Flipper game that will play an aux- +iliary role in our proofs. The main difference is that, while in the usual Flipper game, the entire +graph is replaced by the subgraph induced by a ball of radius r, after each move of Connec- +tor, and the balls are evaluated in smaller and smaller graphs, in the Pseudo-Flipper game, we +are keeping the original graph G, and keeping track of a set of vertices (called the arena) from +which Connector can pick the center of the next ball. Moreover, the ball is measured with re- +spect to the distance induced by |⌣ +r +P in the original graph G, where P is a partition picked by +Pseudo-Flipper. The precise definition follows. +Fix a radius r ∈ N. The Pseudo-Flipper Game of radius r is played by two players, Pseudo- +Flipper and Connector, as follows on a graph G. Let A0 = V(G) and F0 = {V(G)}. For +k = 1, 2, . . . , the kth round proceeds as follows. +– If |Ak−1| = 1 then Pseudo-Flipper wins. +– Otherwise, Connector picks ck ∈ Ak−1 and we set +Ak := Ak−1 − {w | w +r +|⌣ +Fk−1 +ck} = Ak−1 ∩ Br +Fi−1(ck) +15 + +( |⌣ +r +Fk−1 and Br +Fi−1(ck) are evaluated in the original graph G). +– Then Pseudo-Flipper chooses a partition Fk, obtained by taking Fk−1 and splitting one +part F ∈ Fk−1 into two non-empty, disjoint sets F1, F2 with F1 ∪ F2 = F. +Here, the sets Ak are called arenas. +Similarly to the case of the Flipper game, it might be convenient to consider the extended +variant of Pseudo-Flipper game in which Pseudo-Flipper refines Fi−1 to obtain Fi by splitting +g(i) sets in Fi−1 for some function g : N → N. Again, it doesn’t change the game significantly, +as whenever Pseudo-Flipper wins the extended game in m rounds, Pseudo-Flipper also wins +the standard Pseudo-Flipper game in ∑m +i=1 g(i) rounds. +Separation game. +A variant of the Pseudo-Flipper game in which the paritions are partitions +into S-classes, is called the Separation game, and is defined below. +Fix a radius r ∈ N. The Separation Game of radius r is played by two players, Separator +and Connector, on a graph G, as follows. Let A0 = V(G) and S0 = ∅. For k = 1, 2, . . . , the kth +round proceeds as follows. +– If |Ak−1| = 1, then Separator wins. +– Otherwise, Connector picks ck ∈ Ak−1 and we set +Ak := Ak−1 − +� +w +��� w +r +|⌣ +Sk−1 +ck +� +(where separation is evaluated in the graph G). +– Then Separator picks sk ∈ V(G) and we set Sk := Sk−1 ∪ {sk}, and proceed to the next +round. +Again as in the case of Flipper and Pseudo-Flipper games, we may allow Separator to add +g(i) vertices to Si−1 in the ith round, where g: N → N is some fixed function. Again, if Sep- +arator can win this new game in m rounds, then Separator can also win the original game in +∑m +i=1 g(i) rounds. +3.1 +Relating the game variants +We now prove easy relations between the variants of the game. First, we prove that that Pseudo- +Flipper can use Separator’s strategy to win the Pseudo-Flipper game. Then we prove that Flip- +per can use Pseudo-Flipper’s strategy to win the Flipper game. +Lemma 3.1 For every radius r and graph G the following holds. If Separator wins the Separation game +of radius r on G in at most k rounds, then Pseudo-Flipper wins the Pseudo-Flipper game of radius r on +G in at most 2k+1 rounds. +Proof. Consider the extended version of the Pseudo-Flipper game in which in ith move Pseudo- +Flipper can refine Fi−1 to obtain Fi by splitting at most 2i sets. If we show that Pseudo-Flipper +can win this game in k rounds, so he can win the standard game in ∑k +i=1 2i ⩽ 2k+1 rounds. +Consider the following strategy of Pseudo-Flipper, which will be just simulating Separator’s +strategy. When Connector makes her first move, Pseudo-Flipper asks Separator what he would +16 + +play. Separator answers that he would play a vertex v, thus obtaining S1 = {v}. Pseudo-Flipper +then splits F0 = {V(G)} into three S1-classes: {v}, vertices adjacent to v, and the remaining +vertices (non-adjacent to v). Then the game continues in this way – after ith move of Connector, +Pseudo-Flipper asks Separator what he would play, and then splits some sets of Fi−1 to Fi, so +that Fi is a partition into Si-classes. He can do it as follows. Assume by induction that Fi−1 is +a partition into Si−1-classes, so in particular |Fi−1| ⩽ 2i−1 + (i − 1) ⩽ 2i − 1. Assume also that +in ith move Separator picks a vertex u. Then, Pseudo-Flipper can split every set of Fi−1 so that +now Fi is a partition into Si−1 ∪ {u}-classes. This can be performed within at most |Fi−1| + 1 +splits: this entails splitting each Si−1-class into at most two parts, and then distinguishing {u} +as a separate class. Since |Fi−1| + 1 ⩽ 2i, that finishes the proof. +We now show that Flipper can use a winning strategy of Pseudo-Flipper from the game with +larger radius and at the cost of playing a few extra rounds. +Lemma 3.2 There exists a function f : N → N such that for every radius r and every graph G the +following holds. If Pseudo-Flipper wins the Pseudo-Flipper game of radius 2r on G in at most k rounds, +then Flipper wins the Flipper game of radius r on G in at most f (k) rounds. +Note that the converse direction, allowing to translate a winning strategy of Flipper in the +Flipper game, to a winning strategy in the Pseudo-Flipper game, is not immediately clear. The +reason is that in the Flipper game, Flipper has additional power since the balls are measured +in the current graph Gi, and are therefore potentially smaller than if they were measured in +(some flip) of the original graph G. In fact, we do not know of a direct proof of the converse +implication, but the equivalence of the two games ultimately follows from Theorem 1.4. +Before proving Lemma 3.2, we will state a simple lemma which will be used in its proof. Let +G be a graph and H an F-flip of G. It is easy to check that any two F-flips H, H′ of G are F-flips +of each other. More generally, since the sets used in atomic flips do not have to be subsets of the +graph we flip on, we have the following: +Lemma 3.3 Let G be a graph and H, H′ two F-flips of G. For every B ⊆ V(G) we have H′[B] is a +F-flip of H[B]. +We now proceed with the proof of Lemma 3.2. +Proof of Lemma 3.2. We will show how to use a winning strategy of Pseudo-Flipper in the Pseudo- +Flipper game of radius 2r played on G to win the Flipper game of radius r played on G for +suitably chosen f. We will play the two games on G in parallel, in such a way that one move in +the Pseudo-Flipper game corresponds to several moves in the Flipper game. +We remark that in the proof we will play the extended Flipper game, meaning that not only +we will play several rounds in the Flipper game for each round of the Pseudo-Flipper game, but +each round in the Flipper game actually consists of flipping between several pairs of subsets of +the graph we currently play on. From the proof it will follow that in kth round of the Flipper +game we play an F-flip of the current graph such that |F| is upper bounded by a number +depending only on k, as required. +For j ∈ N0 we denote by R(j) the maximum possible number of different F-flips of any +graph G and any F ⊆ P(V(G)), provided that |F| ⩽ j. For the sake of readability, we will +denote by Gi,b the graph in the Flipper game after ∑i +j=1 R(j − 1) + b moves. Note that with this +notation Gi+1,0 is the graph obtained from Gi,0 after R(i) moves. +The strategy of Flipper will be such that the following invariant is maintained: If Ai is the +arena in the Pseudo-Flipper game after i rounds, then Gi,0 is an induced subgraph of G[Ai]. +17 + +Note that at the beginning of the game this condition is satisfied, as G0,0 = G and A0 = V(G). +Showing that this invariant can be maintained proves the lemma, as after at most k − 1 rounds in +the Pseudo-Flipper game we have |Ak| = 1, which by our invariant implies |Gk,0| = 1, meaning +that Flipper wins. +For any i we now describe Flipper’s strategy to obtain Gi+1,0 from Gi,0. Let Ai be the arena +of the Pseudo-Flipper game after i rounds, Fi the partition chosen by Pseudo-Flipper in the ith +move and Gi,0 an induced subgraph of G[Ai]. Let p := R(i) be the number of all Fi-flips of G. +Flipper’s strategy will be chosen in such a way that after p rounds of play starting from Gi,0, +irrespective of Connector’s moves, we arrive at graph Gi,p = Gi+1,0 and a vertex c ∈ Ai such +that Gi+1,0 is an induced subgraph of G[Ai ∩ B2r +Fi(c)] = G[Ai+1], thus satisfying our invariant. +Intuitively, Flipper’s strategy is to play every possible F-flip of Gi,0 (with Connector making +moves in between) and then in the final round to flip back to (an induced subgraph of) Gi,0. +Formally, we get the following: +▷ Claim 3.4 Let H0, . . . , Hp be a sequence of graphs such that H0 = Hp = G and H1, . . . , Hp−1 are all +possible nontrivial Fi-flips of G. There is a strategy for Flipper in the p-round Flipper game with radius +r starting from Gi,0 such that irrespective of replies of Connector, if Gi,j is the graph after the jth move, +then Gi,j is an induced subgraph of Hj. +Proof of the claim. This clearly holds for j = 0, since Gi,0 is an induced subgraph of G[Ai+1] +which is an induced subgraph of G = H0. We now argue that assuming that Gi,j is an induced +subgraph of Hj, Flipper can guarantee that Gi,j+1 is an induced subgraph of Hj+1: First, going +from Gi,j to Gloc +i,j (Connector’s move) is done by taking an induced subraph of Gi,j. Therefore, +Gloc +i,j = Gi,j[B] = Hj[B] for some B ⊆ V(G). By Lemma 3.3 we have that Hj+1[B] is an Fi-flip of +Hj[B], and so Flipper can set Gi,j+1 to be Hj+1[B]. +◁ +Now consider p rounds of play of the Flipper game starting from Gi,0 in which Flipper plays +according to the strategy given above, ending with graph Gi+1,0 = Gi,p. It remains to define +the choice of vertex c. We set c to be the last vertex played by Connector in the p round play. +That is, c is the vertex played by Connector in Gi,p−1 to obtain Gloc +i,p−1 = Br(c), where the ball of +radius r is taken in Gi,p−1. Since Gi+1,0 = Gi,p is obtained by performing flips on Gloc +i,p−1, the two +graphs have the same vertex sets and we have c ∈ V(Gi+1,0). +We now prove that Gi+1,0 is an induced subgraph of G[Ai+1]. Since Gi+1,0 is an induced +subgraph of G, it suffices to show that V(Gi+1,0) ⊆ Ai+1. Because Ai+1 = Ai ∩ B2r +Fi(c) and +V(Gi+1,0) ⊆ Ai, we need to show that V(Gi+1,0) ⊆ B2r +Fi(c). We will prove this by contraposition +– we will show that for any w ̸∈ B2r +Fi(c) we have w ̸∈ V(Gi+1,0). From the definition, w ̸∈ B2r +Fi(c) +means there exists an Fi-flip H of G such that distH(c, w) > 2r. We know that H has to be one +of H0, . . . , Hp−1 from Claim 3.4, say Hj, meaning that distHj(c, w) > 2r. From Claim 3.4 we have +that Gi,j is an induced subgraph of Hj, and so in Gi,j the distance between c and w is also more +than 2r. Then, if v is the vertex chosen by Connector in Gi,j to obtain Gloc +i,j = Br(v) (here the +ball around v is taken in Gi,j), it cannot be the case that both c and w are in Gloc +i,j , because Gloc +i,j +has radius at most r and so one could join c with w by a path of length at most 2r going from +c to v and from there to w, contradicting that distGi,j(c, w) > 2r. This means that at most one +of c and w can be in V(Gloc +i,j ) and consequently in V(Gi,j+1) (because the two graphs have the +same vertex set). But since we know that c ∈ V(Gi+1,0) and V(Gi+1,0) ⊆ V(Gi,j+1), we have +18 + +c ∈ V(Gi,j+1) and therefore w ̸∈ V(Gi,j+1). Since V(Gi+1,0) ⊆ V(Gi,j+1) for j ⩽ p, we have +w ̸∈ V(Gi+1,0), which finishes the proof. +4 +Relations to flip-wideness and existential monadic stability +In this section we prove that the existence of short winning strategies in variants of the Flipper +game implies two properties that are known to be equivalent to monadic stability: flip-wideness +and existential monadic stability. +4.1 +From Pseudo-Flipper game to flip-wideness +This subsection is devoted to proving the following lemma. +Lemma 4.1 Let C be a class of graphs such that for every r there exists k such that Pseudo-Flipper wins +the Pseudo-Flipper game of radius r on any G ∈ C , in at most k rounds. Then C is flip-wide. +Together with Lemma 3.1, this yields the following corollary, proving the implication (5)→(7) +in Theorem 1.4. +Corollary 4.2 Let C be a class of graphs such that for every r there exists k such that Separator wins the +Separation game of radius r on any G ∈ C , in at most k rounds. Then C is flip-wide. +Proof of Lemma 4.1. Let r ∈ N be arbitrary. Let k be a number of rounds such that Pseudo- +Flipper wins the Pseudo-Flipper game of radius r on any G ∈ C in at most k rounds; such k +exists by our assumptions. Let c be the maximal possible number of different F-flips on any +graph G with respect to any F with |F| ⩽ k; it is easily seen that such number exists (the +upper bound does not depend on any particular G and F, only on k). For m ∈ N denote by +R(m) the least number N such that any coloring of the edges of a clique with N vertices using +c colors yields a monochromatic clique of size m. Such a number exists by Ramsey’s theorem. +Set sr := k2 and for m ∈ N set Nr(m) := R(m)k. Let m ∈ N and let G ∈ C and A ⊆ V(G) with +|A| > Nr(m). From the definition of the Pseudo-Flipper game of radius r follows that whenever +Pseudo-Flipper can win such game in k rounds, then he can also win in k rounds if we decide +that the initial arena A0 is a subset of V(G). Indeed, in every round we intersect the previous +arena with something that depends only on the partition played by Pseudo-Flipper. Therefore, +set A0 := A. We will extract a partition F and an F-flip H of G with the desired properties +from a play of the Pseudo-Flipper game played on G. We define a strategy for Connector in the +Pseudo-Flipper game played on G as follows. Let Fi denote the partition of V(G) resulting from +Pseudo-Flipper’s choices after i rounds. For i > 0, in the ith round, if there is a vertex v ∈ Ai−1 +such that |Br +Fi−1(v) ∩ Ai−1| > R(m)k−i then Connector picks v as ci, otherwise Connector picks +any vertex in Ai−1. In other words, Connector tries to make sure that |Ai| > R(m)k−i in ith +round whenever possible. +Consider now a play of the Pseudo-Flipper game played on G in which Connector plays +according to the strategy described above and Pseudo-Flipper plays according to an optimal +strategy which leads to a win in at most k rounds. We then get the following. +▷ Claim 4.3 There exists i < k such thatAi−1 ⩾ R(m)k−i+1 and |Br +Fi−1(v) ∩ Ai−1| ⩽ R(m)k−i for +each v ∈ Ai−1. +Proof of the claim. Since Pseudo-Flipper wins in k rounds, there has to exist i < k such that +|Br +Fi−1(v) ∩ Ai−1| ⩽ R(m)k−i for each v ∈ Ai−1 (otherwise after k − 1 rounds we would have +19 + +that |Ak−1| > R(m) ⩾ 1, which would contradict Pseudo-Flipper winning after k rounds). +Consider the smallest i with this property. This means that no matter which vertex v ∈ Ai−1 +Connector plays, the arena Ai has size less than R(m)k−i. Since i was the smallest such number, +we have that Ai−1 ⩾ R(m)k−i+1. +◁ +Let i be the number from Claim 4.3. We greedily construct a subset A′ of Ai−1 by repeatedly +picking a vertex v in Ai−1 and removing Br +Fi−1(v) ∩ Ai−1 from Ai−1 for as long as possible. Be- +cause of the bounds given by Claim 4.3, set A′ will have size at least R(m), and by construction +all vertices in A′ will be pairwise r-disconnected over F in G. +Assign to each unordered pair u, v of distinct vertices in A′ a color which represents an F- +flip H of G such that distH(u, v) > r (if there is more than one such F-flip, pick one arbitrarily). +This way we assign at most c colors to pairs of vertices from a set of size at least R(m), and +therefore by the definition of R(m) there exists a subset A′′ of A′ with |A′′| ⩾ m such that each +pair u, v ∈ A′′ is assigned the same F-flip H. Thus we have distH(u, v) > r for each u, v ∈ A′′, +and so F and H have the properties required by the definition of flip-wideness. This means +that C is flip-wide, which finishes the proof. +4.2 +From Flipper game to existential monadic stability +In this subsection we will show that a winning strategy for Flipper in the Flipper game implies +existential monadic stability. Since existential monadic stability implies stability of the edge +relation, and existential monadic dependence, this will prove the implication (6)→(2) in Theo- +rem 1.4. For simplicity, we will concentrate on the case when we can define an infinite ladder in +a single graph. +We will say that a formula ϕ(x, y) defines an infinite ladder in a structure M if there is a +sequence (ai, bi)i∈N of pairs of elements of M such that for every i, j ∈ N +M |= ϕ(ai, bj) ⇐⇒ i < j. +Lemma 4.4 Let M be a graph and let � +M be a monadic lift of M. Let ϕ(x, y) be an existential formula +that defines an infinite ladder in � +M. Then there exists r ∈ N such that Connector can play infinitely +many rounds in the Flipper game of radius r without losing. +The statement of Lemma 4.4 can be restated for the case when we can define arbitrarily long +ladders in graphs from a given class. +Lemma 4.5 (Finitary version of Lemma 4.4) Let C be a class of graphs and let ϕ(x, y) be an existential +formula such that for every k ∈ N there exists a graph Mk ∈ C and a monadic lift � +Mk in which ϕ(x, y) +defines a ladder of length at least k. Then there exists r ∈ N such that for every ℓ ∈ N there is a graph +Nℓ ∈ C with the following property: in the Flipper game of radius r with the initial current graph Nℓ +Connector can play at least ℓ rounds without losing. +Of course Lemma 4.5 gives us implication (6)→(2) from Theorem 1.4. For the rest of this +section we will concentrate on proving Lemma 4.4. It can be easily observed that the proof can +be restated for Lemma 4.5. +Let M be a monadic lift of a graph. For an integer r, we say that a set A of vertices of M +is r-close if the vertices of A are pairwise at distance at most r. In particular, if (ai, bi)i∈N are +the vertices of a ladder defined by some formula, we say that this ladder is r-close if the set +{ai | i ∈ N} ∪ {bi | i ∈ N} is. +20 + +A useful tool for the proof of Lemma 4.4 will be an easy corollary from Gaifman’s locality +theorem. The theorem was originally proven in [Gai82], but we will use a corollary of it, similar +to the one from [BDG+22, Lemma 2.1]. +Theorem 4.6 (Corollary of Gaifman’s locality theorem) Let ϕ(x, y) be an FO formula in the vocab- +ulary of graphs with a number of additional unary predicates. Then there are numbers t, s ∈ N with t +depending only on the quantifier rank of ϕ such that for every graph G with a number of additional unary +predicates, G can be vertex-colored using s colors in such a way that for any two vertices u, v ∈ V(G) at +distance more than t, whether or not ϕ(u, v) holds depends only on the color of u and the color of v. +Using Theorem 4.6 we can prove that whenever ϕ defines an infinite ladder, then it also +defines an infinite d-close ladder. +Lemma 4.7 Let M be a monadic lift of a graph and let ϕ(x, y) be a formula which defines an infinite +ladder in M. Then, there is an integer d depending only on the quantifier rank of ϕ(x, y) such that +ϕ(x, y) defines in M an infinite d-close ladder. +Proof. Observe, that by Theorem 4.6, there exists a constant t depending only on the quantifier +rank of ϕ and a vertex-coloring of M with s colors with the following property: if a, b, a′, b′ are +four vertices of M such that dist(a, b), dist(a′, b′) > t, and colors of a and a′ are equal, and colors +of b and b′ are equal, then +M |= ϕ(a, b) ⇐⇒ M |= ϕ(a′, b′). +Take an infinite ladder (ai, bi)i∈N defined by ϕ. Let us assign to every pair of natural num- +bers (i, j) with i < j with one of three colors which correspond to three possible scenarios (if +more than one scenario holds, we pick an arbitrary one): +– dist(ai, bj) ⩽ t, +– dist(bi, aj) ⩽ t, +– dist(ai, bj) > t and dist(bi, aj) > t. +By Ramsey’s theorem, there exists an infinite subladder (aij, bij)j∈N defined by ϕ such that every +pair (ij, ik) for j < k has the same color. It is straightforward to see that in the first two cases the +vertices {aij | j ⩾ 2} ∪ {bij | j ⩾ 2} are pairwise at distance at most 3t. It remains to deal with +the last case – we will show that, in fact, it cannot hold. +Color the vertices of the graph into a finite number of colors according to Theorem 4.6. By +the pigeonhole principle, we can assume that all aij have been assigned the same color and, +separately, all bij have been assigned the same color. However, that means +M |= ϕ(aij, bik) ⇐⇒ M |= ϕ(aik, bij) +for any j ̸= k. This is clearly a contradiction, so we conclude it is enough to take d := 3t. +To prove Lemma 4.4, we will present a strategy for Connector. We start with an infinite +ladder defined in M by an existential formula ϕ(x, y) in prenex normal form of quantifier rank at +most q (i.e., ϕ(x, y) ≡ ∃¯z.α(x, y, ¯z) where |¯z| ⩽ q and α is a quantifier-free formula). The strategy +will maintain the following invariant: after each round there is an existential formula ϕ′(x, y) +which defines an infinite ladder in some monadic lift of the current graph. The proof will follow +from two lemmas, corresponding to the moves of Flipper and Connector. +21 + +Lemma 4.8 Let M be a graph and let � +M be a monadic lift of M. Let ϕ(x, y) be an existential formula of +quantifier rank q in prenex normal form which defines an infinite ladder in � +M. Let N be a flip of M. Then +there exists a monadic lift �N of N and a formula ψ(x, y) of quantifier rank q in prenex normal which +defines an infinite ladder in �N. +Proof. Let (ai, bi)i∈N be an infinite ladder defined by ϕ(x, y) in � +M. We define �N by adding the +same unary predicates as in � +M and two additional unary predicates that mark the sets that were +flipped in M. We also define ψ(x, y) by changing the atomic check E(u, v) in ϕ(x, y) to a more +complicated quantifier-free formula verifying if there exists an edge between u and v in M and +whether u and v were included in the flipped sets. Of course, ψ(x, y) has the same quantifier +rank as ϕ(x, y) and is in prenex normal form. Moreover, for every i, j ∈ N, +� +M |= ϕ(ai, bj) ⇐⇒ �N |= ψ(ai, bj). +Lemma 4.9 Let M be a graph and let � +M be a monadic lift of M. Let ϕ(x, y) be an existential formula +of quantifier rank q in prenex normal form which defines an infinite ladder in � +M. There exists an integer +r depending only on q, a formula ψ(x, y) of quantifier rank at most q in prenex normal form, a monadic +lift � +M′ of M, and an element m ∈ M such that ψ(x, y) defines an infinite ladder in � +M′[Br(m)]. +Proof. By Lemma 4.7 we can assume that ϕ(x, y) defines a d-close ladder (ai, bi)i∈N. As ϕ(x, y) ≡ +∃¯z.α(x, y, ¯z), for every i < j we can find a tuple ¯cij such that α(ai, bj, ¯cij) holds in � +M. +For every tuple ¯cij we define its profile as a function +πij : ¯z → [q] ∪ {∞}, +where πij(z) for a variable z ∈ ¯z is defined as follows. Let cij +z be the element of the tuple ¯cij +corresponding to the variable z. For ℓ ∈ [q], we set πij(z) = ℓ if the distance between ai and cij +z +is at least 10(ℓ − 1)d and at most 10ℓd − 1. However, if the distance between ai and cij +z is at least +10qd, we set πij(z) = ∞. +By a standard Ramsey argument, we can assume that for every i < j all tuples (ai, bj, ¯cij) +have the same atomic type τ and all ¯cij have the same profile function π (possibly by going to +an infinite subladder of (ai, bi)i∈N). If π does not assign ∞ to any coordinate, then all ¯cij are in +the ball of radius r := 10qd + d around a1. Therefore, after restricting M to the ball of radius r +around a1 we still have for all i, j ∈ N that +� +M[Br(a1)] |= ϕ(ai, bj) ⇐⇒ i < j. +(Note that it is important here that ϕ is an existential formula; if not for that assumption, ϕ(ai, bj) +could have become true in � +M[Br(a1)] for some i ⩾ j.) +Now assume that π does assign ∞ to some coordinate. Therefore, by the pigeonhole prin- +ciple, there exists s ∈ [q] such that no variable is assigned s by π. We split the variables z ∈ ¯z +into two parts – these where π(z) < s (call these variables close), and those where π(z) > s (call +those far). Observe now that if i < j and z is close, then dist(ai, cij +z ) < 10(s − 1)d by definition, +and thus dist(a1, cij +z ) < (10s − 9)d by triangle inequality. On the other hand, if z is far, then +dist(ai, cij +z ) ⩾ 10sd, and therefore dist(a1, cij +z ) ⩾ (10s − 1)d, again by triangle inequality. In par- +ticular, the atomic type τ specifies that there are no edges between the vertices assigned to close +22 + +variables and the vertices assigned to far variables: if z is close and z′ is far, then dist(cij +z , cij +z′) > 1. +Let us create � +M′ by adding to � +M one more unary predicate U satisfied for the vertices that +are at distance at most (10s − 9)d from a1. By construction, every vertex of the ladder is in U, +and for every pair i < j and every variable z ∈ ¯z, the vertex cij +z is in U if and only if z is close. +Let τ′ be the atomic type obtained from τ by additionally specifying which variables should +satisfy U. Consider also ϕ′(x, y) ≡ ∃¯z.τ′(x, y, ¯z). It is immediate that for any i, j ∈ N, +� +M′ |= ϕ′(ai, bj) ⇐⇒ i < j. +Now, simplify ϕ′ to ϕ′′ by removing the quantifiers that correspond to far variables (of course, +we also need to simplify τ′ to τ′′ by removing the same variables). It is again obvious that +� +M′ |= ϕ′′(ai, bj) for i < j. We will show that � +M′ ̸|= ϕ′′(ai, bj) for i ⩾ j. +Assume by contradiction that for some i ⩾ j we have a tuple ¯c such that � +M′ |= τ′′(ai, bj, ¯c). +By the construction of τ′′, the vertices ai and bj and each element of ¯c must satisfy U and thus +each of them is at distance at most (10s − 9)d from a1. Consider now extending this tuple to +¯c′ by adding the vertices of ¯c12 corresponding to the far variables of ¯z; recall that these vertices +are at distance at least (10s − 1)d from a1. Clearly, � +M′ |= τ′(ai, bj, ¯c′), as the vertices which +we used for extending ¯c do not neighbor any vertex from ¯c, ai or bj. This is a contradiction to +� +M′ ̸|= ϕ′(ai, bj). Therefore, again we have +� +M′ |= ϕ′′(ai, bj) ⇐⇒ i < j, +and by using the same argument as previously we have +� +M′[Br(a1)] |= ϕ′′(ai, bj) ⇐⇒ i < j. +Using Lemma 4.8 and Lemma 4.9 we can easily prove Lemma 4.4. +Proof of Lemma 4.4. Rewrite formula ϕ(x, y) to a formula ϕ′(x, y) in prenex normal form. Denote +its quantifier rank by q and take r as in Lemma 4.9. Assume that we consider the Flipper game +of radius r on M. +We call a graph A winning if there exists a monadic lift �A of A and a formula ψ(x, y) of quan- +tifier rank at most q in prenex normal form which defines an infinite ladder in �A. Obviously, if +a graph is winning, then it cannot be a single vertex. +Observe that by our assumption, the initial current graph M is winning. Then, by Lemma 4.8, +if Flipper does a flip on a winning current graph, the resulting current graph is also winning. +Finally, by Lemma 4.9, for every winning current graph there exists a move of Connector such +that the resulting current graph is also winning. Therefore, if Connector always picks such +moves, she plays infinitely many rounds without losing. +Part II +Model-theoretic proof +In this part we prove the implications (2)→(3)→(4)→(5) of Theorem 1.4 using elementary tech- +niques from model theory. +23 + +In Section 5 we give additional model-theoretic preliminaries: we discuss models, theories, +compactness, the Tarski-Vaught test, definability of types, a variant of Morley sequences, and +some basic lemmas about ladders. +In Section 6 we introduce pattern-free classes, and prove several simple facts about them. +In particular, we prove that every existentially monadically dependent class is pattern-free, +proving the implication (2)→(3) in Theorem 1.4. Section 7 presents the main technical step, +the implication (3)→(4). In a nutshell, from pattern-freeness and edge-stability we derive a +model-theoretic property, a variant of definability of types, which allows us to control elements +in elementary extensions through finite sets in the ground model. +In Section 8, we use this definability property together with compactness of first-order logic +to give a strategy for Separator that ensures victory in the Separation game in a bounded num- +ber of rounds. This is the implication (4)→(5). +We start with introducing the necessary tools. +5 +Additional model-theoretic preliminaries +A theory T (over Σ) is a set of Σ-sentences. A model of a theory T is a model M such that M |= ϕ +for all ϕ ∈ T. When a theory has a model, it is said to be consistent. The theory of a class of +Σ-structures C is the set of all Σ-sentences ϕ such that M |= ϕ for all M ∈ C . The elementary +closure C of C is the set of all models M of the theory of C . Thus C ⊆ C , and C and C have +equal theories. +We will use compactness for first-order logic and the Tarski-Vaught test, recalled below. +Theorem 5.1 (Compactness) A theory T is consistent if and only if every finite subset T′ of T is +consistent. +Let M and N be two structures with M ⊆ N, that is, the domain of M is contained in the +domain of N. Then N is an elementary extension of M, written M ≺ N, if for every formula ϕ( ¯x) +(without parameters) and tuple ¯m ∈ M ¯x, the following equivalence holds: +M |= ϕ( ¯m) ⇐⇒ N |= ϕ( ¯m). +We also say that M is an elementary substructure of N. In other words, M is an elementary +substructure of N if M is an induced substructure of N, where we imagine that M and N are +each equipped with every relation Rϕ of arity k (for k ∈ N) that is defined by any fixed first- +order formula ϕ(x1, . . . , xk). In this intuition, formulas of arity 0 correspond to Boolean flags, +with the same valuation for both M and N. +Theorem 5.2 (Tarski-Vaught Test) The following conditions are equivalent for any structures M and +N with M ⊆ N. +– The structure N is an elementary extension of M. +– For every formula ϕ(y; ¯x) and tuple ¯m ∈ M ¯x, if N |= ϕ(n; ¯m) holds for some n ∈ N, then +N |= ϕ(n′; ¯m) holds for some n′ ∈ M. +Fix a model M over a signature Σ. A Σ-formula ϕ( ¯x) with parameters from a set A ⊆ M is a +formula ϕ( ¯x) over the signature Σ ⊎ A, where the elements of A are treated as constant symbols +(which are interpreted by themselves). If ϕ( ¯x) is a formula (with or without parameters) and +U ⊆ M, then by ϕ(U) we denote the set of all ¯x-tuples ¯u ∈ U ¯x such that M |= ϕ( ¯u). Now let +24 + +A, B ⊆ M be sets, and let ϕ(x; y) a formula (here x and y are single variables). A ϕ-type of A +over B is an equivalence class of the relation ∼ on A such that for a, a′ ∈ A we have a ∼ a′ if +and only if ϕ(a; B) = ϕ(a′; B), that is, +M |= ϕ(a; b) ⇐⇒ ϕ(a′; b) +for all b ∈ B. +We denote the set of ϕ-types of A over B by Typesϕ(A/B). +The following result is a fundamental fact about stable formulas (see e.g. [Pil96, Lemma 2.2 +(i)]). +Theorem 5.3 (Definability of types) Let M ≺ N be two models and ϕ(x; y) be a formula that is +stable in M. For every n ∈ N there is some formula ψ(x) with parameters from M, which is a positive +boolean combination of formulas of the form ϕ(x; m) for m ∈ M, such that the following conditions are +equivalent for every a ∈ M: +– N |= ϕ(n; a) holds, +– M |= ψ(a) holds. +The following lemma is reminiscent of the classic notion of Morley sequences from model +theory (see e.g. [Pil96, Def. 2.27]). +Lemma 5.4 Let M ≺ N be graphs, let ¯n ∈ N ¯y, and let A ⊆ M be a finite set. There is an infinite +sequence ¯b0, ¯b1, . . . ∈ M ¯y such that +– for each i ∈ N, ¯bi has the same atomic type as ¯n over A ∪ ¯b0 ∪ · · · ∪ ¯bi−1; and +– the atomic types of the tuples ¯bi¯bj are the same for all i < j. +If moreover N has a stable edge relation, one may choose the sequence (¯bi)i∈N so that the atomic types of +¯bi¯bj are the same for all i ̸= j. +Proof. We construct the sequence ¯b0, ¯b1, · · · ∈ M ¯y satisfying the first condition by induction. +Let i ∈ N and assume that the elements ¯b0, . . . , ¯bi−1 have already been constructed and follow +the first condition (this assumption is vacuous for i = 0). Let α( ¯y) be the conjunction of all +(finitely many) formulas in the atomic type of ¯n over A ∪ ¯b0 ∪ · · · ∪ ¯bi−1. Since N |= α( ¯n), and +A ∪ ¯b0 ∪ · · · ∪ ¯bi−1 ⊆ M, and M ≺ N, it follows that there exists a tuple in M ¯y satisfying α( ¯y). +(Formally, let ¯u be a tuple enumerating the elements of A ∪ ¯b0 ∪ · · · ∪ ¯bi−1, and ϕ( ¯y, ¯z) be the +formula without parameters for which ϕ( ¯y, ¯u) = α( ¯y). We apply the definition of an elementary +extension to the formula ∃ ¯y.ϕ( ¯y, ¯u) and infer that the formula holds in M. Denote by ¯bi ∈ M ¯y +any tuple satisfying ϕ(¯bi, ¯u), concluding the induction step.) +By construction, the sequence we have constructed satisfies the first item in the statement of +the lemma. For the second item, it suffices to apply the infinite Ramsey theorem, as there are +finitely many possible atomic types for ¯bi¯bj’s. +Assume now that N has a stable edge relation. By the second item, we may assume the +atomic types of tuples ¯bi¯bj for i < j are all equal to each other, and hence, the atomic types of tu- +ples ¯bj¯bi for i < j are also equal to each other; assume for contradiction that these are two differ- +ent types. Since the atomic types of all ¯bi’s are identical, this implies that there are two different +coordinates y1, y2 of ¯y such that for all i < j, the corresponding coordinates bi,1, bi,2, bj,1, bj,2 ∈ M +of ¯bi and ¯bj are such that +bi,1 and bj,2 are adjacent in M, whereas bi,2 and bj,1 are not. +25 + +Thus the semi-induced bipartite graph in M between the bi,1’s and the bi,2’s is an infinite ladder, +a contradiction to the stability of N’s edge relation. +For two sets A, B of vertices in a given graph, we say that A dominates B if for any b ∈ B +there is a ∈ A such that ab is an edge. Likewise, we say that A antidominates B if it dominates B +in the complement graph: for each b ∈ B there is a ∈ A such that ab is not an edge. +Lemma 5.5 Let M be a graph with a stable edge relation, and A, B ⊆ M be two sets of vertices. Then +one of the following holds: +– there is a finite subset S ⊆ A that dominates B, +– there is a finite subset S ⊆ B that antidominates A. +We note that a variant of the lemma holds even in dependent models [Sim15, Corollary +6.13], [BDG+22, Theorem 2.4], under the additional assumption that A and B are finite. We +only require it for (monadically) stable models, for which it admits a simple proof given below. +Proof. We describe an iterative process that either gives one of the two desired outcomes, or +constructs an infinite ladder. Let i ⩾ 1, assume that a1, . . . , aℓ−1 as well as b1, . . . , bℓ−1 are already +constructed so that for i, j < ℓ, aibj is an edge if and only if i ⩾ j. Then +– if {a1, . . . , aℓ−1} dominates B, then we are done; otherwise there is b ∈ B − {b1, . . . , bℓ−1} +with no adjacencies to {a1, . . . , aℓ−1}. Pick such a b and call it bℓ; then +– if {b1, . . . , bℓ} antidominates A, then we are done; otherwise there is a ∈ A − {a1, . . . , aℓ−1} +adjacent to each b1, . . . , bℓ. Pick such an a and call it aℓ. +This extends our ladder, as required. +An infinite matching is the bipartite graph on vertices {ai, bi : i ∈ N} such that aibj is an edge +if and only if i = j. An infinite co-matching is defined in the same way, except there is an edge +aibj if and only if i ̸= j. (Recall that an infinite ladder is again defined similarly, but with the +condition i < j.) The following result is folklore. An equivalent, finitary formulation can be +found e.g. in [DOOV96, Corollary 2.4]. +Theorem 5.6 Let E ⊆ A × B be an infinite bipartite graph. Then one of the following cases holds: +1. TypesE(A/B) is finite, +2. E contains an infinite induced matching, +3. E contains an infinite induced co-matching, +4. E contains an infinite induced ladder. +Lemma 5.7 Let C be a graph class with a stable edge relation. Then every graph M in the elementary +closure C of C has a stable edge relation. +Proof. For k ∈ N, let ϕk be the sentence that holds in a graph G if and only if there are vertices +a1, . . . , ak and b1, . . . , bk in G such that for all 1 ⩽ i, j ⩽ k, G |= E(ai, bj) if and only if i ⩽ j. Since +C has a stable edge relation, there is a number k such that G |= ¬ϕk for all G ∈ C . Hence, +M |= ¬ϕk, proving that M has a stable edge relation. +The following lemmas will be useful in simplifying the inductive proof of Theorem 7.1. +26 + +Lemma 5.8 Let M and N be graphs such that N is an elementary extension of M. Further, let S ⊆ M +be any finite set and N′ be any S-flip of N. Then N′ is an elementary extension of the subgraph of N′ +induced by the domain of M. +Proof. Let M′ denote the subgraph of N′ induced by the domain of M. Observe first that there +is a quantifier-free formula η(x, y) with parameters from S such that for all u, v ∈ N, +N |= η(u, v) ⇐⇒ N′ |= E(u, v). +Let ϕ( ¯x) be a formula and ¯m ∈ M ¯x a tuple. We need to show that +M′ |= ϕ( ¯m) ⇐⇒ N′ |= ϕ( ¯m). +(1) +Rewrite ϕ( ¯x) to a formula ϕ′( ¯x) with parameters from S, by replacing each occurrence of an +atom E(x, y) with η(x, y). Then we have that +N′ |= ϕ( ¯m) ⇐⇒ N |= ϕ′( ¯m), +and similarly, +M′ |= ϕ( ¯m) ⇐⇒ M |= ϕ′( ¯m). +Since N is an elementary extension of M and S ⊆ M, we have that +M |= ϕ′( ¯m) ⇐⇒ N |= ϕ′( ¯m). +Putting together the equivalences yields (1). +6 +Pattern-free classes +If ϕ(x, y) is a first-order formula and G is a structure, by ϕ(G) we denote the graph with vertices +V(G) and edges uv, for distinct u, v ∈ V(G) such that G |= ϕ(u, v) ∨ ϕ(v, u). If C is a class of +structures, then denote ϕ(C ) := {ϕ(G) | G ∈ C }. Say that a class C of graphs transduces a +class D of graphs if there is a unary expansion � +C of C and a formula ϕ(x, y) in the signature +of � +C , such that for every H ∈ D there is some �G ∈ +� +C such that H is an induced subgraph of +ϕ( �G). If above, the formula ϕ is existential, resp. quantifier-free, then we say that C existentially +transduces D, resp. quantifier-free transduces D. +The r-subdivision of a graph G, denoted G(r) below, is obtained by replacing each edge of +G by a path of length r + 1. A (1, r)-subdivision of a graph G is a graph obtained from G by +replacing each edge of G by a path of length at least 2, and at most r + 1. +Definition 2. Say that a graph class C is pattern-free if for every r ⩾ 1, unary expansion � +C of C , +and quantifier-free formula ϕ(x, y) in the signature of � +C , there is some n ⩾ 1 such that ϕ( � +C ) +avoids the r-subdivision of Kn as an induced subgraph. Say that a graph M is pattern-free if the +class {M} is pattern-free. +Note that C is not pattern-free if and only if C quantifier-free transduces the class of r- +subdivisions of all cliques, for some fixed r ⩾ 1. +In this section we study pattern-free classes. In Proposition 6.1 we prove that every simply +existentially monadically dependent class is pattern-free, proving implication (2)→(3) in Theo- +rem 1.4. In Lemma 6.5 we prove that if C is pattern-free then every M ∈ C is pattern-free. In +27 + +Lemma 6.10 we prove that if M is a pattern-free graph and M′ is obtained from M by applying +a finite set of flips, then M′ is also pattern-free. In Lemma 6.9 we describe certain obstructions +that are forbidden in pattern-free graphs M. +Proposition 6.1 If a graph class C is simply existentially monadically dependent, then C is pattern-free. +Proposition 6.1 immediately follows from the next three lemmas. +Lemma 6.2 If C is not pattern-free, then for some r ⩾ 1, the class C quantifier-free transduces the class +{G(r) | G is a graph} of r-subdivisions of all graphs. +Lemma 6.3 Fix r ⩾ 0. Let C be a hereditary graph class that quantifier-free transduces the class of +r-subdivisions of all graphs. Then C existentially transduces the class of all graphs. +Lemma 6.4 Let C be a graph class that existentially transduces the class of all graphs. Then C is not +simply existentially monadically dependent. +Proof of Lemma 6.2. Suppose C is not pattern-free. Then there is a unary expansion � +C and a +quantifier-free formula ϕ(x, y), such that for every n ⩾ 1 there is some �Gn ∈ � +C of G, such that +K(r) +n +is an induced subgraph of ϕ( �G). +We argue that C quantifier-free transduces the class of r-subdivisions of all graphs. +Let G be a graph with vertices {1, . . . , n}. Let Gn and �Gn be as above, so that K(r) +n +is an +induced subgraph of ϕ( �G). Since G(r) is an induced subgraph of K(r) +n , it is also an induced +subgraph of ϕ( �G). Therefore, C quantifier-free transduces the class of r-subdivisions of all +graphs. +Proof of Lemma 6.3. Assume that C quantifier-free transduces the class of r-subdivisions of all +graphs. Then there is a unary expansion � +C of C and a quantifier-free formula ϕ(x, y) such that +for every graph G for some �HG ∈ C such that G(r) is an induced subgraph of ϕ( �HG). Without +loss of generality, the class � +C is hereditary, since C is. +We may assume that V( �HG) = V(G(r)). This is because V(G(r)) ⊆ V( �HG), and if �H′ +G +denotes the substructure of �HG induced by V(G(r)), then ϕ( �H′ +G) = ϕ( �HG)[V(G(r))] = G(r), +where the first equality holds since ϕ is quantifier-free. Moreover, �H′ +G ∈ � +C since � +C is hereditary. +We therefore assume that V( �HG) = V(G(r)), for all graphs G. We may also assume that the +formula ϕ(x, y) is symmetric and irreflexive, that is, ∀x, y.ϕ(x, y) → (x ̸= y) ∧ ϕ(y, x) holds in +every k-colored graph. (Otherwise replace ϕ(x, y) by the formula (ϕ(x, y) ∨ ϕ(y, x)) ∧ x ̸= y.) +We show that C existentially transduces the class of all graphs. +Fix any finite graph G. Then for u, v ∈ G(r) we have that �HG |= ϕ(u, v) if and only if +G(r) |= E(u, v). +Let W ⊆ V(G(r)) denote the set of vertices of G(r) that correspond to the original vertices of +G. In particular, if η(x, y) denotes the formula +η(x, y) := ∃x1 . . . xr.E(x, x1) ∧ E(x1, x2) ∧ · · · ∧ E(xr−1, xr) ∧ E(xr, y), +then η(G(r))[W] is isomorphic to G. Now consider the formula +ψ(x, y) := ∃x1 . . . xr.ϕ(x, x1) ∧ ϕ(x1, x2) ∧ · · · ∧ ϕ(xr−1, xr) ∧ ϕ(xr, y). +It follows that for u, v ∈ W we have that �HG |= ψ(u, v) if and only if G(r) |= η(u, v). In particular, +ψ( �HG)[W] is isomorphic to G. +28 + +Therefore, the existential formula ψ(x, y) (which does not depend on G), witnesses that C +existentially transduces the class of all graphs. +Proof of Lemma 6.4. Suppose that C existentially transduces the class of all graphs. Then there +is an existential formula ψ(x, y) such that every finite graph G is an induced subgraph of ψ( �H), +for some �H ∈ � +C . For n ⩾ 1, let Gn be the bipartite graph with parts {1, . . . , n} and 2{1,...,n}, and +edges iJ such that i ∈ J. Let �Hn ∈ +� +C be such that Gn is an induced subgraph of ψ( �Hn). This +means that there are vertices v1, . . . , vn and wJ, for J ⊆ {1, . . . , n}, such that �Hn |= ψ(vi, wJ) if +and only if i ∈ J, for i ∈ {1, . . . , n} and J ⊆ {1, . . . , n}. Since n is arbitrary, this shows that C is +not simply existentially monadically dependent. +Proof of Proposition 6.1. We may assume without loss of generality that C is hereditary, since +if a class C is simply existentially monadically dependent, then so is the class of all induced +subgraphs of C . This is because that taking an induced subgraph may be simulated by using a +unary predicate. Proposition 6.1 now follows from Lemmas 6.2, 6.3, and 6.4. +We now prove that every model in the elementary closure of a pattern-free class, is pattern- +free as well. +Lemma 6.5 Let C be a hereditary pattern-free graph class and let M ∈ C . Then M is pattern-free. +Proof. Let M ∈ C , and let �Σ be a unary expansion of the signature of graphs. Fix an integer +r ⩾ 1 and a quantifier-free formula ϕ(x, y) in the signature �Σ. Let � +C be the class of all monadic +lifts of graphs in C in the signature �Σ. By assumption, there is a number n such that K(r) +n +is not +an induced subgraph of ϕ(�A), for all �A ∈ � +C . +Let � +M be a monadic lift of M in the signature of ϕ. We show that ϕ(� +M) does not contain K(r) +n +as an induced subgraph. Suppose otherwise. +Consider the substructure �A of � +M induced by the elements of V(K(r) +n ) ⊆ V(� +M). Then �A is a +finite �Σ-structure. Let A be the graph such that �A is a monadic lift of A. +We claim that A is an induced subgraph of some graph B ∈ C . Write a sentence ϕA that +holds in a graph B if and only if A is isomorphic to an induced subgraph of B. Since M |= ϕA +and M ∈ C , there is some B ∈ C such that B |= ϕA, proving the claim. +Since C is hereditary, we conclude that A ∈ C as well, and hence �A ∈ � +C . In particular, ϕ(�A) +is isomorphic to K(r) +n , a contradiction. +It is convenient to use a relaxed form of patterns, where instead of r-subdivisions, we have +(1, r)-subdivisions. +Lemma 6.6 Fix r ⩾ 1, and let C be a graph class that quantifier-free transduces a class D that contains +some (1, r)-subdivision of every clique. Then C is not pattern-free. +Lemma 6.6 follows easily by Ramsey’s theorem which we recall below. +Theorem 6.7 (Ramsey’s theorem) Fix k ⩾ 1. There is a monotone, unbounded function f : N → N, +such that for every n ⩾ 1, if the edges of Kn are colored using k colors, then there is a set W ⊆ V(Kn) +with |W| ⩾ f (n) such that W is monochromatic, that is, all edges uv of Kn with u, v ∈ W, have the +same color. +Lemma 6.6 follows immediately from the next lemma. +Lemma 6.8 Fix r ⩾ 1. Let D be a hereditary class that contains some (1, r)-subdivision of every clique. +Then there is some s ∈ [1, r] such that D contains the s-subdivision of every clique. +29 + +Proof. We want to assign to each n ⩾ 1 a subdivision number cn ∈ [1, r] as follows. Let Gn be +some (1, r)-subdivision of Kn that belongs to D. For each edge e of Kn, let ℓ(e) ∈ [1, r] be the +number of times it was subdivided in Gn. Then ℓ is a coloring of the edges of Kn using r colors. +By Ramsey’s theorem, there is a subset Wn ⊆ V(Kn) which is monochromatic, that is, there is +a color cn ∈ [1, r] such that for every pair u, v of distinct vertices of Wn, we have ℓ(uv) = cn. +Moreover, the size of Wn is at least f (n), where f : N → N is some fixed unbounded, monotone +function (depending only on r). +By the pigeonhole principle, in the infinite sequence (c1, c2, . . .), some element s ∈ [1, r] +occurs infinitely many times. We claim that D contains the s-subdivision of every clique Km. +Indeed, pick any m ⩾ 1, and let n ∈ N be such that f (n) ⩾ m and cn = s. Then Wn ⊆ V(Kn) +is such that |Wn| ⩾ m and ℓ(u, v) = s for all distinct u, v ∈ Wn. Recall that Gn ∈ D is a (1, r)- +subdivision of Kn, and for all distinct u, v ∈ Wn we have that edges uv of Kn are subdivided +s many times in Gn. As |Wn| ⩾ m, it follows that Gn contains, as an induced subgraph, the +s-subdivision of Km. +The following lemma describes certain obstructions (depicted in Fig. 3) that are forbidden +in pattern-free graphs. +Lemma 6.9 Fix r ⩾ 2. Let M be a graph such that for any k ∈ N, there is an infinite collection of +pairwise disjoint finite sets A, B0, B1, B2, . . . of vertices of M with the following properties: +(i) the set A has cardinality k; +(ii) there is a semi-induced matching between A and a subset Ci of Bi for all i; +(iii) there are no edges between A and Bi − Ci for all i; +(iv) there are no edges between Bi − Ci and Bj for i ̸= j; and +(v) for all i and every pair of disjoint vertices u, v ∈ Ci, there is a path of length ⩾ 2 and ⩽ r that +connects u and v and whose all internal vertices belong to Bi − Ci +Then M is not pattern-free. +FIGURE 3: The obstruction pattern, and the setting of the proof of Lemma 6.9. We assume that Bab = B0. +30 + +Proof. We construct a quantifier-free formula ϕ(x, y) using three additional unary predicates, +denoted A, B, C, and for each k a lift � +Mk of M, such that ϕ(� +Mk) contains an (1, r + 1)-subdivision +of Kk, as an induced subgraph. Then Lemma 6.6 implies that {M} is not pattern-free. +Fix k ⩾ 1 and consider sets A, B0, B1, . . . given by the assumption of the lemma, so that +in particular |A| = k. Pick a subfamily of the sets B0, B1, . . . containing exactly (|A| +2 ) sets, and +reindex those sets as Bab, for ab ∈ (A +2). Likewise, denote the distinguished subsets Cab ⊆ Bab, so +that each Cab semi-induces a matching with A. +We now describe the construction of � +Mk. Mark the vertices in A with the unary predicate A, +the vertices in Bab, for some ab ∈ (A +2), with the unary predicate B, and the vertices in Cab ⊆ Bab, +for some ab ∈ (A +2), with the unary predicate C. Define a formula ϕ0(x, y) as follows: +ϕ0(x, y) +≡ +(A(x) ∧ C(y)) +∨ +(A(y) ∧ C(x)) +∨ +(B(x) ∧ B(y) ∧ ¬(C(x) ∧ C(y))). +Let ϕ(x, y) ≡ ϕ0(x, y) ∧ E(x, y). We argue that ϕ(� +Mk) contains an (1, r + 1)-subdivision of Kk, +as an induced subgraph. +For each ab ∈ (A +2), let a′, b′ be the unique elements in Cab connected to a and b, respectively. +Consider a path πab between a′ and b′, of length ⩾ 2 and ⩽ r, that is internally contained in +Bab − Cab. Let W denote the vertices in A ∪ {V(πab) | ab ∈ (A +2)}. Now, it is easy to check that +ϕ(� +Mk)[W] is a (1, r + 1)-subdivision of Kk (see Figure 3). +Finally, we state two lemmas that will be useful later. +Lemma 6.10 For any pattern-free graph M, any graph M′ obtained from M by applying a finite set of +flips, is also pattern-free. +Proof. It suffices to consider the case when M′ is obtained from M by applying an atomic flip, +specified by a pair (A, B) of subsets of M. Add the sets A and B as unary predicates UA and +UB to the graph M, obtaining a unary expansion � +M of M. Then the quantifier-free formula +η(x, y) := E(x, y)△(UA(x) ∧ UB(y)) is such that for any so that for every a, b ∈ M, +M′ |= E(a, b) ⇐⇒ � +M |= η(a, b). +Suppose M′ is not pattern-free. Then there is r ⩾ 1, a class � +C of unary expansions of {M′}, +and a quantifier-free formula ϕ(x, y) such that ϕ( � +C ) contains the r-subdivision of every clique, +as an induced subgraph. Replace each atom E(z, t) by the formula η(z, t) in ϕ(x, y), yielding a +quantifier-free formula ϕ′(x, y). Then for every � +M′ ∈ +� +C there is a unary expansion � +M′′ of � +M, +such that � +M′ |= ϕ(a, b) +⇐⇒ +� +M′′ |= ϕ′(a, b) for all a, b ∈ M. It follows that there is some +class � +C ′ of unary expansions of {� +M} such that ϕ′( � +C ′) contains the r-subdivision of every clique +as an induced subgraph. Hence, by transitivity of unary expansions, M is not pattern-free, a +contradiction. +A similar statement holds for graphs with a stable edge relation. +Lemma 6.11 For any graph M with a stable edge relation, any graph M′ obtained from M by applying +a finite set of flips, also has a stable edge relation. +Proof. Again it is enough to assume that M′ is obtained from M by performing a single atomic +flip F = (A, B). We prove the lemma by contraposition – we argue that if M′ contains ladders +of arbitrary length, then so does M. +31 + +Let a1, a2, . . . , ak, b1, b2, . . . , ak be the vertices of a ladder of length k in M′ (we have aibj ∈ +E(M) iff i ⩽ j). Since each ai has four possibilities of being included or non-included in the +sets A, B and the same holds for bi, there exists a subset Z of {1, . . . , k} of size at least k′ := +k +16 +such that for all i, i′, j, j′ ∈ Z we have (ai, bj) ∈ (A × B) ∪ (B × A) if and only if (ai′, bj′) ∈ +(A × B) ∪ (B × A). Without loss of generality assume that Z = {1, . . . , k′} (this can be achieved +by forgetting all ai, bi such that i ̸∈ Z and renaming the indices which are left). Then there +are two options – either (a1, b1) ̸∈ (A × B) ∪ (B × A) and then a1, a2, . . . , ak′, b1, b2, . . . , ak′ is a +ladder in M′ ⊕ F of length k +16, or (a1, b1) ∈ (A × B) ∪ (B × A) and then the edges between the +sides of the ladder semi-induced by a1, a2, . . . , ak′, b1, b2, . . . , ak′ in M′ get complemented, which +results in a ladder of length k +16 − 1 in M′ ⊕ F given by vertices c1, . . . , ck′−1, d1, . . . , dk′−1, where +ci = ak′−i+1 and di = bk′−i. Since by our assumption on M′ we can choose k to be arbitrarily +large, this means that there are arbitrarily large ladders in M′ ⊕ F, which finishes the proof. +7 +Finite separators in pattern-free stable models +Say that a graph M is r-separable if for every elementary extension N of M, and every v ∈ N − M, +there is a finite set S ⊆ M such that v and M are r-separated over S in N. This section is +dedicated to proving the following theorem. +Theorem 7.1 Let M be a pattern-free graph with a stable edge relation. Then M is r-separable, for every +r ∈ N. +By Lemmas 5.7 and 6.5, we immediately get the following corollary, proving the implication +(3)→(4) in Theorem 1.4. +Corollary 7.2 If C is a pattern-free class of graphs with stable edge relation and r ∈ N, then every +M ∈ C is r-separable. +We will prove Theorem 7.1 by induction on r. +7.1 +Case of finitely many types +The following lemma is a generalization of the case r = 1 of Theorem 7.1, where instead of one +vertex v we have a set of vertices U with a bounded number of types over M. +Lemma 7.3 For any graphs M and N with M ≺ N and such that the edge relation is stable in N, and +for any set U ⊆ N − M such that TypesE(U/M) is finite, there exists a finite set S ⊆ M and an S-flip +which: +– 1-separates U from M; and +– does not flip the S-class T := {v ∈ N : ∀s ∈ S. ¬E(v, s)} with any other S-class (including +itself), as long as T ∩ U is nonempty. +Proof. We will construct the sought set S in a series of steps. +▷ Claim 7.4 There is a finite set SU ⊆ M such that any two vertices in M with the same SU-class also +have the same U-class. +Proof of the claim. For each set in TypesE(U/M), select one vertex v ∈ U of that type. Then +apply Theorem 5.3 to the formula E(x, y) and the element v ∈ N: there is a positive boolean +32 + +combination ψ(x) of formulas of the form E(x, m) for m ∈ M, and for every u ∈ M, +N |= E(v, u) ⇐⇒ M |= ψ(u). +It follows that there is a finite set Sv ⊆ M, namely the set of parameters of ψ(x), so that whether +or not a vertex u ∈ M is adjacent to v depends only on its Sv-class. The finite set SU := � +v Sv +has the desired properties. +◁ +▷ Claim 7.5 There is a finite set SM ⊆ M such that any two vertices in U with the same SM-class also +have the same M-class. +Proof of the claim. Notice that since TypesE(U/M) is finite, TypesE(M/U) is also finite. Thus we +can obtain SM by including one vertex from each set in TypesE(M/U). +◁ +Now, for each of the finitely many ordered pairs (A, B) of distinct (SU ∪ SM)-classes, apply- +ing Lemma 5.5 on A ∩ M and B ∩ M yields a finite set SA,B which is either contained in A ∩ M +and dominates B ∩ M, or contained in B ∩ M and antidominates A ∩ M. Note that it may be +that SA,B and SB,A are different. We now set S to be +S := SU ∪ SM ∪ +� +(A,B) +SA,B. +Note that S is finite and contained in M, as required. +Now we define an S-flip N′ of N. Let N′ be the graph obtained from N by flipping between +S-classes A and B (possibly with A = B) if there exists an edge in N which has one end in M +and the other end in U, and at the same time has one end in A and the other end in B. +We will now show that N′ satisfies the required properties. We begin by considering the +S-class T := {v ∈ N : ∀s ∈ S. ¬E(v, s)} of vertices non-adjacent to all of S. +▷ Claim 7.6 If T ∩ U is nonempty, then T is not flipped with any other S-class. +Proof of the claim. Assume by contradiction that there is some T′ (possibly with T′ = T) such +that T and T′ are flipped. Moreover, let w be some vertex of T ∩ U. +First, observe that there is no edge in N between w and M. Indeed, as SM contains a rep- +resentative of every equivalence class in TypesE(M/U) and yet w is not connected in N to any +vertex in SM, then it is not connected to any vertex in M. Therefore, there is no edge in N be- +tween T ∩ U and M, so in particular there is no edge in N between T ∩ U and T′ ∩ M. However, +a flip was made between T and T′, so there must be an edge in N between T ∩ M and T′ ∩ U. +Denote by v and u two vertices connected by an edge in N such that v ∈ T ∩ M and u ∈ T′ ∩ +U. We know that the neighborhood of u in M can be defined by a positive boolean combination +of the formulas of the form E(x, s) for s ∈ S. Since v satisfies none of these formulas (and yet +is connected to u), we infer that the neighborhood of u must be described by a formula that is +always true. Therefore, the neighborhood of u contains all of M. It follows that U contains two +vertices u and w such that in N one of them is connected to every vertex in M and the other is +disconnected from every vertex in M. +However, this cannot happen. Indeed, by Lemma 5.5 applied to A = B = M there is a finite +R ⊆ M that either dominates every vertex in M or antidominates every vertex in M. As M is an +elementary substructure of N, then R also dominates every vertex in N or antidominates every +vertex in N. This is a contradiction, as R neither dominates w nor antidominates u. Therefore, +T is not flipped with any other S-class (including itself). +◁ +33 + +To complete the proof, we now show that N′ has no edge connecting a vertex in M with a +vertex in U. Suppose towards a contradiction that N′ has an edge ab so that a ∈ M and b ∈ U. +Let A and B denote the S-classes of N which contain a and b, respectively (so maybe A = B). If +ab ∈ E(N), then we would have flipped between A and B, removing the edge. Thus ab /∈ E(N), +and we flipped between A and B. So there must have been some other edge a′b′ ∈ E(N) such +that a′ ∈ A, b′ ∈ B, one of a′, b′ is in M, and the other is in U. +▷ Claim 7.7 Let x ∈ M, y ∈ U. Assume that a and x have the same SU ∪ SM-class, and that b and y +have the same SU ∪ SM-class. Then xy /∈ E(N). +Proof of the claim. Since, in N, the vertices a and x have the same SU ∪ SM-class, they also have +the same SU-class and therefore the same U-class. As ab /∈ E(N), a ∈ M and b ∈ U, we infer +that xb /∈ E(N). Likewise, b and y have the same M-class. Therefore, xy /∈ E(N). +◁ +First suppose that a′ ∈ M and b′ ∈ U. Then Claim 7.7 applies with (x, y) = (a′, b′), contra- +dicting the fact that a′b′ ∈ E(N). +Thus we may assume that a′ ∈ U and b′ ∈ M. Let A0 and B0 denote the SU ∪ SM-classes +of a′ and b′, respectively; note that A ⊆ A0 and B ⊆ B0. If A0 = B0, then Claim 7.7 applies +with (x, y) = (b′, a′), again contradicting that a′b′ ∈ E(N). Thus A0 ̸= B0, and a, b, a′, b′ are four +different vertices. +FIGURE 4: Illustration for the end of the proof of Lemma 7.3. +▷ Claim 7.8 S does not contain a dominating set of B0 ∩ M entirely contained in A0 ∩ M. +Proof of the claim. Consider the intersection Ra of S and A0; see Figure 4 for an illustration. If +Ra = ∅, then the claim is trivial. Otherwise, let us pick an arbitrary element r of Ra, aiming to +show that rb′ is a nonedge in M. As b′ ∈ B0 ∩ M, this immediately finishes the proof. +Since r and a belong to M and have the same SU-class, they have the same U-class and +therefore rb is a non-edge in N, just as ab. Now since b and b′ have the same S-class (as they +both belong to B), they have the same connection to r ∈ S, and therefore rb′ is a nonedge as +well. +◁ +However, we also have that: +▷ Claim 7.9 S does not contain an antidominating set of A0 ∩ M entirely contained in B0 ∩ M. +Proof of the claim. Similarly, we let Rb′ be the intersection of S and B0. A symmetric argument +shows that there are all possible edges between a and Rb′, and thus S cannot contain an an- +tidominating set for A0 ∩ M which is contained in B0 ∩ M. +◁ +34 + +Claims 7.8 and 7.9 directly contradict the construction of SA0,B0. This finishes the proof. +7.2 +Balls have finitely many types over a model +Observe that the base case of r = 1 of Theorem 7.1, which follows by Lemma 7.3, only assumes +that the edge relation in M is stable. In the inductive argument we will use the assumption +that M is pattern-free. By Lemma 6.10, every graph obtained from M by performing some flips +remains pattern-free. Using Lemma 6.9, we prove the following lemma. +Lemma 7.10 Fix r ∈ N. Let M be pattern-free graph with a stable edge relation, let N be its elemen- +tary extension, and let v ∈ N be such that the r-ball Br(v) around v in N is disjoint from M. Then +TypesE(Br(v)/M) is finite. +Proof. First note that N is pattern-free and a stable edge relation. This is because N ∈ {M}, and +we can apply Lemma 6.5 and Lemma 5.7, respectively. +Suppose, going for a contradiction, that TypesE(Br(v)/M) is infinite. By Theorem 5.6, the +bipartite graph semi-induced in N between Br(v) and M either contains an infinite induced +matching, an infinite induced co-matching, or an infinite induced ladder. Since M has a sta- +ble edge relation, the last case is excluded. Moreover, up to performing a flip (over ∅) which +exchanges edges and non-edges, we may assume without loss of generality (again thanks to +Lemmas 5.8 and 6.10 and 6.11) that we are in the first case: there is an infinite induced match- +ing between Br(v) and M. We now show that the assumptions of Lemma 6.9 are satisfied with +radius 2r: for any k ∈ N, we will construct A and the Bi’s as in the statement of the lemma. By +Lemma 6.9, this implies that N is not pattern-free, a contradiction. +Let A ⊆ M and C ⊆ Br(v) be sets of cardinality k that semi-induce a matching in N. Now +for each c ∈ C, consider a path of length ⩽ r from v to c, let B ⊆ Br(v) be the union of these +paths (as sets of vertices); note that |B| ⩽ kr + 1. Let ¯y be a tuple of variables with | ¯y| = |B|, +and let ¯n ∈ N ¯y be a ¯y-tuple comprised of all elements of B as its components. Note that since +Br(v) avoids M, there can be no edges in N between M and vertices at distance < r from v, +therefore C is exactly the set of vertices which are at distance r from v in N[B] (and they are also +at distance r from v in N). Observe that for every pair of distinct vertices c1, c2 in C, there exists +a simple path of length at least 2 and at most 2r internally contained in B − C, constructed as +the concatenation of a nonempty suffix of the path from v to c1 and a nonempty suffix of the +path from v to c2. +We now apply Lemma 5.4, which yields a sequence of tuples ¯b0, ¯b1, . . . in M such that for +each i, ¯bi has the same atomic type as ¯n over A ∪ {¯b0, . . . , ¯bi−1}, and such that the atomic types +of (¯bi, ¯bj) are the same for i ̸= j. For each i, we let Bi ⊆ N be the set of vertices appearing as +components of ¯bi. Note that in particular, the ¯bi’s have same atomic type as ¯n (in other words, +the N[Bi]’s are all isomorphic copies of N[B]). Thus for each i, we let vi ∈ Bi be the vertex +corresponding to v ∈ B, and we let Ci ⊆ Bi be the set of vertices at distance r from vi in N[Bi], +which are also the vertices corresponding to C ⊆ B. Note that conditions (i) and (v) from the +statement of Lemma 6.9 hold. +By definition, there is a semi-induced matching between A ⊆ M and C in N, and moreover +there are no edges between B − C and M. Now since each ¯bi has the same atomic type as ¯n +over A, there is also a semi-induced matching between A and Ci for all i and there are no edges +between Bi − Ci and A: conditions (ii) and (iii) are satisfied. +Recall from Lemma 5.4 that for i ̸= j, the atomic type of (¯bi, ¯bj) is the same as the atomic +type of (¯b1, ¯b0). Moreover, ¯b1 and ¯n have the same atomic type over A ∪ ¯b0, so the atomic type +35 + +of (¯b1, ¯b0) is the same as the atomic type of ( ¯n, ¯b0). Now as noticed above, there are no edges in +N between M and B − C. As also ¯b0 is contained in M, there are no edges between the vertices +of ¯n (outside of C) and the vertices of ¯b0. By the equality of atomic types of (¯bi, ¯bj) and ( ¯n, ¯b0), +it follows that there are no edges between the vertices of ¯bi (outside of Ci) and the vertices of ¯bj. +Hence, there are no edges in N between Bi − Ci and Bj. Therefore, (iv) holds. +Therefore Lemma 6.9 tells us that N is not pattern-free, a contradiction, which proves that +TypesE(Br(v)/M) is finite. +7.3 +Inductive proof +An inductive proof of Theorem 7.1 now follows by putting together Lemma 7.3 and Lemma 7.10. +Proof of Theorem 7.1. We proceed by induction on r. The base case r = 0 is immediate as we +may take S to be ∅ since v /∈ M. In the inductive step, assume that the result is proved for the +distance r ∈ N; that is, there is a finite S ⊆ M such that v |⌣ +r +S M. Stated differently, there is an +S-flip N′ of N in which the r-ball around v is disjoint from M. By working in N′ instead of N, +we may assume, thanks to Lemmas 5.8 and 6.10 and 6.11, that the r-ball Br(v) around v in N is +disjoint from M. By Lemma 7.10, TypesE(Br(v)/M) is finite. Now Lemma 7.3 applied to Br(v) +finishes the inductive step and the proof (we are using the fact that we obtain a set S and an +S-flip which doesn’t flip the S-class which contains Br−1(v)). +8 +Separator wins in monadically stable classes +The goal of this section is to prove the following theorem. +Theorem 8.1 Fix r ∈ N, and let C be a class of graphs such that every G ∈ C is r-separable. Then +there exists k ∈ N such that Separator wins the Separation Game with radius r in k rounds on every +G ∈ C . +This proves the implication (4)→(5) in Theorem 1.4. By Corollary 7.2 and Lemmas 3.1 and +3.2 we immediately get the following. +Corollary 8.2 Let C be a monadically stable class of graphs. Then for any r ∈ N, there exists k ∈ N +such that Flipper wins the Flipper game with radius r in k rounds on every G ∈ C . +The rest of Section 8 is devoted to the proof of Theorem 8.1. Fix an enumeration ϕ1, ϕ2, . . . +of all formulas (in the signature of graphs) of the form ϕ(y, x1, . . . , xℓ), with ℓ ⩾ 0. We define +a strategy of Separator in any graph G. In the kth round, after Connector picks ck ∈ Ak−1, +Separator first sets S := Sk−1 ∪ {ck} and marks ck. Then, for every i = 1, . . . , k, for the formula +ϕi(y, ¯x), Separator does the following. +For each valuation ¯a ∈ S ¯x such that G |= ∃y.ϕi(y, ¯a), Separator marks any vertex +b ∈ V(G) such that G |= ϕi(b, ¯a). +We say that any strategy of Separator which has this property is Connector-complete. The marked +vertices form Separator’s response in the kth round, and we set Sk to be the union of Sk−1 and +all the marked vertices. Note that there is a function f : N → N such that |Sk| ⩽ f (k) for all +k ∈ N, regardless of which vertices Connector picks or which of the formulas ∃y.ϕi(y, ¯a) hold. +We prove that there is a number k ∈ N such that when Separator plays according to any +Connector-complete strategy on a graph G ∈ C , then he wins in at most k rounds. Assume +36 + +that the conclusion of the theorem does not hold. Then, there exists a sequence of graphs +G1, G2, . . . ∈ C , where in Gn Connector has a strategy ensuring that Separator does not win +for at least n rounds. We shall now prove that there is some graph G in the elementary closure +of C and a vertex in the graph that survives in the arena indefinitely, when Separator plays ac- +cording to a Connector-complete strategy. We will then show that this contradicts r-separability +of G. +▷ Claim 8.3 There exists a graph G ∈ C , a strategy of Connector, and a Connector-complete strategy +of Separator for which the Separation Game on G lasts indefinitely and � +n<ω An is nonempty. +Proof of the claim. For every graph Gn ∈ C , choose any Connector-complete strategy of Sepa- +rator, and any strategy of Connector ensuring the game continues for more than n rounds. We +define a new class C ′ of structures by adding constants to graphs G1, G2, . . . as follows. +– Add constant symbols c1, c2, . . . , cω. For each graph Gn, interpret ck as: +– Connector’s move in the kth round if k ⩽ n; +– any vertex remaining in the arena after n rounds if k = ω; +– an arbitrary vertex of Gn otherwise. +– Add constant symbols sn,i for 1 ⩽ n < ω and 1 ⩽ i ⩽ f (n). For each graph Gn ∈ C , +interpret sk,1, . . . , sk,f (k) as: +– the vertices marked by Separator in the kth round if k ⩽ n. We allow these symbols +to be interpreted as the same vertex if Separator plays fewer than f (k) vertices, and +we ensure that sk,1 is interpreted in the same way as ck; +– arbitrary vertices of Gn otherwise. +Now, for convenience, for each 1 ⩽ n < ω, we write Sn−1 := {sk,i | k < n, 1 ⩽ i ⩽ f (k)}. +Finally, let T be the theory which is obtained by including: +– the sentences ϕ which hold in all structures G ∈ C ′ (that is, the theory of C ′); +– sentences which express that cn is a valid move in the nth round, namely, that cn ̸ |⌣ +r +Sn−1 ck +for 1 ⩽ k < n (we remark that these sentences are existential); +– analogous sentences which express that cω is a valid move in each round n; +– sentences which express that Separator plays according to a Connector-complete strategy; +that is, sn,1 = cn, and for each 1 ⩽ i ⩽ n, for the formula ϕi(y, ¯x) and for each valuation +¯c ∈ (Sn−1 ∪ {sn,1}) ¯x, the sentence ∃y.ϕi(y, ¯c) =⇒ ϕi(sn,2, ¯c) ∨ . . . ∨ ϕi(sn,f (n), ¯c). +We now show that T is consistent. To this end, pick a finite subset T′ of T. Then there is +a number n0 ∈ N such that no constants cn or sn,i, with n0 < n < ω, occur in a sentence in +T′. Since Connector avoids losing in Gn0+1 for at least n0 + 1 rounds in our fixed strategies, the +structure corresponding to Gn0+1 in C ′ models T′. It follows by compactness (Theorem 5.1) that +T is consistent. +Since T is consistent, there exists a model G′ of T, which is a graph equipped with constant +symbols c1, c2, . . . , cω and sk,i. Let G be the graph obtained from G′ by forgetting these constants. +As T contains the theory of C ′, which in turn contains the theory of C , we infer that G ∈ C . +Now consider the instance of the Separation Game where in the nth round, Connector picks the +37 + +vertex cn and Separator picks the vertices sn,i with 1 ⩽ i ⩽ f (n). This is a valid strategy for +Connector by the fact that G′ |= T. For the same reason, the vertex cω remains in the arena after +each round. Finally, in G, Separator’s strategy as defined above is Connector-complete. +◁ +Let G ∈ C be the graph produced by Claim 8.3, along with the strategies of Connector and +Separator. By assumption, G is r-separable. Recall that A0 ⊇ A1 ⊇ . . . is the sequence of arenas +in the play, c1, c2, . . . is the sequence of moves of Connector, and S0 ⊆ S1 ⊆ . . . is the sequence +of sets of vertices marked by Separator. Denote Aω := � +n<ω An, and Sω := � +n<ω Sn. We will +get a contradiction with the previous claim by proving the following claim: +▷ Claim 8.4 Aω is empty. +Proof of the claim. Observe that for each k ∈ N, we have ck /∈ Sk−1: as soon as Connector plays +ck in Sk−1, the arena Ak shrinks to a single vertex and Separator wins in the following round. +Then, Ak is disjoint from Sk−1: since Connector plays ck outside of Sk−1, each vertex of Sk−1 +becomes separated from ck and thus is removed from the arena. It follows that Aω ∩ Sω = ∅. +Since Separator follows a Connector-complete strategy, Sω induces an elementary substruc- +ture of G by the Tarski-Vaught test (Theorem 5.2). We also have that c1, c2, . . . ∈ Sω by con- +struction. Now suppose for a contradiction that there exists some cω ∈ Aω. We remark that +cω /∈ Sω. +By Theorem 7.1, there exists a finite set S ⊆ Sω such that cω |⌣ +r +S Sω. As S is finite, there is +some n < ω such that S ⊆ Sn, so in particular, cω |⌣ +r +Sn Sω. On the other hand, cω ̸ |⌣ +r +Sn cn+1, as +cω ∈ An+1. This is a contradiction since cn+1 ∈ Sω. +◁ +However, this means that there exists a graph G ∈ C and strategies of Connector and Sep- +arator, for which Aω is simultaneously nonempty (Claim 8.3) and empty (Claim 8.4). This +contradicts the existence of the graphs G1, G2, . . . ∈ C and completes the proof of Theorem 8.1. +Part III +Algorithmic Flipper game +9 +Outline +In this part we prove Theorem 1.5, recalled below for convenience. +Theorem 1.5 Let C be a monadically stable class of graphs. Then for every radius r ∈ N there exist +k ∈ N and a Flipper strategy flip⋆ such that the following holds: +– When playing according to flip⋆ in the Flipper game of radius r on any graph G ∈ C , Flipper +wins within at most k rounds. +– The moves of flip⋆ on an n-vertex graph G ∈ C can be computed in time OC ,r(n2). +Recall here that notation OC ,r(·) hides multiplicative factors that depend only on the class +C and the radius r. +Let us first sketch a natural approach to use the flip-wideness characterization of monadic +stability (see Definition 1) to derive a winning strategy for Flipper. Consider the radius-r Flipper +game on a graph G from a monadically stable class C . For convenience we may assume for now +38 + +that we work with an extended version of the game where at each round Flipper can apply a +bounded (in term of the round’s index) number of flips, instead of just one (see the discussion in +Section 3). As making a vertex isolated requires one flip — between the vertex in question and +its neighborhood — we can always assume that the flips applied by Flipper in round i make all +the i vertices previously played by Connector isolated. Hence, Connector needs to play a new +vertex in each round, thus building a growing set X of her moves. +Fix some constant m ∈ N. According to flip-wideness, there exists some number N := +N2r(m) with the property that once X has grown to the size N, we find a set of flips F — whose +size is bounded independently of m — and a set Y of m vertices in X that are pairwise at distance +greater than 2r in G ⊕ F. It now looks reasonable that Flipper applies the flips from F within +his next move. Indeed, since after applying F the vertices of Y are at distance more than 2r +from each other, the intuition is that F robustly “disconnects” the graph so that the subsequent +move of the Connector will necessarily localize the game to a simpler setting. This intuition +is, however, difficult to capture: flip-wideness a priori does not provide any guarantees on the +disconnectedness of G ⊕ F other than that the vertices of Y are far from each other. +The main idea for circumventing this issue is to revisit the notion of flip-wideness and +strengthen it with an additional predictability property. Intuitively, predictability says that being +given any set of 5 vertices in Y as above is sufficient to uniquely reconstruct the set of flips F. +Formally, in Section 10 we prove the following strengthening of the results of [DMST22]. Here +and later on, O(G) denotes the set of linear orders on the vertices of G. +Theorem 9.1 (Predictable flip-wideness) Fix radius r ∈ N and a monadically stable class of graphs C . +Then there exist the following: +– An unbounded non-decreasing function αr : N → N and a bound λr ∈ N. +– A function FWr that maps each triple (G ∈ C , ≼ ∈ O(G), X ⊆ V(G)) to a pair (Y, F) such that: +– F is a set of at most λr flips in G, and +– Y is a set of αr(|X|) vertices of X that is distance-r independent in G ⊕ F. +– A function Predictr that maps each triple (G ∈ C , ≼ ∈ O(G), Z ⊆ V(G)) with |Z| = 5 to a set +F of flips in G such that the following holds: +– For every X ⊆ V(G), if (Y, F) = FWr(G, ≼, X) and Z ⊆ Y, then F = Predictr(G, ≼, Z). +Moreover, given G, ≼, and Z, Predictr(G, ≼, Z) can be computed in time OC ,r(|V(G)|2). +Let us explain the intuition behind the mappings FWr and Predictr provided by Theorem 9.1. +The existence of bounds αr and λr and of the function FWr with the properties as above is +guaranteed by the standard flip-wideness, see Definition 1 and Theorem 2.8. However, in the +proof we pick the function FWr in a very specific way, so that the flip set F is defined in a +somewhat minimal way with respect to a given vertex ordering ≼. This enables us to predict +what the flip set F should be given any set of 5 vertices from Y. This condition is captured by +the function Predictr. +We remark that the predictability property implies the following condition, which we call +canonicity, and which may be easier to think about. (We assume the notation from Theorem 9.1.) +– For every G ∈ C , ≼∈ O(G), and X, X′ ⊆ V(G), if we denote (Y, F) = FWr(G, ≼, X) and +(Y′, F′) = FWr(G, ≼, X′), then |Y ∩ Y′| ⩾ 5 entails F = F′. +39 + +Indeed, to derive canonicity from predictability note that F = Predictr(G, ≼, Z) = F′, where Z +is any 5-element subset of Y ∩ Y′. Predictability strengthens canonicity by requiring that the +mapping from 5-element subsets to flip sets is governed by a single function Predictr, which is +moreover efficiently computable. +We now outline how Flipper can use predictable flip-wideness for radius 2r to win the +radius-r Flipper game in a bounded number of rounds. Suppose the game is played on a +graph G; we also fix an arbitrary ordering ≼ of vertices of G. Flipper will keep track of a +growing set X of vertices played by the Connector. The game proceeds in a number of eras, +where at the end of each era X will be augmented by one vertex. In an era, Flipper will spend +2 · (|X| +5 ) rounds trying to robustly disconnect the current set X. To this end, for every 5-element +subset Z of X Flipper performs a pair of rounds: +– In the first round, Flipper computes F := Predict2r(G, ≼, Z) and applies the flips from F. +Subsequently, Connector needs to localize the game to a ball of radius r in the F-flip of the +current graph. +– In the second round, Flipper reverses the flips by applying F again, and Connector again +localizes. +Thus, after performing a pair of rounds as above, we end with an induced subgraph of the +original graph, which moreover is contained in a ball of radius r in the F-flip. Having performed +all the (|X| +5 ) pairs of rounds as above, Flipper makes the last round of this era: he applies flips +that isolates all vertices of X, thus forcing Connector to play any vertex outside of X that is still +available. This adds a new vertex to X and a new era begins. +Let us sketch why this strategy leads to a victory of Flipper within a bounded number of +rounds. Suppose the game proceeds for N eras, where N is such that α2r(N) ⩾ 7. Then we +can apply predictable flip-wideness to the set X built within those eras, thus obtaining a pair +(Y, F) := FW2r(G, ≼, X) such that |Y| = 7 and F is a set of flips such that Y is distance-2r inde- +pendent in G ⊕ F. Enumerate Y as {v1, . . . , v7}, according to the order in which they were added +to X during the game. Let Z := {v1, . . . , v5} and note that F = Predict2r(G, ≼, Z). Observe that +in the era following the addition of v5 to X, Flipper considered Z as one of the 5-element subsets +of the (current) set X. Consequently, within one of the pairs of rounds in this era, he applied +flips from F and forced Connector to localize the game subsequently. Since v6 and v7 are at +distance larger than 2r in G ⊕ F, this necessarily resulted in removing v6 or v7 from the graph. +This is a contradiction with the assumption that both v6 and v7 were played later in the game. +In Section 10 we prove Theorem 9.1. In Section 11 we formalize the strategy outline pre- +sented above and analyze the time complexity needed to compute the moves. This will amount +to proving Theorem 1.5. +10 +Predictable flip-wideness +This section is devoted to proving Theorem 9.1: monadic stability implies predictable flip- +wideness. We will first collect some facts from the work of Dreier et al. [DMST22] and shape +them to our convenience. +40 + +10.1 +Classifiers +The main idea behind the proof of [DMST22] is to perform gradual classification of vertices +while showing that this classification needs to satisfy very rigid conditions, imposed by monadic +stability. To formalize this we will rely on the notion of a classifier, presented below. +For a vertex u, N(u) denotes the (open) neighborhood of u, that is, the set comprising all the +neighbors of u. The neighborhood of u in a set of vertices B is N(u) ∩ B. Further, s is adjacent to +B if N(u) ∩ B ̸= ∅. Usually, the graph, in which neighborhoods and adjacencies are evaluated, +will be clear from the context. Otherwise, we specify it in the subscript. +Definition 3. A classifier in a graph G is a quadruple B = (B, S, exc, rep), where B is a family of +pairwise disjoint vertex subsets of G, called further blobs, S is a non-empty subset of vertices +of G, and exc: V(G) → B ∪ {⊥} and rep: V(G) → S are mappings satisfying the following +properties: +(a) S ∩ � B = ∅; that is, no vertex of S belongs to any blob. +(b) Every s ∈ S is adjacent either to all the blobs in B or to no blob in B. +(c) For all distinct s, s′ ∈ S and each blob B ∈ B, N(s) ∩ B ̸= N(s′) ∩ B. +(d) For each v ∈ � B, we have exc(v) ̸= ⊥ and v ∈ exc(v). +(e) For all v ∈ V(G) and B ∈ B − {exc(v)}, we have N(v) ∩ B = N(rep(v)) ∩ B. +The size of a classifier (B, S, exc, rep) is |B|, and its order is |S|. +Let us give some intuition. In a classifier we have a family of disjoint blobs B and a set +of representative vertices S. Further, with every vertex v we can associate its exceptional blob +exc(v) ∈ B and its representative rep(v) ∈ S. The key condition (e) says the following: every +vertex v behaves in the same way as its representative rep(v) with respect to all the blobs in B, +except for its (single) exceptional blob exc(v). We allow exc(v) to be equal to ⊥, which indicates +that v has no exceptional blob (this will be convenient in notation). Condition (d) says that if v +is contained in some blob B ∈ B, then in fact B must be the exceptional blob of v. Conditions +(a), (b), and (c) are technical assertions that expresses that the representative set S is reasonably +chosen. +A classifier naturally partitions the vertex set of the graph, as formalized below. +Definition 4. For a classifier B = (B, S, exc, rep), the partition raised by B is the partition ΠB of +the vertex set of G defined as follows: +ΠB := {rep−1(s): s ∈ S}. +For s ∈ S, we write ΠB(s) := rep−1(s) to indicate the part of ΠB associated with s. +The following observation is easy, but will be the key to our use of classifiers. +Observation 10.1 Let B = (B, S, exc, rep) be classifier of size at least five in a graph G. Then for every +pair of vertices u, v of G, the following conditions are equivalent. +(i) u and v are in the same part of ΠB. +(ii) u and v have the same neighborhood in at least three blobs from B. +(iii) u and v have different neighborhoods in at most two blobs from B. +41 + +Proof. Implication (i)⇒(iii) follows by observing that since rep(u) = rep(v), u and v must have +exactly the same neighborhood in every blob, possibly except for exc(u) and exc(v). Implication +(iii)⇒(ii) is immediate due to |B| ⩾ 5. +Finally, for implication (ii)⇒(i), observe that u and rep(u) have the same neighborhood in +all but at most one blob from B, and similarly for v and rep(v). Since u and v have the same +neighborhood in at least three blobs from B, it follows that rep(u) and rep(v) have the same +neighborhood in at least one blob from B. By condition (c) of Definition 3, this means that +rep(u) = rep(v), so u and v belong to the same part of ΠB. +From Observation 10.1 we can derive a canonicity property for classifiers: whenever two +classifiers share at least five blobs in common, the associated partitions are the same. In the next +lemma we show an even stronger property: (efficient) predictability for classifiers. +Lemma 10.2 There exists an algorithm that given a graph G and family B◦ consisting of five pairwise +disjoint subsets of V(G), computes a partition Π◦ of V(G) with the following property: for every classi- +fier B = (B, S, exc, rep) in G with B◦ ⊆ B, we have Π◦ = ΠB. The running time of the algorithm is +O(|Π◦| · n2), where n = |V(G)|. +Proof. We first present the construction of Π◦. Along the way we also construct a set of repre- +sentatives S, and at each point vertices s ∈ S are in one-to-one correspondence with parts Π◦(s) +of Π◦. We start with Π◦ = ∅ and S = ∅. Then we iterate through the vertices of G in any order, +and when considering the next vertex v we include it in the partition as follows: +– If there exists s ∈ S such that v and s have the same neighborhood in at least three of the +blobs of B◦, select such s that was added the earliest to S and add v to Π◦(s). +– Otherwise, if no s as above exists, add v to S and associate with v a new part Π◦(v) = {v}. +It is straightforward to implement the algorithm to work in time O(|Π◦| · n2), where |Π◦| = |S| +is the size of the output partition. +We now verify that Π◦ constructed in this manner satisfies the requested properties. Let then +B = (B, S, exc, rep) be any classifier with B◦ ⊆ B; we need to argue that Π◦ = ΠB. Consider +any pair u, v of vertices of G. We need to prove that u, v are in the same part of Π◦ if and only if +they are in the same part of ΠB. +For the forward implication, suppose u and v belong to the same part of Π◦, say Π◦(s) for +some s ∈ S. By construction, u and s have the same neighborhood in at least three of the blobs +of B◦. By Observation 10.1, this implies that u and s are in the same part of ΠB. Similarly, v and +s are in the same part of ΠB. By transitivity, u and v are in the same part of ΠB. +For the other direction, suppose u ∈ Π◦(s) and v ∈ Π◦(s′) for some s ̸= s′. By symmetry, +we may assume that s′ was added to S later than s. Since v was included in Π◦(s′) instead +of Π◦(s), by construction it follows that v and s must have different neighborhoods in at least +three different blobs of B◦. So by Observation 10.1, v and s belong to different parts of ΠB. +Since u and s belong to the same part of Π◦, by the forward implication they also belong to the +same part of ΠB. Hence u and v belong to different parts of ΠB. +We next observe that, at the cost of a moderate loss on the size of a classifier, we may choose +the representatives quite freely. +Lemma 10.3 Let B = (B, S, exc, rep) be a classifier in a graph G and let S′ be any set such that rep is +a bijection from S′ to S. Then there is a classifier B′ = (B′, S′, exc′, rep′) in G such that B′ ⊆ B and +|B′| ⩾ |B| − |S|. +42 + +Proof. Let B′ be obtained from B by removing exc(s′) for each s′ ∈ S′. Note that by condition +(d) of Definition 3, S′ is disjoint from � B′. Next, for each vertex u set exc′(u) := exc(u), except +for the case when exc(u) ∈ B − B′; then set exc′(u) := ⊥. Since rep is a bijection from S′ to S, +for every vertex u there exists exactly one vertex s′ ∈ S′ satisfying rep(u) = rep(s′), and we set +rep′(u) := s′. +We claim that B′ := (B′, S′, exc′, rep′) is a classifier. For this, observe that for every s′ ∈ S′, +since exc(s′) has been removed when constructing B′, we in fact have N(rep(s′)) ∩ B = N(s′) ∩ +B for every B ∈ B′. With this observation in mind, all conditions of Definition 3 for B′ follow +directly from those for B. +Let G be a graph and ≼ be any linear order on V(G). We shall say that a classifier B = +(B, S, exc, rep) is canonical with respect to ≼ if the following condition holds: each s ∈ S is the +≼-minimum element of ΠB(s). We note the following. +Corollary 10.4 Let G be a graph, ≼ be a linear order on G, and B = (B, S, exc, rep) be a classifier in G. +Then there is also a classifier B′ = (B′, S′, exc′, rep′) such that |S′| = |S|, B′ ⊆ B, |B′| ⩾ |B| − |S|, +and B′ is canonical with respect to ≼. +Proof. It suffices to apply Lemma 10.3 to S′ := {min≼ ΠB(s): s ∈ S}. +The following lemma is the main outcome of this section. Here, an r-ball in a graph G is a +set of the form {w ∈ V(G) | distG(v, w) ⩽ r} for some vertex v (the center of the ball). +Lemma 10.5 Fix a monadically stable class of graphs C and r ∈ N. Then there exist a constant +κ ∈ N and an unbounded non-decreasing function β: N → N such that the following holds. For every +G ∈ C , linear order ≼ on V(G), and a non-empty family A of pairwise disjoint r-balls in G, there exists +a classifier B = (B, S, exc, rep) in G such that: +– B has size at least β(|A|) and order at most κ; +– B ⊆ A; and +– B is canonical with respect to ≼. +To prove Lemma 10.5 we need the following lemma, which follows from Theorem 4.2 of +[DMST22] by setting ϕ(x, y) = adj(x, y) and α(x, y) = dist⩽r(x, y), where adj(x, y) is the adja- +cency predicate and dist⩽r(x, y) is the first-order formula expressing that the distance between +x and y is at most r. +Lemma 10.6 (follows from Theorem 4.2 of [DMST22]) Fix a monadically stable class of graphs C +and r ∈ N. Then there exist k ∈ N and a function M: N → N such that for every m ⩾ 3 and family +A of size at least M(m) of pairwise disjoint r-balls in a graph G ∈ C , the following holds. There exists +a subfamily B ⊆ A of size at least m and a set S ⊆ V(G) of at most k vertices such that for every +v ∈ V(G) there exists a single exceptional ball Bv ∈ B and an element sv ∈ S such that for every ball +B ∈ B − {Bv} we have +N(v) ∩ B = N(sv) ∩ B. +Furthermore, if v ∈ B for some B ∈ B, then Bv = B. +In essence, Lemma 10.6 already gives us the needed classifier, except that we need to mas- +sage it using Corollary 10.4 and a pigeonhole argument. +43 + +Proof of Lemma 10.5. Let k and M be the constant and the function provided by Lemma 10.6 for +the class C and the radius r. We may assume that M is non-decreasing. We set κ := k. Further, +for every n ∈ N, set β(n) to be the largest positive integer m such that +n ⩾ M(kk · (k + 1) · (m + k) + k); +if there is no such m, set β(n) := 0. Clearly, β defined in this way is non-decreasing and un- +bounded. +Consider any family A of pairwise disjoint r-balls A in G. Let m := β(|A|). If m = 0, then +we set B to be the unique canonical classifier of size 0 and order 1, that is B := (∅, {v0}, v �→ +⊥, v �→ v0) where v0 = min≼ V(G). So from now on we may assume that m ⩾ 1; thus we have +|A| ⩾ M(kk · (k + 1) · (m + k) + k). +Apply Lemma 10.6 to A, yielding suitable B1, S1, and mappings v �→ Bv and v �→ sv. Note +that we have |S1| ⩽ k and |B1| ⩾ kk · (k + 1) · (m + k) + k. We would like to claim that these +objects constitute a classifier, but for this we have to make sure that conditions (a), (b), and (c) +of Definition 3 hold. This will be done using a pigeonhole argument as follows. +Construct B′ +1 from B1 by removing the exceptional ball Bs for each s ∈ S1; thus |B′ +1| ⩾ +kk · (k + 1) · (m + k) and S1 is disjoint from � B′ +1. For a ball B ∈ B′ +1, let the profile of B be the pair +consisting of: +– the following equivalence relation on S1: s, s′ ∈ S1 are equivalent if they have the same +neighborhood in B; and +– the unique equivalence class of the relation above whose members have empty neighbor- +hood in B, or ⊥ if there is no such equivalence class. +Note that the total number of profiles is at most |S1||S1| · (|S1| + 1) ⩽ kk · (k + 1). Hence, there +exists B2 ⊆ B′ +1 with |B2| ⩾ m + k such that all balls from B2 have the same profile. Say this +profile is (≡, C), where ≡ is an equivalence relation on S1 and C is either an equivalence class +of ≡ or ⊥. +Let S2 ⊆ S1 be any set consisting of one member of each equivalence class of ≡. Note that for +all distinct s, s′ ∈ S2 and B ∈ B2, the neighborhoods of s and s′ in B are different. Furthermore, +every member of S2 is adjacent to all the balls from B2, except possibly for the vertex chosen +from C (if existent), which is non-adjacent to all the balls in B2. +We now define B2 := (B2, S2, exc2, rep2) as follows. For each vertex u of G, set exc2(u) = Bu, +unless Bu /∈ B2, in which case set exc2(u) := ⊥. Finally, set rep2(u) = η(su), where η : S1 → S2 +maps every s ∈ S1 to the unique s′ ∈ S2 such that s ≡ s′. It is now straightforward to verify that +B2 is a classifier. +It now remains to apply Corollary 10.4 to the classifier B2, yielding a classifier B = (B, S, exc, rep) +that is canonical with respect to ≼ and satisfies +|S| = |S2| ⩽ k, +B ⊆ B2 ⊆ A, +and +|B| ⩾ m = β(|A|). +10.2 +Proof of the result +We are ready to prove Theorem 9.1. The proof follows closely the reasoning from [DMST22], +except that we define the flip set somewhat more carefully in order to ensure the predictability +property. +44 + +Proof of Theorem 9.1. Throughout the proof we fix the monadically stable class C . For t ∈ N, by +C [t] we denote the class consisting of all ⩽ t-flips of graphs from C , that is, +C [t] := {G ⊕ F | G ∈ C and F is a flip set of size at most t in G}. +Note that since flips can be simulated by unary predicates, monadic stability of C entails monadic +stability of C [t], for every fixed t ∈ N. +Since we will be using Lemma 10.5 for different radii and different classes, in notation we +follow the convention that κD +s and βD +s denote the constant κ and the function β obtained from +applying Lemma 10.5 to a monadically stable class D and radius s ∈ N. +The proof proceeds by induction on r. Recall that our goal is to define suitable functions +FWr and Predictr, along with bounds λr and αr, for all r ∈ N. For this, we may use functions +FWr′ and Predictr′ and bounds λr′ and αr′ for r′ < r, obtained from the induction assumption. +Case 1: Base Case. +For r = 0, we may simply set +FW0(G, ≼, X) := (X, ∅) +and +Predict0(G, ≼, Z) := ∅. +Thus, we can set the bounds +λ0 := 0 +and +α0(n) := n. +Case 2: Inductive Case. +We first define the function FWr. For this, let us consider any G ∈ C , +≼ ∈ O(G), and X ⊆ V(G). We would like to find a pair (Y, F) satisfying the requirements for +the value FWr(G, ≼, X). +Let +(Yr−1, Fr−1) := FWr−1(G, ≼, X) +and +H := G ⊕ Fr−1. +By induction, Fr−1 consists of at most λr−1 flips. Thus H ∈ D, where +D := C [λr−1]. +Moreover, Yr−1 has size at least αr−1(|X|) and is (r − 1)-independent in H. +For convenience, we denote +r′ := ⌈r/2⌉ − 1. +Let A be the family of r′-balls in H whose centers are the vertices of Yr−1. Note that the balls of +A are pairwise disjoint. +Apply Lemma 10.5 to radius r′, graph H ∈ D, order ≼, and family of r′-balls A, thus obtain- +ing a classifier B = (B, S, exc, rep). Hence we have +|B| ⩾ βD +r′ (|A|) +and +|S| ⩽ κD +r′ . +By Ramsey’s Theorem, we may select a subfamily B⋆ ⊆ B of size at least log |B| satisfying the +following property: either the centers of the balls in B⋆ are pairwise at distance greater than r +in H, or they are pairwise at distance exactly r in H. We set +αr(n) := +� +log βD +r′ (αr−1(n)) +� +45 + +and note that by the construction, we have +|B⋆| ⩾ αr(|X|). +We define Y to be the set of centers of the balls in B⋆; thus |Y| = |B⋆| ⩾ αr(|X|). +It remains to construct a set of flips F such that Y is distance-r independent in G ⊕ F. We +proceed by cases, each time exposing a flip set F′ such that we may set F := Fr−1△F′ and +|F′| ⩽ max(4, k2 · 4k), +where +k := κD +r′ . +Note that thus we will have |F| ⩽ λr, assuming we set +λr := λr−1 + max(4, k2 · 4k). +Also, we will have G ⊕ F = H ⊕ F′. +Case 2.1: The vertices of Y are pairwise at distance greater than r in H. +In this case Y is +already distance-r independent, hence we may simply set F′ := ∅. +Case 2.2: The vertices of Y are pairwise at distance exactly r in H. +We may assume that +|Y| ⩾ 5, for otherwise we can choose F′ to be a set of at most 4 flips that isolate every vertex of Y +in H. In what follows, whenever speaking about adjacencies or distances, we mean adjacencies +and distances in H. +Recall that B is canonical with respect to ≼, hence +s = min +≼ ΠB(s) +for every s ∈ S. +By definition, every vertex of S is adjacent either to all the balls in B⋆, or to none. Further, since +vertices of S have pairwise different neighborhoods in every ball B ∈ B⋆, there is at most one +vertex of S that is not adjacent to any ball of B⋆. Let S′ ⊆ S consist of those vertices of S that are +adjacent to every ball in B⋆; thus either S′ = S or |S − S′| = 1. Let +W := +� +s∈S′ +ΠB(s). +We observe that the vertices of W are the ones that keep the vertices of Y at close distance, in +the following sense. +▷ Claim 10.7 For every vertex v ∈ V(G), the following conditions are equivalent: +1. v belongs to W; +2. v is at distance exactly r′ + 1 from all the vertices of Y, possibly except for one; +3. v is at distance at most r′ + 1 from at least two vertices of Y. +Proof of the claim. Implication (2)→(3) is trivial due to |Y| ⩾ 5. +For implication (1)→(2) we use that B is a classifier. Let s ∈ S′ be such that v ∈ ΠB(s). +By definition, s is adjacent to all the balls in B⋆. Therefore, v is adjacent to all the balls in B⋆, +possibly except for exc(v). Recalling that v ∈ exc(v) in case v ∈ � B⋆, we conclude that v is +46 + +at distance exactly r′ + 1 from the centers of all the balls in B⋆, that is, vertices of Y, possibly +except for one — the center of exc(v). +We are left with implication (3)→(1). Let s ∈ S be such that v ∈ ΠB(s). As v is at distance +at most r′ + 1 from the centers of two balls in B⋆, at least one of them, say B, is different from +exc(v). In particular v /∈ B, so v being at distance r′ + 1 from the center of B means that v has +to be adjacent to B. As B ̸= exc(v), we infer that s is also adjacent to B. It follows that s ∈ S′, +implying that v ∈ W. +◁ +Next, we make a case distinction depending on whether r is odd or even. In both cases, we +use the following notation. For s ∈ S and U ⊆ S, we write +Qs,U := {v ∈ ΠB(s) | NH(v) ∩ S = U}. +Further, we let +Q := {Qs,U : s ∈ S, U ⊆ S}. +Note that Q is a partition of the vertex set of H into at most k · 2k parts, and the definition of Q +only depends on the graph H, partition ΠB, and set S. In order to later prove the predictability +property, it will be crucial that, in both of the following two cases, the definition of the exposed +set of flips F′ only depends on the partition Q (and therefore on H, ΠB, and S), the set S′, and +the order ≼. +Case 2.2.1: r is odd. +We define F′ as the set of all pairs (Qs1,U1, Qs2,U2) ∈ Q2 satisfying the +following conditions: +– s1, s2 ∈ S′; +– Qs1,U1 ̸= ∅ and Qs2,U2 ̸= ∅; +– s1 ∈ U2 or s2 ∈ U1; and +– min≼ Qs1,U1 ≼ min≼ Qs2,U2. +Thus |F′| ⩽ (|Q| +2 ) ⩽ k2 · 4k. As desired, F′ depends only on Q, S′, and ≼. The following claim +explains the flip set F′ in more friendly terms. +▷ Claim 10.8 For any u1, u2 ∈ V(G), applying F′ flips the adjacency between u1 and u2 if and only if +u1, u2 ∈ W and (u2 ∈ NH(rep(u1)) or u1 ∈ NH(rep(u2))). +Proof of the claim. Let s1, U1, s2, U2 be such that u1 ∈ Qs1,U1 and u2 ∈ Qs2,U2; in particular +Qs1,U1 ̸= ∅ and Qs2,U2 ̸= ∅. By symmetry, we may assume that min≼ Qs1,U1 ≼ min≼ Qs2,U2. +By definition, the adjacency between u1 and u2 is flipped when applying F′ if and only if +(Qs1,U1, Qs2,U2) ∈ F′, which in turn is equivalent to the conjunction of conditions s1, s2 ∈ S′ and +(s1 ∈ U2 or s2 ∈ U1). It now remains to note that condition s1, s2 ∈ S′ is equivalent to u1, u2 ∈ W, +and condition (s1 ∈ U2 or s2 ∈ U1) is equivalent to (u2 ∈ NH(rep(u1)) or u1 ∈ NH(rep(u2))). ◁ +Further, we note that the vertices of W may only lie outside the balls of B⋆ or on their +boundaries. +▷ Claim 10.9 If v ∈ W, then for every y ∈ Y we have distH(v, y) ⩾ r′. +47 + +B1 +B2 +y1 +y2 +v1 +v2 +s1 = rep(v1) +FIGURE 5: Case 2.2.1 in a nutshell: Up to symmetry, the (depicted in red) adjacency between v1 and v2 is the same +as between s1 and v2, hence the edge v1v2 is flipped away when applying F′ if and only if it was present. +Proof of the claim. Suppose distH(v, y) ⩽ r′ − 1 for some y ∈ Y. As v ∈ W, by Claim 10.7 there +exists some other y′ ∈ Y, y′ ̸= y, such that distH(v, y′) = r′ + 1. Hence distH(y, y′) ⩽ 2r′ = r − 1. +This is a contradiction with the assumption that Y is (r − 1)-independent in H. +◁ +We are now ready to argue the following: Y is distance-r independent in H ⊕ F′. See Figure 6 +for an illustration. For contradiction, suppose in H ⊕ F′ there exists a path P of length at most r +connecting some distinct y1, y2 ∈ Y. Let B1, B2 ∈ B⋆ be the r′-balls with centers y1, y2, respec- +tively. Since the flips of F′ only affect the adjacency between the vertices of W, and these vertices +have to be at distance at least r′ = r−1 +2 +from y1, y2 due to Claim 10.9, we infer the following: P +can be written as +P = (y1, . . . , v1, v2, . . . , y2), +where (y1, . . . , v1) and (v2, . . . , y2) are paths of length r′ in H that are entirely contained in B1 +in B2, respectively. In particular, P has length exactly 2r′ + 1 = r and v1v2 is the only edge on P +that might have been flipped when applying F′. +Observe that if the edge v1v2 appeared when applying the flip F′, then we necessarily have +v1, v2 ∈ W. Otherwise, if v1v2 was present in H, then path P witnesses that already in H, both +v1 and v2 are at distance at most r′ + 1 from both y1 and y2. By Claim 10.7, this implies that +v1, v2 ∈ W. So in any case, we have v1, v2 ∈ W. +Let s1 := rep(v1) and s2 := rep(v2). Since v1 ∈ B1 and v2 ∈ B2, we have exc(v1) = B1 and +exc(v2) = B2, hence +NH(s1) ∩ B2 = NH(v1) ∩ B2 +and +NH(s2) ∩ B1 = NH(v2) ∩ B1. +In particular, +v1, v2 are adjacent in H +⇔ +v1, s2 are adjacent in H +⇔ +v1 ∈ NH(s2), +and similarly +v1, v2 are adjacent in H +⇔ +s1, v2 are adjacent in H +⇔ +v2 ∈ NH(s1). +Therefore, +v1, v2 are adjacent in H +⇔ +(v1 ∈ NH(s2) or v2 ∈ NH(s1)). +As v1, v2 ∈ W, by Claim 10.8 we conclude that v1 and v2 are adjacent in H if and only if their ad- +48 + +B1 +B2 +y1 +y2 +v1 +v2 +s = rep(u) +u +FIGURE 6: Case 2.2.2 in a nutshell: Up to symmetry, the (depicted in red) adjacency between v1 and u is the same +as between v1 and s, hence the edge uv1 is flipped away when applying F′ if and only if it was present. +jacency gets flipped when applying F′. So v1 and v2 are non-adjacent in H ⊕ F′, a contradiction +with the existence of the edge v1v2 on P. +Case 2.2.2: r is even. +This time, F′ is defined as the set of all pairs (Qs1,U1, Qs2,U2) ∈ Q2 satis- +fying the following conditions: +– Qs1,U1 ̸= ∅ and Qs2,U2 ̸= ∅; +– (s1 ∈ S′ and s1 ∈ U2) or (s2 ∈ S′ and s2 ∈ U1); and +– min≼ Qs1,U1 ≼ min≼ Qs2,U2. +Again, |F′| ⩽ |Q|2 ⩽ k2 · 4k and the definition of F′ depends only on Q, S′, and ≼. Also, we may +similarly explain flipping according to F′ as follows. +▷ Claim 10.10 For any u1, u2 ∈ V(G), applying F′ flips the adjacency between u1 and u2 if and only +if (u1 ∈ W and u2 ∈ NH(rep(u1))) or (u2 ∈ W and u1 ∈ NH(rep(u2))). +Proof of the claim. Analogous to the proof of Claim 10.8, we leave the details to the reader. +◁ +Note that Claim 10.8 implies in particular that whenever the adjacency between two vertices +is flipped when applying F′, at least one of them belongs to W. (However, contrary to the odd +case, there might be flips in F′ that affect vertices outside of W.) In this vein, the following +observation will be convenient. +▷ Claim 10.11 W ∩ � B⋆ = ∅. +Proof of the claim. For contradiction, suppose there exists B ∈ B⋆ and v ∈ B such that v ∈ W. +Letting y be the center of B, we have distH(v, y) ⩽ r′. By Claim 10.7, there exists another y′ ∈ Y, +y′ ̸= y, such that distH(v, y′) ⩽ r′ + 1. Hence distH(y, y′) ⩽ 2r′ + 1 = r − 1, contradicting the +distance-(r − 1) independence of Y in H. +◁ +As in the odd case, we are left with arguing that Y is distance-r independent in H ⊕ F′. +For contradiction, suppose that there exist distinct y1, y2 ∈ Y and a path P of length at most r +that connects y1 and y2 in H ⊕ F′. As before, let B1, B2 ∈ B⋆ be the balls with centers y1, y2, +respectively. +By Claim 10.10, the flips of F′ affect only the vertices of W ∪ � +s∈S′ NH(s). By Claim 10.11 +and as S′ is disjoint with � B⋆, all vertices of W ∪ � +s∈S′ NH(s) are at distance (in H) at least r′ +49 + +from all the vertices of Y. Since r = 2r′ + 2, similarly as in Case 2.2.1 it follows that P has length +2r′ + 1 = r − 1 or 2r′ + 2 = r and can be written as +P = (y1, . . . , v1, v2, . . . , y2) +or +P = (y1, . . . , v1, u, v2, . . . , y2), +where (y1, . . . , v1) and (v2, . . . , y2) are paths of length r′ in H entirely contained in B1 and B2, +respectively. +Consider the first case: P has length r − 1 and is of the form (y1, . . . , v1, v2, . . . , y2). Ob- +serve that edge v1v2 cannot be present in H, because then P would be entirely contained in H, +a contradiction with distance-(r − 1) independence of Y in H. On the other hand, note that +v1, v2 /∈ W due to Claim 10.11, so by Claim 10.10 the adjacency between v1 and v2 is not flipped +when applying F′. We conclude that v1 and v2 remain non-adjacent in H ⊕ F′, a contradiction +with the presence of the edge v1v2 on P. +We are left with the second case: P has length r and is of the form (y1, . . . , v1, u, v2, . . . , y2). +Let us first argue that u ∈ W. If u is adjacent both to v1 and to v2 in H, then u is at distance +at most r′ + 1 from both y1 and y2 in H, hence that u ∈ W follows directly from Claim 10.7. On +the other hand, if u is non-adjacent in H to one of v1 or v2, say to v1, then the adjacency between +u and v1 must get flipped when applying F′. By Claim 10.10 this means that at least one of u +and v1 belongs to W, but it cannot be v1 due to Claim 10.11. So u ∈ W in this case as well. +Let s := rep(u). By symmetry, we may assume that B1 ̸= exc(u). This means that +v1, u are adjacent in H +⇔ +v1, s are adjacent in H +⇔ +v1 ∈ NH(s). +Since u ∈ W and v1 /∈ W (due to Claim 10.11), by Claim 10.10 we conclude that u and v1 are +adjacent in H if and only if their adjacency gets flipped when applying F′. So in any case, u and +v1 are non-adjacent in H ⊕ F′. This is a contradiction with the presence of the edge uv1 on P. +This concludes the construction of the pair (Y, F) that we set for FWr(G, ≼, X). We are +left with defining a suitable inductive case for the function Predictr and showing that it can +be computed efficiently, in time OC ,r(|V(G)|2). For G ∈ C and Z ⊆ V(G) with |Z| = 5, +Predictr(G, ≼, Z) is defined as the flip set F◦ output by the following procedure. +– Let F◦ +r−1 := Predictr−1(G, ≼, Z), where the function Predictr−1 is provided by the induction +assumption. Let H◦ := G ⊕ F◦ +r−1. +– If Z is not distance-(r − 1) independent in H◦, output F◦ := ∅ and terminate. +– Similarly, if Z is distance-r independent in H◦, output F◦ := F◦ +r−1 and terminate. +– Otherwise, let B◦ consist of the five r′-balls in H◦ with centers in vertices of Z; note that +the balls of B◦ are pairwise disjoint. Apply the algorithm of Lemma 10.2 to the family B◦, +thus obtaining a partition Π◦. +– Let S◦ := {min≼ A : A ∈ Π◦} and let S′◦ be the subset of vertices of S◦ that are adjacent +to every ball of B◦. +– Output the flip set F◦ := F◦ +r−1△F′◦, where F′◦ is defined from H◦, Π◦, S◦, and S′◦ exactly +in the way described in Cases 2.2.1 and 2.2.2 above. +50 + +We now argue that provided FWr(G, ≼, X) = (Y, F) and Z ⊆ Y is a set of size 5, we have +Predictr(G, ≼, Z) = F. Adopt the notation from the definition of FWr. We revisit the case +study presented above and show that in each case, the set F◦ output by the procedure defining +Predictr(G, ≼, Z) coincides with the set of flips F constructed in the definition of FWr(G, ≼, X). +By the induction assumption, we have F◦ +r−1 = Fr−1, which implies that H◦ = H. In partic- +ular, as Z ⊆ Y is distance-(r − 1) independent in H, the termination in the second point above +cannot happen. Also, if the vertices of Y are pairwise at distance more than r in H, then so is +the case for Z, and we have F◦ = F◦ +r−1 (termination in the third point above). In the definition +of FWr(G, ≼, X) Case 2.1 applies here, yielding F = Fr−1. So F = Fr−1 = F◦ +r−1 = F◦, as required. +We are left with Case 2.2: the vertices of Y are pairwise at distance exactly r in H. By +Lemma 10.2 we have Π◦ = ΠB, where B = (B, S, exc, rep) is the classifier provided by Lemma 10.5 +that is used in the construction of Y. Since B is canonical with respect to ≼, we have +S = {min +≼ A : A ∈ ΠB} = {min +≼ A : A ∈ Π◦} = S◦. +Similarly, as B is a classifier in H = H◦, we have that a vertex from S = S◦ is adjacent to every +ball of B⋆ ⊆ B if and only if it is adjacent to every ball of B◦, and we conclude that S′ = S′◦. +As now H◦ = H, Π◦ = ΠB, S◦ = S, and S′◦ = S′, the construction presented in Cases 2.2.1 +and 2.2.2 provides the same flip set for H◦, Π◦, S◦, and S′◦, as for H, ΠB, S, and S′: we have +F◦ = F, as required. +Finally, we need to argue that Predictr(G, ≼, Z) can be computed in time OC ,r(|V(G)|2). +For this, observe that the procedure presented above executes r inductive calls, each of which +consists of internal computation that is easy to implement in time OC ,r(|V(G)|2), and one call +to the algorithm of Lemma 10.5. Since we a priori know that the partition Π◦ returned by this +call should be of size OC ,r(1), we may terminate this call once the elapsed running time exceeds +OC ,r(|V(G)|2), and if so, return ∅ as Predictr(G, ≼, Z). Therefore, each of the r inductive calls +runs in time OC ,r(|V(G)|2), giving a total time complexity of OC ,r(|V(G)|2) as well. +11 +Winning strategy +We are almost ready to prove Theorem 1.5. Before commencing to the proof, we will first clarify +the notion of a strategy for Flipper, and what we mean by a running time of a strategy. +Furthermore, in this section, we will work with an extension of the Flipper game, which we +call Induced-Subgraph-Flipper game. In this variant of the game, we allow Connector to localize +the graph to an induced subgraph of an r-ball in the current arena, instead of requiring her +to pick an entire r-ball. Intuitively, “losing” additional vertices on purpose yields no benefit +to Connector. However, the ability to work with induced subgraphs is useful for the design +of algorithms, as exhibited in [DMS]. For this reason we will explicitly prove the algorithmic +winning strategy for Flipper for this variant of the game. As the modification only expands the +move pool of Connector, the proven strategy then also works for the unmodified Flipper game +and implies Theorem 1.5. +11.1 +Strategies and runtimes +Strategies and runs. +Strategies are commonly represented by functions mapping the history +of the game to a new (played) position. In our context, it will be convenient to use the following +51 + +equivalent abstraction, which will fit better to our algorithmic perspective. Fix radius r ∈ N. +Graphs considered in consecutive rounds of the Induced-Subgraph-Flipper game will be often +called arenas, for brevity. A radius-r Connector strategy is a function +con: (Gi) �→ (Gloc +i +) +that maps the arena Gi at round i to a graph Gloc +i +that is an induced subgraph of the ball of +radius r around some vertex v in Gi. +A radius-r Flipper strategy is a function +flip: (Gloc +i +, Ii) �→ (F, Ii+1) +that maps the graph Gloc +i +obtained from Connector’s move to the atomic flip F chosen by Flip- +per; the next arena will be Gi+1 := Gloc +i +⊕ F. Additionally, we allow Flipper to keep an auxil- +iary memory: the strategy takes, as the second argument, an internal state Ii from the previous +round, and outputs an updated internal state Ii+1. The initial state I0 = I0(flip, G) will be +computed from the initial graph at the beginning of the game. The internal states will be used +as memory and to precompute flips for future turns, which makes them convenient from an +algorithmic point of view. Strategies operating with game histories instead of internal states +can simulate the latter in the following sense: knowing the game history, Flipper can compute +the current internal state by replaying the entire game up to the current round. Note that since +we are interested in Flipper’s strategies that work against any behavior of Connector, it is not +necessary to equip Connector’s strategies with memory as well. +Given radius-r Connector and Flipper strategies con and flip, and a graph G, we define the +run R(con, flip, G) to be the infinite sequence of positions +R(con, flip, G) := (G0, I0), (G1, I1), (G2, I2), (G3, I3), . . . +such that G0 = G, I0 = I0(flip, G), and for all i ⩾ 0 we have +Gi+1 = con(Gi) ⊕ F, +where +(Ii+1, F) = flip(con(Gi), Ii). +A winning position is a tuple (Gi, Ii) where Gi contains only a single vertex. A radius-r Flipper +strategy flip is ℓ-winning on a class of graphs C , if for every G ∈ C and for every radius-r +Connector strategy con, the ℓth position of R(con, flip, G) is a winning position. Note that while +R(con, flip, G) is an infinite sequence, once a winning position is reached, it is only followed by +winning positions. +Runtime. +Let r ∈ N and let flip be a radius-r Flipper strategy. For a function f : N → N, we +say that flip has runtime f if the following holds: +– given a graph G, the internal state I0(flip, G) can be computed in time f (|G|); and +– given a graph H and an internal state I, the value flip(H, I) can be computed in time f (|G|). +Note that in the second item above, the time complexity is allowed to depend on the original +graph G, which is possibly much larger than the current arena H. On the other hand, we do +not require a dependence on the size of the encoding of I. Namely, it will always be the case +that in positions that may appear in runs of flip on graphs from the considered class of graphs, +52 + +the encoding size of I will be linear in the encoding size of G. Hence, positions with larger +encoding size of I can be just ignored. (Formally, the algorithm outputs anything on them +while not reading the whole internal state.) +We will often say that a strategy has runtime F for a class of functions F to indicate that it +has runtime f ∈ F. (For instance, we may say that a strategy has runtime OC ,r(n2).) +Playing multiple flips. +As discussed in Section 3, we may also consider the variant of Flipper +game where in every round, Flipper can apply not a single atomic flip F, but a set of flips F +of size at most g(i), where i is the index of the round. Here, g: N → N is a function and we +call this variant of the game g-bounded. We may also speak about the k-bounded variant of the +game where k ∈ N, and by this we mean the g-bounded game for g being the constant function +g(i) = k. Thus, the standard game is 1-bounded. The notions of strategies, runs, and runtimes +translate to the setting of g-bounded Flipper game naturally. +The following simple lemma shows that when designing a strategy for Flipper on a graph +class, it suffices to consider the setting where playing multiple flips in a single move is allowed. +Lemma 11.1 Let C be a class of graphs and r ∈ N be a fixed radius. Suppose that for some ℓ ∈ N and +functions f, g: N → N, Flipper has a strategy in the g-bounded radius-r Induced-Subgraph-Flipper +game that is ℓ-winning on C , and moreover this strategy has runtime f. Then Flipper also has a strategy +in the standard (1-bounded) radius-r Induced-Subgraph-Flipper game that is ℓ′-winning on C , where +ℓ′ = ∑ℓ +i=1 g(i), and this strategy has runtime Og,ℓ( f ). +Proof. Let flip be the assumed strategy in the g-bounded game. We define a strategy flip′ in +the 1-bounded game as follows. When playing flip′, Flipper simulates flip by replacing the +ith move in the g-bounded game by g(i) consecutive moves in the 1-bounded game. More +precisely, supposing that flip proposes to play a flip set F of size at most g(i), in flip′ Flipper +plays the atomic flips of F one by one, in |F| consecutive rounds. Within the formal framework +of strategies, these atomic flips are saved in a queue within the internal state, and popped from +the queue one by one until the queue is empty — and the next move of flip in the simulated +game needs to be computed. The moves of Connector along the way are ignored, except for the +last one, which is considered the next Connector’s move in the simulated g-bounded game for +the purpose of computing the next Flipper’s move proposed by flip. +A straightforward induction argument shows that for every j ∈ N, the arena after ∑ +j +i=1 g(i) +rounds in the 1-bounded game played according to flip′ is an induced subgraph of the arena +after j rounds in the simulated g-bounded game played according to flip. Consequently, flip′ is +ℓ′-winning on C for ℓ′ = ∑ℓ +i=1 g(i). As for the runtime, the algorithm computing the next move +of flip′ either pops the next atomic flip from the queue stored in the internal state, or, in case the +queue is empty, invokes the algorithm to compute the next move of flip. It is straightforward to +see that this can be done in time Og,ℓ( f ). +11.2 +Finalizing the argument +With the definitions above settled, we can now rephrase and prove Theorem 1.5 as follows. +Theorem 11.2 For every monadically stable class of graphs C and radius r ∈ N, there exists ℓ ∈ N +and a radius-r Flipper strategy for the Induced-Subgraph-Flipper game that is ℓ-winning on C and has +runtime OC ,r(n2). +53 + +Proof. In notation, we fix the objects provided by Theorem 9.1 for the class C . Let then +t := α−1 +2r (7), +k := λ2r, +and +ℓ := 2 · +��t +5 +� ++ 1 +�2 +, +where by α−1 +2r (7) we mean the least integer N such that α2r(N) ⩾ 7. We will describe a strategy +flip⋆ for Flipper in the g-bounded radius-r game, where +g(i) := max(i, k). +Strategy flip⋆ will be ℓ-winning on C and will have runtime OC ,r(n2). By Lemma 11.1, this +suffices to prove Theorem 11.2. +We explain now flip⋆ in natural language; the easy translation to the formal layer of strate- +gies with internal states, described in Section 11.1, is left to the reader. We fix the graph G ∈ C +on which the game is played, together with an arbitrary linear order ≼ on V(G). +First, Flipper will always play moves in move pairs: Having constructed some flip set F, Flip- +per first applies F to the current arena H, then lets the Connector localize the game to a radius-r +ball in H ⊕ F, and finally he applies F again. In this way, the following invariant will be satis- +fied: After applying every move pair, the arena is an induced subgraph of G (cf. Observation 2.2 +and Observation 2.5). We will say that a move pair as described above is defined by F. +Second, Flipper proceeds in a sequence of eras, each consisting of a number of consecutive +moves. Along the way, he keeps track of a growing chain of vertex subsets +∅ = X0 ⊊ X1 ⊊ X2 ⊊ X3 ⊊ . . . , +where Xi is obtained from Xi−1 at the end of era i by adding one vertex that is still contained in +the arena, but not contained in Xi−1. Up until Flipper wins the game, we will ensure that such +a vertex always exists, and therefore |Xi| = i for every i ∈ N until the game concludes. +We now describe Flipper’s moves in era i (i = 1, 2, 3, . . .). For every Z ⊆ Xi−1 with |Z| = 5, +we compute the flip set +FZ := Predict2r(G, ≼, Z). +Note that, instead of the current arena, the original graph G is used to compute FZ. First, Flipper +performs (|Xi−1| +5 +) move pairs, each defined by FZ for a different Z as above. Finally, let F be a +flip set of size |Xi−1| = i − 1 such that in G ⊕ F, every vertex of Xi−1 is isolated. (Such F can be +obtained by iteratively isolating vertices of Xi−1 by performing a flip between a vertex and its +neighborhood.) +At the end of the era, Flipper applies the move pair defined by F. After the first application +of F within the move pair, the resulting arena is an induced subgraph of G ⊕ F where all the +vertices of Xi−1 are isolated. Therefore, the induced subgraph chosen as Connector’s response +must contain a vertex x not belonging to Xi−1, otherwise Connector loses immediately after +making her move. Followingly, we may set Xi := Xi−1 ∪ {x} and proceed to the next era. +This concludes the description of flip⋆. Clearly, flip⋆ is a valid strategy in the g-bounded +game. We now argue that following flip⋆ leads to a quick victory. +▷ Claim 11.3 If Flipper follows flip⋆, the game concludes within at most t eras. +Proof of the claim. For contradiction, suppose the game enters era t + 1 without termination. +54 + +Denote X := Xt; we have |X| = t = α−1 +2r (7). Let +(Y, F) := FW2r(G, ≼, X). +Thus, |Y| ⩾ 7 and Y is distance-2r independent in G ⊕ F. Let y1, . . . , y7 be any seven distinct +vertices of Y, where yi was added earlier to X than yj for all i < j. +Let +Z := {y1, . . . , y5} +and +FZ := Predict2r(G, ≼, Z). +Further, let s be the index of the era that concluded with adding y6 to X. (That is, we have +Xs = Xs−1 ∪ {y6} and in particular Xs−1 ∩ {y6, y7} = ∅.) Note that Z ⊆ Xs−1, hence within era +s, Flipper applied the move pair defined by FZ. Let H be the arena during that era right before +the first application of FZ. Note that H is an induced subgraph of G. After the first application +of FZ, Connector responded by restricting the arena to an induced subgraph H′ of some radius-r +ball in H ⊕ FZ. Clearly, the vertex set of H′ is entirely contained in some radius-r ball in G ⊕ FZ. +Since Y is distance-2r independent in G ⊕ FZ and y6, y7 ∈ Y, we conclude {y6, y7} ⊈ V(H′). In +other words, at least one of the vertices y6, y7 got removed from the arena as a consequence of +Connector’s move. This contradicts the assumption that both y6 and y7 were later added to X, +which requires them to both be contained in the arena at the end of era s. +◁ +Note that in era i, Flipper applies exactly (|Xi−1| +5 +) + 1 = (i−1 +5 ) + 1 move pairs. Hence, by +Claim 11.3, the game terminates within at most +t +∑ +i=1 +2 · +��i − 1 +5 +� ++ 1 +� +⩽ ℓ +rounds. +We conclude that flip⋆ is ℓ-winning on C , as promised. Finally, computing Flipper’s moves for +an era boils down to at most (t +5) = OC ,r(1) applications of the algorithm provided by Theo- +rem 9.1, which runs in time OC ,r(n2). It follows that flip⋆ has runtime OC ,r(n2) (in a suitable +formalization of the strategy through internal states). +References +[AA14] +Hans Adler and Isolde Adler. Interpreting nowhere dense graph classes as a +classical notion of model theory. Eur. J. 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[Online; accessed 23- +Jan-2023]. +57 + diff --git a/mNFST4oBgHgl3EQfKDg1/content/tmp_files/load_file.txt b/mNFST4oBgHgl3EQfKDg1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0df2fd7b50a61250877644b5f9003340d6ed2ecb --- /dev/null +++ b/mNFST4oBgHgl3EQfKDg1/content/tmp_files/load_file.txt @@ -0,0 +1,2821 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf,len=2820 +page_content='Flipper games for monadically stable graph classes* Jakub Gajarský† Nikolas Mählmann‡ Rose McCarty§ Pierre Ohlmann† Michał Pilipczuk† Wojciech Przybyszewski† Sebastian Siebertz‡ Marek Sokołowski† Szymon Toru´nczyk† Abstract A class of graphs C is monadically stable if for any unary expansion � C of C , one cannot interpret, in first-order logic, arbitrarily long linear orders in graphs from � C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' It is known that nowhere dense graph classes are monadically stable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' these encompass most of the studied concepts of sparsity in graphs, including classes of graphs that exclude a fixed topological minor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' On the other hand, monadic stability is a property expressed in purely model-theoretic terms and hence it is also suited for capturing structure in dense graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For several years, it has been suspected that one can construct a structure theory for monadically stable graph classes that mirrors the theory of nowhere dense graph classes in the dense setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In this work we provide a next step in this direction by giving a character- ization of monadic stability through the Flipper game: a game on a graph played by Flipper, who in each round can complement the edge relation between any pair of vertex subsets, and Connector, who in each round is forced to localize the game to a ball of bounded radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This is an analog of the Splitter game, which characterizes nowhere dense classes of graphs (Grohe, Kreutzer, and Siebertz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ACM ’17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We give two different proofs of our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The first proof is based on tools borrowed from model theory, and it exposes an additional property of monadically stable graph classes that is close in spirit to definability of types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Also, as a byproduct, we give an alternative proof of the recent result of Braunfeld and Laskowski (arXiv 2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='05120) that monadic stabil- ity for graph classes coincides with existential monadic stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The second proof relies on the recently introduced notion of flip-wideness (Dreier, Mählmann, Siebertz, and Toru´nczyk, arXiv 2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='13765) and provides an efficient algorithm to compute Flipper’s moves in a win- ning strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We thank Patrice Ossona de Mendez for his valuable contributions to this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This paper is part of projects that have received funding from the European Research Council (ERC) (grant agree- ment No 948057 – BOBR) and from the German Research Foundation (DFG) with grant agreement No 444419611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' †University of Warsaw, Poland ‡University of Bremen, Germany §Princeton University, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Supported by European Research Council (ERC) grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 714704 – CUTACOMBS and National Science Foundation (NSF) grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' DMS-2202961.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='13735v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='LO] 31 Jan 2023 erc European Research Council Established by the European CommissionContents 1 Introduction 3 I Prelude 8 2 Preliminaries 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 Model theory .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 44 11 Winning strategy 51 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 Strategies and runtimes .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 51 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2 Finalizing the argument .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 53 2 1 Introduction Monadic stability is a notion of logical tameness for classes of structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Introduced by Baldwin and Shelah [BS85] in the context of model theory1, it has recently attracted attention in the field of structural graph theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We recall the logical definition below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' One of the main contributions of this paper is to provide a purely combinatorial characterization of monadically stable classes of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Our characterization is effective, and can be employed in algorithmic applications, as we explain later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In this paper we focus on (undirected, simple) graphs, rather than arbitrary structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' A graph is modelled as a relational structure with one symmetric binary relation signifying adja- cency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By a class of graphs we mean any set of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For a class of graphs C , a unary expansion of C is any class � C of structures such that each �G ∈ � C is obtained from some graph in G ∈ C by adding some unary predicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Thus, the elements of � C can be regarded as vertex-colored graphs from C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' A class of graphs C is called monadically stable if one cannot interpret, using a fixed formula ϕ( ¯x, ¯y) of first-order logic, arbitrarily long linear orders in any unary expan- sion � C of C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' More precisely, for every unary expansion � C and formula ϕ( ¯x, ¯y) with | ¯x| = | ¯y| (over the signature of � C ) there is a bound ℓ such that there is no structure �G ∈ � C and tuples ¯a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , ¯aℓ ∈ V( �G) ¯x such that �G |= ϕ(¯ai, ¯aj) if and only if i ⩽ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' More generally, C is monadically dependent (or monadically NIP) if one cannot interpret, using a fixed formula ϕ( ¯x, ¯y) of first-order logic, all finite graphs in any unary expansion of C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Thus, from the model-theoretic perspective, the intuition is that being monadically dependent is being non-trivially constrained: for any fixed interpretation, one cannot interpret arbitrarily complicated structures in vertex-colored graphs from the considered class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' On the other hand, graphs from monadically stable classes are “orderless”, in the sense that one cannot totally order any large part of them using a fixed first-order formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Baldwin and Shelah proved that in the definitions, one can alternatively rely on formu- las ϕ(x, y) with just a pair of free variables, instead of a pair of tuples of variables [BS85, Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='3, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Moreover, they proved that monadically stable theories are tree decomposable [BS85, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='17], providing a structure theorem for such theories, although one of a very infinitary nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' A more explicit, combinatorial structure theorem for monad- ically stable and monadically dependent is desirable for obtaining algorithmic results for the considered classes, as we discuss later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' On the other hand, Braunfeld and Laskowski [BL22] very recently proved that for hereditary classes of structures C that are not monadically stable or monadically dependent, the required obstructions (total orders or arbitrary graphs) can be exhibited by an existential formula ϕ( ¯x, ¯y) in the signature of C , without any additional unary predicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Among other things, this shows that for hereditary classes of structures, the notions of monadic stability coincides with the more well-known notion of stability, and similarly, monadic dependence coincides with depen- dence (NIP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Furthermore, since the formulas are existential, this result can be seen as a rather explicit, combinatorial non-structure theorem for hereditary classes that are not monadically stable (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' monadically dependent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Still, they do not provide explicit structural results for classes that are monadically stable or monadically dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Explicit, combinatorial and algorithmic structural results for monadically dependent and monadically stable classes are not only desired, but also expected to exist, based on the known 1Formally, Baldwin and Shelah [BS85], as well as Braunfeld and Laskowski [BL22], study monadically dependent and monadically stable theories, rather than classes of structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Some of their results transfer to the more general setting of monadically dependent/stable classes of structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 3 examples of such classes that have been studied in graph theory and computer science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' As observed by Adler and Adler [AA14] based on the work of Podewski and Ziegler [PZ78], all nowhere dense graph classes are monadically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' A class C is nowhere dense if for every fixed r ∈ N, one cannot find r-subdivisions2 of arbitrarily large cliques as subgraphs of graphs in C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In particular, every class excluding a fixed topological minor (so also the class of planar graphs, or the class of subcubic graphs) is monadically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In fact, it follows from the results of Adler and Adler [AA14] and of Dvoˇrák [Dvo18] that monadic stability and monadic dependence are both equivalent to nowhere denseness if one assumes that we work with a class of sparse graphs (formally, with a class of graphs that excludes a fixed biclique as a subgraph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' However, since they are defined in only model-theoretic terms, monadic stability and monadic dependence are not bound to sparsity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' they can be used to understand and quantify structure in dense graphs as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The pinnacle of the theory of nowhere dense graph classes is the result of Grohe, Kreutzer, and Siebertz [GKS17] that the model-checking problem for first-order logic is fixed-parameter tractable on any nowhere dense class of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 ([GKS17]) For every nowhere dense graph class C , first-order sentence ϕ, and ε > 0, there exists an algorithm that given an n-vertex graph G ∈ C decides whether G |= ϕ in time OC ,ϕ,ε(n1+ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Monadically dependent classes include all monadically stable classes, in particular all nowhere dense classes, but also for instance all classes of bounded twin-width [BKTW20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' An analogous result, with 1 + ε replaced by 3, holds for all classes C of ordered graphs3 of bounded twin- width [BGOdM+22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In light of the discussion above, monadic stability and monadic dependence seem to be well-behaved generalizations of nowhere denseness that are defined in purely model-theoretic terms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' hence these concepts may be even better suited for treating the model-checking problem for first-order logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This motivated the following conjecture [war16], which has been a subject of intensive study over the last few years4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 Let C be a monadically dependent graph class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' There exists a constant c ∈ N depending only on C and, for every first-order sentence ϕ, an algorithm that, given a n-vertex graph G ∈ C , decides whether G |= ϕ in time OC ,ϕ(nc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 is not even resolved for monadically stable classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' To approach this conjec- ture, it is imperative to obtain explicit, combinatorial structure theorems for monadically stable and in monadically dependent graph classes, with a particular focus on finding analogs of the tools used in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Our work contributes in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We provide cer- tain tree-like decompositions for graphs in monadically stable graph classes, which can be most intuitively explained in terms of games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' On the one hand, our decompositions generalize a sim- ilar result for nowhere dense classes, recalled below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' On the other hand, they are remininiscent of the tree decomposability property proved by Baldwin and Shelah, but are more explicit and finitary in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Splitter game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The cornerstone of the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 is a game-theoretic characteriza- tion of nowhere denseness, through the Splitter game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This game has a fixed radius parameter 2The r-subdivision of a graph H is the graph obtained from H by replacing every edge with a path of length r + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 3Ordered graphs are graphs equipped with a total order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 4To the best of our knowledge the conjecture was first explicitly discussed during the open problem session of the Algorithms, Logic and Structure Workshop in Warwick, in 2016, see [war16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 4 r ∈ N and is played on a graph G between two players, Splitter and Connector, who make moves in rounds alternately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In each round, Splitter first chooses any vertex u and removes it from the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Next, Connector has to select any other vertex v, and the game gets restricted to the subgraph induced by the ball of radius r with center at v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The game ends with Splitter’s victory when there are no vertices left in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2 ([GKS17]) A class C of graphs is nowhere dense if and only if for every r ∈ N there exists k ∈ N such that for every G ∈ C , Splitter can win the radius-r Splitter game on G within k rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Very roughly speaking, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2 shows that any graph from a nowhere dense class can be hierarchically decomposed into smaller and smaller parts so that the decomposition has height bounded by a constant k depending only on the class and the locality parameter r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This decomposition is used in the algorithm of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 to guide model-checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Flipper game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In this work we introduce an analog of the Splitter game for monadically stable graph classes: the Flipper game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Similarly to before, the game is played on a graph G and there is a fixed radius parameter r ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' There are two players, Flipper and Connector, which make moves in rounds alternately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In her move, Flipper selects any pair of vertex subsets A, B (possi- bly non-disjoint) and applies the flip between A and B: inverts the adjacency between any pair (a, b) of vertices with a ∈ A and b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Connector’s moves are exactly as in the Splitter game;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' in every round, he needs to select a ball of radius r, and the game is restricted to the subgraph induced by this ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The game is won by Flipper once there is only one vertex left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We remark that the Flipper game is a radius-constrained variant of the natural game for graph parameter SC-depth, which is functionally equivalent to shrubdepth, in the same way that the Splitter game is a radius-constrained variant of the natural game for treedepth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' SC-depth and shrubdepth were introduced and studied by Ganian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' in [GHN+12, GHN+19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Our main result is the following analog of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2 for monadically stable classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='3 A class C of graphs is monadically stable if and only if for every r ∈ N there exists k ∈ N such that for every graph G ∈ C , Flipper can win the radius-r Flipper game on G within k rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let us compare Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='3 with another recent characterization of monadic stability, pro- posed by Gajarský and Kreutzer, and proved by Dreier, Mählmann, Siebertz, and Toru´nczyk [DMST22], through the notion of flip-wideness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This notion is an analog of uniform quasi-wideness, introduced by Dawar [Daw10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Without going into technical details, a class of graphs C is uni- formly quasi-wide if for any graph G ∈ C and any large enough set of vertices A in G, one can find many vertices in A that are pairwise far from each other after the removal of a constant number of vertices from G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' As proved by Nešetˇril and Ossona de Mendez [NO11], a class of graphs is uniformly quasi-wide if and only if it is nowhere dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Flip-wideness is an analog of uniform quasi-wideness obtained similarly to the Flipper game: by replacing the concept of deleting a vertex with applying a flip;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' see Definition 1 for a formal definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The fact that monadic stability is equivalent to flip-wideness (as proved in [DMST22]) and to the existence of a short winning strategy in the Flipper game (as proved in this paper) suggests the following: the structural theory of monadically stable graph classes mirrors that of nowhere dense graph classes, where the flip operation is the analog of the operation of removing a vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We give two very different proofs of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The first proof is based on elementary model-theoretic techniques, and it provides new insight into the properties of monadically sta- ble graph classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' As a side effect, it gives a new (though non-algorithmic) proof of the main result of [DMST22]: equivalence of monadic stability and flip-wideness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' On the other hand, the 5 second proof relies on the combinatorial techniques developed in [DMST22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' It has the advan- tage of being effective, and provides an efficient algorithm for computing Flipper’s moves in a winning strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Model-theoretic proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The following statement lists properties equivalent to monadic stabil- ity uncovered in our model-theoretic proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Notions not defined so far will be explained later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4 Let C be a class of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then the following conditions are equivalent: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' C is monadically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' C has a stable edge relation and is monadically dependent with respect to existential formulas ϕ(x, y) with two free variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' C has a stable edge relation and is pattern-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For every r ∈ N every model G of the theory of C , every elementary extension H of G, and every vertex v ∈ V(H) − V(G), there is a finite set S ⊆ V(G) that r-separates v from G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For every r ∈ N there is k ∈ N such that Separator wins the Separation Game of radius r on every G ∈ C in at most k rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For every r ∈ N there is k ∈ N such that Flipper wins the Flipper game of radius r on every G ∈ C in at most k rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' C is flip-wide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' (1) mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' stable (2) stable edge relation + ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' dependent (3) stable edge relation + pattern-free (4) separable (5) Separator wins (6) Flipper wins (7) flip-wide trivial Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 6 Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 7 Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 8 Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2 [DMST22] FIGURE 1: The implications that constitute Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4, together with the sections in which they are proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The implications that constitute Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4 are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Note that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='3 is the equivalence (1)↔(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let us give a brief overview of the presented conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Conditions (1) and (2), respectively, are monadic stability and a weak form of existential monadic stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Recall that Baldwin and Shelah proved that it is sufficient to consider formu- las ϕ(x, y) with two free variables in the definition of monadic stability (instead of formulas ϕ( ¯x, ¯y)), whereas Braunfeld and Laskowski proved that it is sufficient to consider existential formulas ϕ( ¯x, ¯y) that do not involve additional unary predicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The condition (2) lies some- where in between: it implies that it is sufficient to consider existential formulas ϕ(x, y) with two variables, possibly involving additional unary predicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In particular, it implies the re- sult of Baldwin and Shelah (in the case of graph classes) and is incomparable with the result of Braunfeld and Laskowski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Our proof uses different techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 6 Condition (3) is a rather explicit combinatorial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Roughly, a class C is pattern-free if it does not encode, using a quantifier-free formula ϕ(x, y), the class of r-subdivided cliques, for any fixed r ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' See Definition 2 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Condition (4) is phrased in the language of model theory and serves a key role in our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' It resembles a fundamental property called “definability of types”, and in essence it says the following: whenever working with a model G of the theory of C , every element of any elemen- tary extension of G can be robustly “controlled” by a finite subset of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We believe that the new notion of r-separation used here is of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' It refers to non-existence of short paths after applying some flips governed by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The implication (3)→(4) is the core part of our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Conditions (5) and (6) assert the existence of a short winning strategy in the Flipper game and its technical variant, the Separation game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In essence, in the Separation game we restrict the moves of Flipper (now called Separator) by requiring that the flips are definable using a bounded number of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Also, we measure distances somewhat differently, so that intuitively we take into account all possible flips that Separator could have made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The implication (4)→(5) is proved by proposing a strategy for Separator and using compactness combined with (4) to argue that it leads to a victory within a bounded number of rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Finally, condition (7) is the notion of flip-wideness, whose equivalence with monadic sta- bility was proved by Dreier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' [DMST22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We prove the implication (5)→(7) by (essentially) providing a strategy for Connector in the Separator game when the class is not flip-wide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then we rely on the implication (7)→(1) from [DMST22] to close the circle of implications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' this proves the equivalence of (1)-(7) with the exception of (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We remark that (7)→(1) is the easy implica- tion of [DMST22], hence our reasoning can also serve as an alternative proof of the flip-wideness characterization given in [DMST22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' To put the Flipper game into the picture, we separately prove the implications (5)→(6)→(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The implication (5)→(6) relies on a conceptually easy, but technically not-so-trivial translation of the strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In the implication (6)→(2) we use obstructions to existential monadic stability to give a strategy for Connector in the Flipper game that enables her to endure for arbitrarily long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Algorithmic proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We also give a purely combinatorial proof of (the forward implication of) Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='3, which in particular provides a way to efficiently compute Flipper’s moves in a winning strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Formally, we show the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='5 Let C be a monadically stable class of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then for every radius r ∈ N there exist k ∈ N and a Flipper strategy flip⋆ such that the following holds: – When playing according to flip⋆ in the Flipper game of radius r on any graph G ∈ C , Flipper wins within at most k rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' – The moves of flip⋆ on an n-vertex graph G ∈ C can be computed in time OC ,r(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The main idea behind the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='5 is to rely on the result of Dreier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' that monadically stable graph classes are flip-wide [DMST22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Using the combinatorial tools devel- oped in [DMST22], we strengthen this property: we prove that the set of flips F whose applica- tion uncovers a large scattered set Y (a set of vertices that are pairwise far from each other) can be selected in a somewhat canonical way, so that knowing any 5-tuple of vertices in Y is enough to uniquely determine F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We can then use such strengthened flip-wideness to provide a winning strategy for Flipper;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' this roughly resembles the Splitter’s strategy used by Grohe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' in their proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2, which in turn relies on uniform quasi-wideness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 7 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='5, the algorithmic version of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='3, is the key to any algorithmic applica- tions of the Flipper game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In particular, it was very recently already used by Dreier, Mählmann, and Siebertz [DMS] to approach the first-order model checking problem on monadically stable graph classes and even solve it on structurally nowhere dense classes, an important subclass of monadically stable classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' On a high level, the proof of [DMS] follows the approach of Grohe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' [GKS17] on nowhere dense classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Essentially, by Gaifman’s Locality Theorem the model checking problem reduces to computing which formulas up to a certain quantifier rank q are true in the local r-neighborhoods of the input graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Here, the numbers q and r de- pend only on the input formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The set of formulas of quantifier rank q that are true in a local neighborhood is called the local q-type of the neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The computation of local q-types is done recursively in a recursion guided by the Splitter game, which hence terminates after a bounded number of steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' A naive branching into all local neighborhoods, however, leads to a too high running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This issue is solved by grouping close by elements into clusters and computing the local types of all neighborhoods that are grouped in one cluster in one recursive call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Whenever the clusters can be collected into a sparse neighborhood cover, which is the case in nowhere dense graph classes, this leads to an efficient model checking algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Dreier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' [DMS] showed that the Splitter game in the above approach can be replaced by the Flipper game and present an efficient model checking algorithm on all monadically stable classes that admit sparse (weak) neighborhood covers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Their result does not fully solve the model check- ing problem on monadically stable classes as the question whether sparse weak neighborhood covers for monadically stable classes exist remains an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Dreier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' only showed the existence of such covers for structurally nowhere dense classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We refer to [DMS] for the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The paper is split into three parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Part I is devoted to introducing the basic notions and relations between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In particular, we prove the implications (5)→(6)→(2) of and (5)→(7) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In Part II we present the model-theoretic proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' More precisely, we prove the implications (2)→(3)→(4)→(5) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Together with the implication (5)→(7) proved in Part I, the (easy) implication (7)→(1) proved in [DMST22], and the trivial implication (1)→(2), this closes the cycle of implications between the conditions (1), (2), (3), (4), (5), (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' And with the implications (5)→(6)→(2) proved in Part I, this completes the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In Part III we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Part I Prelude In this part, we introduce the basic notions of interest: monadic stability, variants of the Flipper game, flip-wideness, and prove some relations between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In Section 2 we establish notation and common definitions, and present basic model-theoretic background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Also, we discuss the notions of flips, of flip-connectivity, and of flip-wideness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In Section 3 we define three variants of the Flipper game, and prove results relating some of them to the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In particular, we define the Separation game, which is a variant of the Flip- per game in which Flipper’s moves are required to be definable, so Flipper is more constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 8 On the other hand, the balls are measured in different ways in the two games, so it is not imme- diately clear how the two games compare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' However, the implication (5)→(6) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4 is relatively easy, and is proved in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In Section 4 we relate variants of the Flipper game to other notions around monadic stability: flip-wideness and existential monadic stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In particular, we derive flip-wideness from a strategy of Separator in the Separation game (this is the implication (5)→(7) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Also we derive existential monadic stability from a strategy of Flipper in the Flipper game (this is the implication (6)→(2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 2 Preliminaries All graphs in this paper are simple and loopless but not necessarily finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For a vertex v of a graph G, we write N(v) for the (open) neighborhood of v in G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' so N(v) := {u ∈ V(G) | uv ∈ E(G)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For a set of vertices X ⊆ V(G), we write G[X] for the subgraph of G induced by X, and G − X for the subgraph of G induced by V(G) − X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For two sets X, Y ⊆ V(G) the bipartite graph semi-induced by X and Y in G, denoted G[X, Y], is the bipartite graph with parts X and Y, and edges uv for u ∈ X, v ∈ Y with uv ∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By |G| we denote the cardinality of the vertex set of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For vertices a, b ∈ V(G), an (a, b)-path is a path with ends a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Similarly, for sets A, B ⊆ V(G), an (A, B)-path is a path where one end is in A and the other end is in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We write △ for the symmetric difference of two sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Similarly, for two graphs G and G′ on the same vertex-set, we write G△G′ for the graph with vertex-set V(G) = V(G′) and edge-set E(G)△E(G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 Model theory We work with first-order logic over a fixed signature Σ that consists of (possibly infinitely many) constant symbols and of relation symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' A model is a Σ-structure, and is typically denoted M, N, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We usually do not distinguish between a model and its domain, when writing, for instance, m ∈ M or f : M → X, or X ⊆ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' A graph G is viewed as a model over the signature consisting of one binary relation denoted E, indicating adjacency between vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' If ¯x is a finite set of variables, then we write ϕ( ¯x) to denote a first-order formula ϕ with free variables contained in ¯x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We may also write ϕ( ¯x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , ¯xk) to denote a formula whose free variables are contained in ¯x1 ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ∪ ¯xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We will write x instead of {x} in case of a singleton set of variables, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ϕ(x, y) will always refer to a formula with two free variables x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We sometimes write ϕ( ¯x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ¯y) to distinguish a partition of the set of free variables of ϕ into two parts, ¯x and ¯y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' this partition plays an implicit role in some definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' If U is a set and ¯x is a set of variables, then U ¯x denotes the set of all valuations ¯a: ¯x → U of ¯x in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Such a valuation is also called an ¯x-tuple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For a formula ϕ( ¯x) and an ¯x-tuple ¯m ∈ M ¯x, we write M |= ϕ( ¯m), or M, ¯m |= ϕ( ¯x), if the valuation ¯m satisfies the formula ϕ( ¯x) in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' A quantifier-free formula is a formula that does not use quantifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' An existential formula is a formula of the form ∃y1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ∃yl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='α, where α is quantifier-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We will use the notion of an atomic type only in the context of a finite relational signature Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In this case, an atomic type with variables ¯x is a formula α( ¯x) that is a conjunction of clauses of the form R(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , xk) or ¬R(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , xk), where R ∈ Σ ∪ {=} is a relation symbol of arity k and x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , xk are variables from ¯x, such that each possible such clause or its negation occurs as a 9 conjunct in α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The atomic type of a tuple ¯a ∈ M ¯x is the unique atomic type α( ¯x) that is satisfied by ¯a in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Stability and dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' A formula ϕ( ¯x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ¯y) is stable in a class C of structures if there exists k ∈ N such that for every M ∈ C , there are no sequences ¯a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ¯ak ∈ M ¯x and ¯b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , ¯bk ∈ M ¯y such that M |= ϕ(¯ai;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ¯bj) ⇐⇒ i < j, for 1 ⩽ i, j ⩽ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We say that a class C of graphs has a stable edge relation if the formula E(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' y) is stable in C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Equivalently, C excludes some ladder as a semi-induced subgraph, where a ladder (often called also half-graph) of order k is the graph with vertices a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , ak, b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , bk and edges aibj for all 1 ⩽ i < j ⩽ k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Note that replacing < by ⩽ in the above definitions does not change them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' a1 b1 a2 b2 a3 b3 a4 b4 a5 b5 a6 b6 FIGURE 2: A ladder (half-graph) of order 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' A formula ϕ( ¯x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ¯y) is dependent, or NIP (standing for “not the independence property”) in a class C if there exists k ∈ N such that for every M ∈ C , there are no tuples ¯a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , ¯ak ∈ M ¯x and ¯bJ ∈ M ¯y for J ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , k} such that M |= ϕ(¯ai;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ¯bJ) ⇐⇒ i ∈ J, for 1 ⩽ i ⩽ k and J ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Observe that a formula which is stable is also dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' A class C is stable (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' dependent) if every formula ϕ( ¯x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ¯y) is stable (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' dependent) in C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let Σ be a signature and let �Σ be a signature extending Σ by (possibly infinitely many) unary relation symbols and constant symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' A �Σ-structure � M is a lift of a Σ-structure M if M is obtained from � M by forgetting the symbols from �Σ − Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' A class of �Σ-structures � C is a unary expansion of a class of Σ-structures C if every structure � M ∈ � C is a lift of some structure M ∈ C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' A class C of structures is monadically stable if every unary expansion � C of C is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Similarly, C is monadically dependent (or monadically NIP) if every unary expansion � C of C is dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' A class C is simply existentially monadically dependent (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' simply existentially monadically sta- ble) if every existential formula ϕ(x, y) (with two free variables) is dependent (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' stable), in every unary expansion � C of C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' A single structure M is monadically stable (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' monadically dependent) if the class {M} is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Note that a class which is monadically stable (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' monadically dependent) is stable (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' dependent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Also, a class which is simply existentially monadically stable is simply existentially monadically dependent, and has a stable edge relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We remark that even though monadic dependence and monadic stability are defined in terms of formulas ϕ( ¯x, ¯y), where ¯x and ¯y are tuples of free variables, rather than single variables, by the results of Baldwin and Shelah [BS85, Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='3, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='8], it is enough to con- sider formulas ϕ(x, y) with two free variables instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Our definitions of existential monadic stability and existential monadic dependence involves existential formulas ϕ(x, y) with two 10 free variables only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' As follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4, for graph classes monadic stability is equiva- lent to existential monadic stability, thus strengthening the result of Baldwin and Shelah, when restricted to graph classes, by additionally showing that it suffices to consider existential for- mulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2 Flips For the purpose of the following definition, it is convenient to assume that all considered graphs have a domain contained in some fixed, universe Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Atomic flips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' An atomic flip is an operation F specified by a pair (A, B) of (possibly intersect- ing) subsets of Ω, which complements the adjacency relation between the sets A and B in a given graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Formally, for a graph G, the graph obtained from G by applying the atomic flip F is the graph denoted G ⊕ F with vertex set V(G), where, for distinct vertices u, v in V(G), uv ∈ E(G ⊕ F) ⇐⇒ � uv /∈ E(G), if (u, v) ∈ (A × B) ∪ (B × A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' uv ∈ E(G), otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In the remainder of this section, it will be convenient to identify an atomic flip F with the pair (A, B) of sets defining it and to write G ⊕ (A, B) instead of G ⊕ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Note that, in our definition, we do not require A and B to be subsets of V(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Instead, we will allow A and B to be any subset of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In particular, G ⊕ (A, B) = G ⊕ (A ∩ V(G), B ∩ V(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This is useful as we often work with induced subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' As an example, for every graph G and vertex v ∈ V(G), the atomic flip Fv = ({v}, N(v)) isolates the vertex v in G, meaning that v is an isolated vertex of G ⊕ Fv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We denote by Flip(G) = {(A, B) | A, B ⊆ V(G)} the set of all the atomic flips defined by subsets of vertices of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 For every graph G and atomic flips F1, F2 we have G ⊕ F1 ⊕ F2 = G ⊕ F2 ⊕ F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Furthermore, when considering a sequence of atomic flips, we may restrict to the case where no atomic flip appears more than once in the sequence since performing twice the same atomic flip twice leaves the graph unchanged, thanks to the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2 For every graph G and atomic flip F we have G ⊕ F ⊕ F = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This justifies the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Sets of flips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' A set of flips {F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , Fk} defines an operation F that, given a graph G, results in the graph G ⊕ F := G ⊕ F1 ⊕ · · · ⊕ Fk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Note that the order in which we carry out the atomic flips does not matter, according to Ob- servation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 and that it would be useless to consider mutlisets, according Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Abusing terminology, we will often just say that the operation F is a set of flips, and write F = {F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , Fk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='3 If F and F′ are sets of flips, and F △ F′ is the symmetric difference of F and F′, then G ⊕ F ⊕ F′ = G ⊕ (F △ F′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The next observation is elementary but crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 11 Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4 Let G be a graph, let X be a subset of the vertices of G, and let F = (A, B) be an atomic flip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Suppose two vertices u, v have the same neighborhood on X in G and have the same membership in A and B (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' u ∈ A ⇐⇒ v ∈ A and u ∈ B ⇐⇒ v ∈ B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then u and v have the same neighborhood on X in G ⊕ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Finally, flips commute with vertex deletions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' As a consequence, we have Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='5 For every graph G, every subset A of V(G), and every set of flips F, we have G[A] ⊕ F = (G ⊕ F)[A].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Specifying an atomic flip by a pair of subsets of Ω will be convenient for carrying out our proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We note that we could also define an atomic flip by a single set A ⊆ Ω producing the graph G ⊕ (A, A) that is obtained by complementing all adjacencies within A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Up to a constant factor, both definitions yield the same expressive power: every atomic flip (A, B) is equivalent to the set of flips defined by {(A ∪ B, A ∪ B), (A, A), (B, B)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Flip terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For conciseness, we will use the term flip to refer to an atomic flip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Therefore, a flip may be identified with a pair (A, B) of vertex sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' F-flips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let F be a family of subsets of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then an F-flip is a set of flips of the form {F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , Fk}, where each flip Fi is a pair (A, B) with A, B ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Note that there are at most 2|F|2 different F- flips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In our context, the family F will usually be a partition of the set V(G) ⊆ Ω of vertices of some graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' An F-flip of a graph G, where F is a family of subsets of V(G), is a graph G′ obtained from G after applying an F-flip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Whenever we speak about an F-flip, it will be always clear from the context whether we mean a graph or the family of flips used to obtain it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' S-classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let G be a graph, and let S ⊆ V(G) be a finite set of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Consider the equiva- lence relation ∼S on V(G), in which two vertices a, b are equivalent if either a, b ∈ S and a = b, or a, b /∈ S and N(a) ∩ S = N(b) ∩ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' An S-class is an equivalence class of ∼S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In other words, it is a set of vertices either of the form {s} for some s ∈ S, or of the form {v ∈ V(G) − S | N(v) ∩ S = T} for some T ⊆ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The S-class of a vertex v ∈ V(G) is the unique S-class which contains v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Hence, V(G) is partitioned into S-classes, and the number of S-classes is at most |S| + 2|S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' S-flips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' An S-flip of a graph is an F-flip G′ of G, where F is the partition of V(G) into S-classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Note that there are 22O(|S|) many S-flips of a given graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Note that a direct consequence of Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4 is that S-flips preserve S-classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' More- over, S-flips satisfy the following transitivity property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='6 If G is a graph, S and T are finite subsets of V(G), G′ is an S-flip of G, and G′′ is a T-flip of G′, then G′′ is an (S ∪ T)-flip of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We first consider the case where S = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4, the S-classes of G′ are the same as the S-classes of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Thus, G′′ is an S-flip of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Assume S ̸= T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The partition into (S ∪ T)-classes is a common refinement of the partitions into S-classes and into T-classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' It follows that G′ is an (S ∪ T)-flip of G and G′′ an (S ∪ T)-flip of G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Thus, the statement follows from the case where S = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 12 Finally, the S-flips behave well with the extraction of an induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='7 Let G be a graph and S ⊆ X ⊆ V(G) be sets with S finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' If G′ is an S-flip of G, then G′[X] is an S-flip of G[X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let F be the set of flips (between S-classes of G) used to obtain G′ from G, so that G′ = G ⊕ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='5 we have G′[X] = G[X] ⊕ F′, where F′ is obtained from F by replacing each S-class C used in any flip by C ∩ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Now just note that the S-classes of G[X] are exactly the sets of the form C ∩ X, where C is an S-class of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Therefore, F′ applies flips between S-classes in G[X], so G′[X] is an S-flip of G[X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='3 Flip-wideness The following notion of flip-wideness was introduced in [DMST22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Given a graph G and a set of vertices A ⊆ V(G) we call A distance-r independent if all vertices in A are pairwise at distance greater than r in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Definition 1 (Flip-wideness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' A class of graphs C is flip-wide if for every r ∈ N there exists a function Nr : N → N and a constant sr ∈ N such that for all m ∈ N, G ∈ C , and A ⊆ V(G) with |A| ⩾ Nr(m), there exists a set F of flips with |F| ⩽ sr and B ⊆ A with |B| ⩾ m such that B is distance-r independent in G ⊕ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Flip wideness is known to be equivalent to monadic stability, with a polynomial time algo- rithmic version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='8 ([DMST22]) Let C be a class of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then, the following are equivalent: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' C is monadically stable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' C is flip-wide;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' for every r ∈ N there exists a function Nr : N → N and a constant sr ∈ N such that for all m ∈ N, G ∈ C with n vertices, and A ⊆ V(G) with |A| ⩾ Nr(m), we can compute in time fC (r) · n3 (for some function fC ) a set F of flips with |F| ⩽ sR and a set B ⊆ A with |B| ⩾ m such that B is distance-r independent in G ⊕ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4 Flip-connectivity Let G be a graph and P a partition of V(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We will mostly be interested in the case when P is the partition of V(G) into S-classes, for some finite S ⊆ V(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In this case, in the notation below we will write S instead of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The following notion is central in our proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We say that vertices a and b of G are r-separated over P, denoted by5 a r |⌣ P b, if there exists a P-flip H of G such that distH(a, b) > r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We set Br P(a) := {b | b is not r-separated from a over P}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 5The symbol |⌣ denotes forking independence in stable theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Its use here is justified by the relationship of r-separation and forking independence in monadically stable theories, see next footnote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 13 We define a more general notion below, where instead of single vertices a and b we may have sets of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We also justify the use of the notation Br P(a), by observing that this is indeed a ball in some metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For any r ∈ N and any graph G, finite partition P of V(G), and sets A, B ∈ V(G), we write A r |⌣ P B if there exists a P-flip H of G such that H has no (A, B)-path of length at most r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' If A |⌣ r P B we say that A and B are r-separated over P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Note that when A ∩ B ̸= ∅, A and B are not r-separated over any partition P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Note also that in the case when P is the partition into S-classes, we can assume that every vertex in S is isolated in H, because this can be achieved through an S-flip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' If A consists of a single vertex a and B of a single vertex b, then we write a |⌣ r P b for A |⌣ r P B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We use similar notation for a |⌣ r P B and A |⌣ r P b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Note that A |⌣ r P B is a stronger condition than a |⌣ r P b for all a ∈ A and b ∈ B, since we require that the same set P and the same P-flip H are used for all a ∈ A and b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We write ̸ |⌣ r P to denote the negation of the relation |⌣ r P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' If A ̸ |⌣ r P B we say that A and B are r-connected over P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For two vertices u, v, denote by distP(u, v) the smallest number r ∈ N such that u and v are r-connected over P, or +∞ if no such number exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This can be equivalently described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For u, v ∈ V(G), a P-flip-path from u to v in G is a collection consisting of one (u, v)- path in H for each P-flip H of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Note that a P-flip-path might not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The length of a P-flip path is the supremum of the lengths of its paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then, equivalently, distP(u, v) is the infimum of the lengths of all P-flip-paths (where if there is no P-flip-path then distP(u, v) = +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Now we prove that distP(·, ·) is a metric, where we allow metrics to take on the value of +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Thus, in particular, the relation distP(·, ·) < +∞ is an equivalence relation on V(G): the set of all pairs (u, v) ∈ V(G)2 such that distP(u, v) < +∞ is transitive, reflexive, and symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='9 For any graph G and finite partition P of V(G), distP(·, ·) is a metric on V(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' First of all, notice that for any different u, v ∈ V(G) it holds that u |⌣ 1 P v, because we have distH(u, v) > 1 either for H = G or for H being the complement of G, which is a P-flip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' So in general distP(a, b) = 0 if and only if a = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' From the definition we also have that distP(·, ·) is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Finally, a P-flip-path from a to b can be naturally composed with a P-flip-path from b to c in a path-by-path manner, yielding a P-flip-path from a to c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This proves the triangle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Now that we know distP(·, ·) is a metric, it is sensible to define an analog of the “ball of radius r around a vertex”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This is exactly the notion Br P(v) defined earlier, as Br P(v) = {w ∈ V(G) | distP(v, w) ⩽ r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In case P is the partition into S-classes, for some finite S ⊆ V(G), we write A |⌣ r S B and Br S(v) to denote A |⌣ r P B and Br P(v), respectively 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This notion behaves well under taking subsets of 6We comment on the relationship between r-separation and forking independence in monadically stable graphs, thus justifying the use of the symbol |⌣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' As a subset of authors will show in subsequent work, for a monadically stable graph G, its elementary superstructure H, and two vertices a, b ∈ V(H), a and b are forking independent over H if and only if for every r ∈ N there is a finite S ⊆ G such that a |⌣ r S b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This is related to a result of Ivanov, characterising forking independence in nowhere dense (or superflat) graphs [Iva93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 14 S as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='10 For any r ∈ N ∪ {+∞}, graph G, and finite sets T ⊆ S ⊆ V(G), if a is r-separated from b over T, then a is also r-separated from b over S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Since a |⌣ r T b, there exists a T-flip H of G such that H contains no (a, b)-path of length at most r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' As T ⊆ S, H is also an S-flip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Hence a |⌣ r S b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 3 Variants of the Flipper game We define three variants of the Flipper game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Flipper game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Fix a radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The Flipper game of radius r is played by two players, Flipper and Connector, on a graph G as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' At the beginning, set G0 := G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In the ith round, for i > 0, the game proceeds as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' – If |Gi−1| = 1 then Flipper wins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' – Connector chooses a vertex v in Gi−1 and we set Gloc i−1 to be the subgraph of Gi−1 induced by the ball Br(v) of radius r around v in Gi−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' – Flipper chooses an atomic flip F and applies it to produce Gi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Gi = Gloc i−1 ⊕ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' A k-round play of the Flipper game can be represented as a sequence of graphs G0, Gloc 0 , G1, Gloc 1 , G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , Gloc k−1, Gk, where G0 = G and Gloc i is the subgraph of Gi induced by the ball of radius r around some v ∈ V(G) and Gi+1 is a (X, Y)-flip of Gloc i for some X, Y ⊆ V(Gloc i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Note that V(Gi) ⊇ V(Gj) whenever i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' One can consider an extended variant of the Flipper game where Flipper in the ith move applies a set F of flips to Gloc i−1 to obtain Gi, where |F| ⩽ g(i) for some function g: N → N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This does not change the game significantly – if Flipper wins this extended game in m rounds, then Flipper wins the standard Flipper game in ∑m i=1 g(i) rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Pseudo-Flipper game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We now introduce a variant of the Flipper game that will play an aux- iliary role in our proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The main difference is that, while in the usual Flipper game, the entire graph is replaced by the subgraph induced by a ball of radius r, after each move of Connec- tor, and the balls are evaluated in smaller and smaller graphs, in the Pseudo-Flipper game, we are keeping the original graph G, and keeping track of a set of vertices (called the arena) from which Connector can pick the center of the next ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Moreover, the ball is measured with re- spect to the distance induced by |⌣ r P in the original graph G, where P is a partition picked by Pseudo-Flipper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The precise definition follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Fix a radius r ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The Pseudo-Flipper Game of radius r is played by two players, Pseudo- Flipper and Connector, as follows on a graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let A0 = V(G) and F0 = {V(G)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , the kth round proceeds as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' – If |Ak−1| = 1 then Pseudo-Flipper wins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' – Otherwise, Connector picks ck ∈ Ak−1 and we set Ak := Ak−1 − {w | w r |⌣ Fk−1 ck} = Ak−1 ∩ Br Fi−1(ck) 15 ( |⌣ r Fk−1 and Br Fi−1(ck) are evaluated in the original graph G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' – Then Pseudo-Flipper chooses a partition Fk, obtained by taking Fk−1 and splitting one part F ∈ Fk−1 into two non-empty, disjoint sets F1, F2 with F1 ∪ F2 = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Here, the sets Ak are called arenas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Similarly to the case of the Flipper game, it might be convenient to consider the extended variant of Pseudo-Flipper game in which Pseudo-Flipper refines Fi−1 to obtain Fi by splitting g(i) sets in Fi−1 for some function g : N → N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Again, it doesn’t change the game significantly, as whenever Pseudo-Flipper wins the extended game in m rounds, Pseudo-Flipper also wins the standard Pseudo-Flipper game in ∑m i=1 g(i) rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Separation game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' A variant of the Pseudo-Flipper game in which the paritions are partitions into S-classes, is called the Separation game, and is defined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Fix a radius r ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The Separation Game of radius r is played by two players, Separator and Connector, on a graph G, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let A0 = V(G) and S0 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , the kth round proceeds as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' – If |Ak−1| = 1, then Separator wins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' – Otherwise, Connector picks ck ∈ Ak−1 and we set Ak := Ak−1 − � w ��� w r |⌣ Sk−1 ck � (where separation is evaluated in the graph G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' – Then Separator picks sk ∈ V(G) and we set Sk := Sk−1 ∪ {sk}, and proceed to the next round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Again as in the case of Flipper and Pseudo-Flipper games, we may allow Separator to add g(i) vertices to Si−1 in the ith round, where g: N → N is some fixed function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Again, if Sep- arator can win this new game in m rounds, then Separator can also win the original game in ∑m i=1 g(i) rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 Relating the game variants We now prove easy relations between the variants of the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' First, we prove that that Pseudo- Flipper can use Separator’s strategy to win the Pseudo-Flipper game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then we prove that Flip- per can use Pseudo-Flipper’s strategy to win the Flipper game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 For every radius r and graph G the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' If Separator wins the Separation game of radius r on G in at most k rounds, then Pseudo-Flipper wins the Pseudo-Flipper game of radius r on G in at most 2k+1 rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Consider the extended version of the Pseudo-Flipper game in which in ith move Pseudo- Flipper can refine Fi−1 to obtain Fi by splitting at most 2i sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' If we show that Pseudo-Flipper can win this game in k rounds, so he can win the standard game in ∑k i=1 2i ⩽ 2k+1 rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Consider the following strategy of Pseudo-Flipper, which will be just simulating Separator’s strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' When Connector makes her first move, Pseudo-Flipper asks Separator what he would 16 play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Separator answers that he would play a vertex v, thus obtaining S1 = {v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Pseudo-Flipper then splits F0 = {V(G)} into three S1-classes: {v}, vertices adjacent to v, and the remaining vertices (non-adjacent to v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then the game continues in this way – after ith move of Connector, Pseudo-Flipper asks Separator what he would play, and then splits some sets of Fi−1 to Fi, so that Fi is a partition into Si-classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' He can do it as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Assume by induction that Fi−1 is a partition into Si−1-classes, so in particular |Fi−1| ⩽ 2i−1 + (i − 1) ⩽ 2i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Assume also that in ith move Separator picks a vertex u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then, Pseudo-Flipper can split every set of Fi−1 so that now Fi is a partition into Si−1 ∪ {u}-classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This can be performed within at most |Fi−1| + 1 splits: this entails splitting each Si−1-class into at most two parts, and then distinguishing {u} as a separate class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Since |Fi−1| + 1 ⩽ 2i, that finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We now show that Flipper can use a winning strategy of Pseudo-Flipper from the game with larger radius and at the cost of playing a few extra rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2 There exists a function f : N → N such that for every radius r and every graph G the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' If Pseudo-Flipper wins the Pseudo-Flipper game of radius 2r on G in at most k rounds, then Flipper wins the Flipper game of radius r on G in at most f (k) rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Note that the converse direction, allowing to translate a winning strategy of Flipper in the Flipper game, to a winning strategy in the Pseudo-Flipper game, is not immediately clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The reason is that in the Flipper game, Flipper has additional power since the balls are measured in the current graph Gi, and are therefore potentially smaller than if they were measured in (some flip) of the original graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In fact, we do not know of a direct proof of the converse implication, but the equivalence of the two games ultimately follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Before proving Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2, we will state a simple lemma which will be used in its proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let G be a graph and H an F-flip of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' It is easy to check that any two F-flips H, H′ of G are F-flips of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' More generally, since the sets used in atomic flips do not have to be subsets of the graph we flip on, we have the following: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='3 Let G be a graph and H, H′ two F-flips of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For every B ⊆ V(G) we have H′[B] is a F-flip of H[B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We now proceed with the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We will show how to use a winning strategy of Pseudo-Flipper in the Pseudo- Flipper game of radius 2r played on G to win the Flipper game of radius r played on G for suitably chosen f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We will play the two games on G in parallel, in such a way that one move in the Pseudo-Flipper game corresponds to several moves in the Flipper game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We remark that in the proof we will play the extended Flipper game, meaning that not only we will play several rounds in the Flipper game for each round of the Pseudo-Flipper game, but each round in the Flipper game actually consists of flipping between several pairs of subsets of the graph we currently play on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' From the proof it will follow that in kth round of the Flipper game we play an F-flip of the current graph such that |F| is upper bounded by a number depending only on k, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For j ∈ N0 we denote by R(j) the maximum possible number of different F-flips of any graph G and any F ⊆ P(V(G)), provided that |F| ⩽ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For the sake of readability, we will denote by Gi,b the graph in the Flipper game after ∑i j=1 R(j − 1) + b moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Note that with this notation Gi+1,0 is the graph obtained from Gi,0 after R(i) moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The strategy of Flipper will be such that the following invariant is maintained: If Ai is the arena in the Pseudo-Flipper game after i rounds, then Gi,0 is an induced subgraph of G[Ai].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 17 Note that at the beginning of the game this condition is satisfied, as G0,0 = G and A0 = V(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Showing that this invariant can be maintained proves the lemma, as after at most k − 1 rounds in the Pseudo-Flipper game we have |Ak| = 1, which by our invariant implies |Gk,0| = 1, meaning that Flipper wins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For any i we now describe Flipper’s strategy to obtain Gi+1,0 from Gi,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let Ai be the arena of the Pseudo-Flipper game after i rounds, Fi the partition chosen by Pseudo-Flipper in the ith move and Gi,0 an induced subgraph of G[Ai].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let p := R(i) be the number of all Fi-flips of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Flipper’s strategy will be chosen in such a way that after p rounds of play starting from Gi,0, irrespective of Connector’s moves, we arrive at graph Gi,p = Gi+1,0 and a vertex c ∈ Ai such that Gi+1,0 is an induced subgraph of G[Ai ∩ B2r Fi(c)] = G[Ai+1], thus satisfying our invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Intuitively, Flipper’s strategy is to play every possible F-flip of Gi,0 (with Connector making moves in between) and then in the final round to flip back to (an induced subgraph of) Gi,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Formally, we get the following: ▷ Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4 Let H0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , Hp be a sequence of graphs such that H0 = Hp = G and H1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , Hp−1 are all possible nontrivial Fi-flips of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' There is a strategy for Flipper in the p-round Flipper game with radius r starting from Gi,0 such that irrespective of replies of Connector, if Gi,j is the graph after the jth move, then Gi,j is an induced subgraph of Hj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This clearly holds for j = 0, since Gi,0 is an induced subgraph of G[Ai+1] which is an induced subgraph of G = H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We now argue that assuming that Gi,j is an induced subgraph of Hj, Flipper can guarantee that Gi,j+1 is an induced subgraph of Hj+1: First, going from Gi,j to Gloc i,j (Connector’s move) is done by taking an induced subraph of Gi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Therefore, Gloc i,j = Gi,j[B] = Hj[B] for some B ⊆ V(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='3 we have that Hj+1[B] is an Fi-flip of Hj[B], and so Flipper can set Gi,j+1 to be Hj+1[B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ◁ Now consider p rounds of play of the Flipper game starting from Gi,0 in which Flipper plays according to the strategy given above, ending with graph Gi+1,0 = Gi,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' It remains to define the choice of vertex c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We set c to be the last vertex played by Connector in the p round play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' That is, c is the vertex played by Connector in Gi,p−1 to obtain Gloc i,p−1 = Br(c), where the ball of radius r is taken in Gi,p−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Since Gi+1,0 = Gi,p is obtained by performing flips on Gloc i,p−1, the two graphs have the same vertex sets and we have c ∈ V(Gi+1,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We now prove that Gi+1,0 is an induced subgraph of G[Ai+1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Since Gi+1,0 is an induced subgraph of G, it suffices to show that V(Gi+1,0) ⊆ Ai+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Because Ai+1 = Ai ∩ B2r Fi(c) and V(Gi+1,0) ⊆ Ai, we need to show that V(Gi+1,0) ⊆ B2r Fi(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We will prove this by contraposition – we will show that for any w ̸∈ B2r Fi(c) we have w ̸∈ V(Gi+1,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' From the definition, w ̸∈ B2r Fi(c) means there exists an Fi-flip H of G such that distH(c, w) > 2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We know that H has to be one of H0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , Hp−1 from Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4, say Hj, meaning that distHj(c, w) > 2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' From Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4 we have that Gi,j is an induced subgraph of Hj, and so in Gi,j the distance between c and w is also more than 2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then, if v is the vertex chosen by Connector in Gi,j to obtain Gloc i,j = Br(v) (here the ball around v is taken in Gi,j), it cannot be the case that both c and w are in Gloc i,j , because Gloc i,j has radius at most r and so one could join c with w by a path of length at most 2r going from c to v and from there to w, contradicting that distGi,j(c, w) > 2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This means that at most one of c and w can be in V(Gloc i,j ) and consequently in V(Gi,j+1) (because the two graphs have the same vertex set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' But since we know that c ∈ V(Gi+1,0) and V(Gi+1,0) ⊆ V(Gi,j+1), we have 18 c ∈ V(Gi,j+1) and therefore w ̸∈ V(Gi,j+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Since V(Gi+1,0) ⊆ V(Gi,j+1) for j ⩽ p, we have w ̸∈ V(Gi+1,0), which finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 4 Relations to flip-wideness and existential monadic stability In this section we prove that the existence of short winning strategies in variants of the Flipper game implies two properties that are known to be equivalent to monadic stability: flip-wideness and existential monadic stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 From Pseudo-Flipper game to flip-wideness This subsection is devoted to proving the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 Let C be a class of graphs such that for every r there exists k such that Pseudo-Flipper wins the Pseudo-Flipper game of radius r on any G ∈ C , in at most k rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then C is flip-wide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Together with Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1, this yields the following corollary, proving the implication (5)→(7) in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2 Let C be a class of graphs such that for every r there exists k such that Separator wins the Separation game of radius r on any G ∈ C , in at most k rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then C is flip-wide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let r ∈ N be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let k be a number of rounds such that Pseudo- Flipper wins the Pseudo-Flipper game of radius r on any G ∈ C in at most k rounds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' such k exists by our assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let c be the maximal possible number of different F-flips on any graph G with respect to any F with |F| ⩽ k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' it is easily seen that such number exists (the upper bound does not depend on any particular G and F, only on k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For m ∈ N denote by R(m) the least number N such that any coloring of the edges of a clique with N vertices using c colors yields a monochromatic clique of size m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Such a number exists by Ramsey’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Set sr := k2 and for m ∈ N set Nr(m) := R(m)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let m ∈ N and let G ∈ C and A ⊆ V(G) with |A| > Nr(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' From the definition of the Pseudo-Flipper game of radius r follows that whenever Pseudo-Flipper can win such game in k rounds, then he can also win in k rounds if we decide that the initial arena A0 is a subset of V(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Indeed, in every round we intersect the previous arena with something that depends only on the partition played by Pseudo-Flipper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Therefore, set A0 := A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We will extract a partition F and an F-flip H of G with the desired properties from a play of the Pseudo-Flipper game played on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We define a strategy for Connector in the Pseudo-Flipper game played on G as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let Fi denote the partition of V(G) resulting from Pseudo-Flipper’s choices after i rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For i > 0, in the ith round, if there is a vertex v ∈ Ai−1 such that |Br Fi−1(v) ∩ Ai−1| > R(m)k−i then Connector picks v as ci, otherwise Connector picks any vertex in Ai−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In other words, Connector tries to make sure that |Ai| > R(m)k−i in ith round whenever possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Consider now a play of the Pseudo-Flipper game played on G in which Connector plays according to the strategy described above and Pseudo-Flipper plays according to an optimal strategy which leads to a win in at most k rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We then get the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ▷ Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='3 There exists i < k such thatAi−1 ⩾ R(m)k−i+1 and |Br Fi−1(v) ∩ Ai−1| ⩽ R(m)k−i for each v ∈ Ai−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Since Pseudo-Flipper wins in k rounds, there has to exist i < k such that |Br Fi−1(v) ∩ Ai−1| ⩽ R(m)k−i for each v ∈ Ai−1 (otherwise after k − 1 rounds we would have 19 that |Ak−1| > R(m) ⩾ 1, which would contradict Pseudo-Flipper winning after k rounds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Consider the smallest i with this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This means that no matter which vertex v ∈ Ai−1 Connector plays, the arena Ai has size less than R(m)k−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Since i was the smallest such number, we have that Ai−1 ⩾ R(m)k−i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ◁ Let i be the number from Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We greedily construct a subset A′ of Ai−1 by repeatedly picking a vertex v in Ai−1 and removing Br Fi−1(v) ∩ Ai−1 from Ai−1 for as long as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Be- cause of the bounds given by Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='3, set A′ will have size at least R(m), and by construction all vertices in A′ will be pairwise r-disconnected over F in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Assign to each unordered pair u, v of distinct vertices in A′ a color which represents an F- flip H of G such that distH(u, v) > r (if there is more than one such F-flip, pick one arbitrarily).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This way we assign at most c colors to pairs of vertices from a set of size at least R(m), and therefore by the definition of R(m) there exists a subset A′′ of A′ with |A′′| ⩾ m such that each pair u, v ∈ A′′ is assigned the same F-flip H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Thus we have distH(u, v) > r for each u, v ∈ A′′, and so F and H have the properties required by the definition of flip-wideness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This means that C is flip-wide, which finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2 From Flipper game to existential monadic stability In this subsection we will show that a winning strategy for Flipper in the Flipper game implies existential monadic stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Since existential monadic stability implies stability of the edge relation, and existential monadic dependence, this will prove the implication (6)→(2) in Theo- rem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For simplicity, we will concentrate on the case when we can define an infinite ladder in a single graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We will say that a formula ϕ(x, y) defines an infinite ladder in a structure M if there is a sequence (ai, bi)i∈N of pairs of elements of M such that for every i, j ∈ N M |= ϕ(ai, bj) ⇐⇒ i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4 Let M be a graph and let � M be a monadic lift of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let ϕ(x, y) be an existential formula that defines an infinite ladder in � M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then there exists r ∈ N such that Connector can play infinitely many rounds in the Flipper game of radius r without losing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The statement of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4 can be restated for the case when we can define arbitrarily long ladders in graphs from a given class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='5 (Finitary version of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4) Let C be a class of graphs and let ϕ(x, y) be an existential formula such that for every k ∈ N there exists a graph Mk ∈ C and a monadic lift � Mk in which ϕ(x, y) defines a ladder of length at least k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then there exists r ∈ N such that for every ℓ ∈ N there is a graph Nℓ ∈ C with the following property: in the Flipper game of radius r with the initial current graph Nℓ Connector can play at least ℓ rounds without losing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Of course Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='5 gives us implication (6)→(2) from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For the rest of this section we will concentrate on proving Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' It can be easily observed that the proof can be restated for Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let M be a monadic lift of a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For an integer r, we say that a set A of vertices of M is r-close if the vertices of A are pairwise at distance at most r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In particular, if (ai, bi)i∈N are the vertices of a ladder defined by some formula, we say that this ladder is r-close if the set {ai | i ∈ N} ∪ {bi | i ∈ N} is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 20 A useful tool for the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4 will be an easy corollary from Gaifman’s locality theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The theorem was originally proven in [Gai82], but we will use a corollary of it, similar to the one from [BDG+22, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='6 (Corollary of Gaifman’s locality theorem) Let ϕ(x, y) be an FO formula in the vocab- ulary of graphs with a number of additional unary predicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then there are numbers t, s ∈ N with t depending only on the quantifier rank of ϕ such that for every graph G with a number of additional unary predicates, G can be vertex-colored using s colors in such a way that for any two vertices u, v ∈ V(G) at distance more than t, whether or not ϕ(u, v) holds depends only on the color of u and the color of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Using Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='6 we can prove that whenever ϕ defines an infinite ladder, then it also defines an infinite d-close ladder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='7 Let M be a monadic lift of a graph and let ϕ(x, y) be a formula which defines an infinite ladder in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then, there is an integer d depending only on the quantifier rank of ϕ(x, y) such that ϕ(x, y) defines in M an infinite d-close ladder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Observe, that by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='6, there exists a constant t depending only on the quantifier rank of ϕ and a vertex-coloring of M with s colors with the following property: if a, b, a′, b′ are four vertices of M such that dist(a, b), dist(a′, b′) > t, and colors of a and a′ are equal, and colors of b and b′ are equal, then M |= ϕ(a, b) ⇐⇒ M |= ϕ(a′, b′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Take an infinite ladder (ai, bi)i∈N defined by ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let us assign to every pair of natural num- bers (i, j) with i < j with one of three colors which correspond to three possible scenarios (if more than one scenario holds, we pick an arbitrary one): – dist(ai, bj) ⩽ t, – dist(bi, aj) ⩽ t, – dist(ai, bj) > t and dist(bi, aj) > t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By Ramsey’s theorem, there exists an infinite subladder (aij, bij)j∈N defined by ϕ such that every pair (ij, ik) for j < k has the same color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' It is straightforward to see that in the first two cases the vertices {aij | j ⩾ 2} ∪ {bij | j ⩾ 2} are pairwise at distance at most 3t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' It remains to deal with the last case – we will show that, in fact, it cannot hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Color the vertices of the graph into a finite number of colors according to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By the pigeonhole principle, we can assume that all aij have been assigned the same color and, separately, all bij have been assigned the same color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' However, that means M |= ϕ(aij, bik) ⇐⇒ M |= ϕ(aik, bij) for any j ̸= k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This is clearly a contradiction, so we conclude it is enough to take d := 3t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' To prove Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4, we will present a strategy for Connector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We start with an infinite ladder defined in M by an existential formula ϕ(x, y) in prenex normal form of quantifier rank at most q (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=', ϕ(x, y) ≡ ∃¯z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='α(x, y, ¯z) where |¯z| ⩽ q and α is a quantifier-free formula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The strategy will maintain the following invariant: after each round there is an existential formula ϕ′(x, y) which defines an infinite ladder in some monadic lift of the current graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The proof will follow from two lemmas, corresponding to the moves of Flipper and Connector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 21 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='8 Let M be a graph and let � M be a monadic lift of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let ϕ(x, y) be an existential formula of quantifier rank q in prenex normal form which defines an infinite ladder in � M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let N be a flip of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then there exists a monadic lift �N of N and a formula ψ(x, y) of quantifier rank q in prenex normal which defines an infinite ladder in �N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let (ai, bi)i∈N be an infinite ladder defined by ϕ(x, y) in � M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We define �N by adding the same unary predicates as in � M and two additional unary predicates that mark the sets that were flipped in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We also define ψ(x, y) by changing the atomic check E(u, v) in ϕ(x, y) to a more complicated quantifier-free formula verifying if there exists an edge between u and v in M and whether u and v were included in the flipped sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Of course, ψ(x, y) has the same quantifier rank as ϕ(x, y) and is in prenex normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Moreover, for every i, j ∈ N, � M |= ϕ(ai, bj) ⇐⇒ �N |= ψ(ai, bj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='9 Let M be a graph and let � M be a monadic lift of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let ϕ(x, y) be an existential formula of quantifier rank q in prenex normal form which defines an infinite ladder in � M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' There exists an integer r depending only on q, a formula ψ(x, y) of quantifier rank at most q in prenex normal form, a monadic lift � M′ of M, and an element m ∈ M such that ψ(x, y) defines an infinite ladder in � M′[Br(m)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='7 we can assume that ϕ(x, y) defines a d-close ladder (ai, bi)i∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' As ϕ(x, y) ≡ ∃¯z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='α(x, y, ¯z), for every i < j we can find a tuple ¯cij such that α(ai, bj, ¯cij) holds in � M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For every tuple ¯cij we define its profile as a function πij : ¯z → [q] ∪ {∞}, where πij(z) for a variable z ∈ ¯z is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let cij z be the element of the tuple ¯cij corresponding to the variable z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For ℓ ∈ [q], we set πij(z) = ℓ if the distance between ai and cij z is at least 10(ℓ − 1)d and at most 10ℓd − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' However, if the distance between ai and cij z is at least 10qd, we set πij(z) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By a standard Ramsey argument, we can assume that for every i < j all tuples (ai, bj, ¯cij) have the same atomic type τ and all ¯cij have the same profile function π (possibly by going to an infinite subladder of (ai, bi)i∈N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' If π does not assign ∞ to any coordinate, then all ¯cij are in the ball of radius r := 10qd + d around a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Therefore, after restricting M to the ball of radius r around a1 we still have for all i, j ∈ N that � M[Br(a1)] |= ϕ(ai, bj) ⇐⇒ i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' (Note that it is important here that ϕ is an existential formula;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' if not for that assumption, ϕ(ai, bj) could have become true in � M[Br(a1)] for some i ⩾ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=') Now assume that π does assign ∞ to some coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Therefore, by the pigeonhole prin- ciple, there exists s ∈ [q] such that no variable is assigned s by π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We split the variables z ∈ ¯z into two parts – these where π(z) < s (call these variables close), and those where π(z) > s (call those far).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Observe now that if i < j and z is close, then dist(ai, cij z ) < 10(s − 1)d by definition, and thus dist(a1, cij z ) < (10s − 9)d by triangle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' On the other hand, if z is far, then dist(ai, cij z ) ⩾ 10sd, and therefore dist(a1, cij z ) ⩾ (10s − 1)d, again by triangle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In par- ticular, the atomic type τ specifies that there are no edges between the vertices assigned to close 22 variables and the vertices assigned to far variables: if z is close and z′ is far, then dist(cij z , cij z′) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let us create � M′ by adding to � M one more unary predicate U satisfied for the vertices that are at distance at most (10s − 9)d from a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By construction, every vertex of the ladder is in U, and for every pair i < j and every variable z ∈ ¯z, the vertex cij z is in U if and only if z is close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let τ′ be the atomic type obtained from τ by additionally specifying which variables should satisfy U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Consider also ϕ′(x, y) ≡ ∃¯z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='τ′(x, y, ¯z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' It is immediate that for any i, j ∈ N, � M′ |= ϕ′(ai, bj) ⇐⇒ i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Now, simplify ϕ′ to ϕ′′ by removing the quantifiers that correspond to far variables (of course, we also need to simplify τ′ to τ′′ by removing the same variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' It is again obvious that � M′ |= ϕ′′(ai, bj) for i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We will show that � M′ ̸|= ϕ′′(ai, bj) for i ⩾ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Assume by contradiction that for some i ⩾ j we have a tuple ¯c such that � M′ |= τ′′(ai, bj, ¯c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By the construction of τ′′, the vertices ai and bj and each element of ¯c must satisfy U and thus each of them is at distance at most (10s − 9)d from a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Consider now extending this tuple to ¯c′ by adding the vertices of ¯c12 corresponding to the far variables of ¯z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' recall that these vertices are at distance at least (10s − 1)d from a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Clearly, � M′ |= τ′(ai, bj, ¯c′), as the vertices which we used for extending ¯c do not neighbor any vertex from ¯c, ai or bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This is a contradiction to � M′ ̸|= ϕ′(ai, bj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Therefore, again we have � M′ |= ϕ′′(ai, bj) ⇐⇒ i < j, and by using the same argument as previously we have � M′[Br(a1)] |= ϕ′′(ai, bj) ⇐⇒ i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='8 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='9 we can easily prove Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Rewrite formula ϕ(x, y) to a formula ϕ′(x, y) in prenex normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Denote its quantifier rank by q and take r as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Assume that we consider the Flipper game of radius r on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We call a graph A winning if there exists a monadic lift �A of A and a formula ψ(x, y) of quan- tifier rank at most q in prenex normal form which defines an infinite ladder in �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Obviously, if a graph is winning, then it cannot be a single vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Observe that by our assumption, the initial current graph M is winning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='8, if Flipper does a flip on a winning current graph, the resulting current graph is also winning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Finally, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='9, for every winning current graph there exists a move of Connector such that the resulting current graph is also winning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Therefore, if Connector always picks such moves, she plays infinitely many rounds without losing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Part II Model-theoretic proof In this part we prove the implications (2)→(3)→(4)→(5) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4 using elementary tech- niques from model theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 23 In Section 5 we give additional model-theoretic preliminaries: we discuss models, theories, compactness, the Tarski-Vaught test, definability of types, a variant of Morley sequences, and some basic lemmas about ladders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In Section 6 we introduce pattern-free classes, and prove several simple facts about them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In particular, we prove that every existentially monadically dependent class is pattern-free, proving the implication (2)→(3) in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Section 7 presents the main technical step, the implication (3)→(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In a nutshell, from pattern-freeness and edge-stability we derive a model-theoretic property, a variant of definability of types, which allows us to control elements in elementary extensions through finite sets in the ground model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In Section 8, we use this definability property together with compactness of first-order logic to give a strategy for Separator that ensures victory in the Separation game in a bounded num- ber of rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This is the implication (4)→(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We start with introducing the necessary tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 5 Additional model-theoretic preliminaries A theory T (over Σ) is a set of Σ-sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' A model of a theory T is a model M such that M |= ϕ for all ϕ ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' When a theory has a model, it is said to be consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The theory of a class of Σ-structures C is the set of all Σ-sentences ϕ such that M |= ϕ for all M ∈ C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The elementary closure C of C is the set of all models M of the theory of C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Thus C ⊆ C , and C and C have equal theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We will use compactness for first-order logic and the Tarski-Vaught test, recalled below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 (Compactness) A theory T is consistent if and only if every finite subset T′ of T is consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let M and N be two structures with M ⊆ N, that is, the domain of M is contained in the domain of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then N is an elementary extension of M, written M ≺ N, if for every formula ϕ( ¯x) (without parameters) and tuple ¯m ∈ M ¯x, the following equivalence holds: M |= ϕ( ¯m) ⇐⇒ N |= ϕ( ¯m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We also say that M is an elementary substructure of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In other words, M is an elementary substructure of N if M is an induced substructure of N, where we imagine that M and N are each equipped with every relation Rϕ of arity k (for k ∈ N) that is defined by any fixed first- order formula ϕ(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In this intuition, formulas of arity 0 correspond to Boolean flags, with the same valuation for both M and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2 (Tarski-Vaught Test) The following conditions are equivalent for any structures M and N with M ⊆ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' – The structure N is an elementary extension of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' – For every formula ϕ(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ¯x) and tuple ¯m ∈ M ¯x, if N |= ϕ(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ¯m) holds for some n ∈ N, then N |= ϕ(n′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ¯m) holds for some n′ ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Fix a model M over a signature Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' A Σ-formula ϕ( ¯x) with parameters from a set A ⊆ M is a formula ϕ( ¯x) over the signature Σ ⊎ A, where the elements of A are treated as constant symbols (which are interpreted by themselves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' If ϕ( ¯x) is a formula (with or without parameters) and U ⊆ M, then by ϕ(U) we denote the set of all ¯x-tuples ¯u ∈ U ¯x such that M |= ϕ( ¯u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Now let 24 A, B ⊆ M be sets, and let ϕ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' y) a formula (here x and y are single variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' A ϕ-type of A over B is an equivalence class of the relation ∼ on A such that for a, a′ ∈ A we have a ∼ a′ if and only if ϕ(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' B) = ϕ(a′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' B), that is, M |= ϕ(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' b) ⇐⇒ ϕ(a′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' b) for all b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We denote the set of ϕ-types of A over B by Typesϕ(A/B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The following result is a fundamental fact about stable formulas (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' [Pil96, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2 (i)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='3 (Definability of types) Let M ≺ N be two models and ϕ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' y) be a formula that is stable in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For every n ∈ N there is some formula ψ(x) with parameters from M, which is a positive boolean combination of formulas of the form ϕ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' m) for m ∈ M, such that the following conditions are equivalent for every a ∈ M: – N |= ϕ(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' a) holds, – M |= ψ(a) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The following lemma is reminiscent of the classic notion of Morley sequences from model theory (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' [Pil96, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4 Let M ≺ N be graphs, let ¯n ∈ N ¯y, and let A ⊆ M be a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' There is an infinite sequence ¯b0, ¯b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ∈ M ¯y such that – for each i ∈ N, ¯bi has the same atomic type as ¯n over A ∪ ¯b0 ∪ · · · ∪ ¯bi−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' and – the atomic types of the tuples ¯bi¯bj are the same for all i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' If moreover N has a stable edge relation, one may choose the sequence (¯bi)i∈N so that the atomic types of ¯bi¯bj are the same for all i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We construct the sequence ¯b0, ¯b1, · · · ∈ M ¯y satisfying the first condition by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let i ∈ N and assume that the elements ¯b0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , ¯bi−1 have already been constructed and follow the first condition (this assumption is vacuous for i = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let α( ¯y) be the conjunction of all (finitely many) formulas in the atomic type of ¯n over A ∪ ¯b0 ∪ · · · ∪ ¯bi−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Since N |= α( ¯n), and A ∪ ¯b0 ∪ · · · ∪ ¯bi−1 ⊆ M, and M ≺ N, it follows that there exists a tuple in M ¯y satisfying α( ¯y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' (Formally, let ¯u be a tuple enumerating the elements of A ∪ ¯b0 ∪ · · · ∪ ¯bi−1, and ϕ( ¯y, ¯z) be the formula without parameters for which ϕ( ¯y, ¯u) = α( ¯y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We apply the definition of an elementary extension to the formula ∃ ¯y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='ϕ( ¯y, ¯u) and infer that the formula holds in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Denote by ¯bi ∈ M ¯y any tuple satisfying ϕ(¯bi, ¯u), concluding the induction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=') By construction, the sequence we have constructed satisfies the first item in the statement of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For the second item, it suffices to apply the infinite Ramsey theorem, as there are finitely many possible atomic types for ¯bi¯bj’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Assume now that N has a stable edge relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By the second item, we may assume the atomic types of tuples ¯bi¯bj for i < j are all equal to each other, and hence, the atomic types of tu- ples ¯bj¯bi for i < j are also equal to each other;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' assume for contradiction that these are two differ- ent types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Since the atomic types of all ¯bi’s are identical, this implies that there are two different coordinates y1, y2 of ¯y such that for all i < j, the corresponding coordinates bi,1, bi,2, bj,1, bj,2 ∈ M of ¯bi and ¯bj are such that bi,1 and bj,2 are adjacent in M, whereas bi,2 and bj,1 are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 25 Thus the semi-induced bipartite graph in M between the bi,1’s and the bi,2’s is an infinite ladder, a contradiction to the stability of N’s edge relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For two sets A, B of vertices in a given graph, we say that A dominates B if for any b ∈ B there is a ∈ A such that ab is an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Likewise, we say that A antidominates B if it dominates B in the complement graph: for each b ∈ B there is a ∈ A such that ab is not an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='5 Let M be a graph with a stable edge relation, and A, B ⊆ M be two sets of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then one of the following holds: – there is a finite subset S ⊆ A that dominates B, – there is a finite subset S ⊆ B that antidominates A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We note that a variant of the lemma holds even in dependent models [Sim15, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='13], [BDG+22, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4], under the additional assumption that A and B are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We only require it for (monadically) stable models, for which it admits a simple proof given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We describe an iterative process that either gives one of the two desired outcomes, or constructs an infinite ladder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let i ⩾ 1, assume that a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , aℓ−1 as well as b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , bℓ−1 are already constructed so that for i, j < ℓ, aibj is an edge if and only if i ⩾ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then – if {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , aℓ−1} dominates B, then we are done;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' otherwise there is b ∈ B − {b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , bℓ−1} with no adjacencies to {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , aℓ−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Pick such a b and call it bℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' then – if {b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , bℓ} antidominates A, then we are done;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' otherwise there is a ∈ A − {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , aℓ−1} adjacent to each b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , bℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Pick such an a and call it aℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This extends our ladder, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' An infinite matching is the bipartite graph on vertices {ai, bi : i ∈ N} such that aibj is an edge if and only if i = j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' An infinite co-matching is defined in the same way, except there is an edge aibj if and only if i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' (Recall that an infinite ladder is again defined similarly, but with the condition i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=') The following result is folklore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' An equivalent, finitary formulation can be found e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' in [DOOV96, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='6 Let E ⊆ A × B be an infinite bipartite graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then one of the following cases holds: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' TypesE(A/B) is finite, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' E contains an infinite induced matching, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' E contains an infinite induced co-matching, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' E contains an infinite induced ladder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='7 Let C be a graph class with a stable edge relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then every graph M in the elementary closure C of C has a stable edge relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For k ∈ N, let ϕk be the sentence that holds in a graph G if and only if there are vertices a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , ak and b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , bk in G such that for all 1 ⩽ i, j ⩽ k, G |= E(ai, bj) if and only if i ⩽ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Since C has a stable edge relation, there is a number k such that G |= ¬ϕk for all G ∈ C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Hence, M |= ¬ϕk, proving that M has a stable edge relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The following lemmas will be useful in simplifying the inductive proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 26 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='8 Let M and N be graphs such that N is an elementary extension of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Further, let S ⊆ M be any finite set and N′ be any S-flip of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then N′ is an elementary extension of the subgraph of N′ induced by the domain of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let M′ denote the subgraph of N′ induced by the domain of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Observe first that there is a quantifier-free formula η(x, y) with parameters from S such that for all u, v ∈ N, N |= η(u, v) ⇐⇒ N′ |= E(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let ϕ( ¯x) be a formula and ¯m ∈ M ¯x a tuple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We need to show that M′ |= ϕ( ¯m) ⇐⇒ N′ |= ϕ( ¯m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' (1) Rewrite ϕ( ¯x) to a formula ϕ′( ¯x) with parameters from S, by replacing each occurrence of an atom E(x, y) with η(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then we have that N′ |= ϕ( ¯m) ⇐⇒ N |= ϕ′( ¯m), and similarly, M′ |= ϕ( ¯m) ⇐⇒ M |= ϕ′( ¯m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Since N is an elementary extension of M and S ⊆ M, we have that M |= ϕ′( ¯m) ⇐⇒ N |= ϕ′( ¯m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Putting together the equivalences yields (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 6 Pattern-free classes If ϕ(x, y) is a first-order formula and G is a structure, by ϕ(G) we denote the graph with vertices V(G) and edges uv, for distinct u, v ∈ V(G) such that G |= ϕ(u, v) ∨ ϕ(v, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' If C is a class of structures, then denote ϕ(C ) := {ϕ(G) | G ∈ C }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Say that a class C of graphs transduces a class D of graphs if there is a unary expansion � C of C and a formula ϕ(x, y) in the signature of � C , such that for every H ∈ D there is some �G ∈ � C such that H is an induced subgraph of ϕ( �G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' If above, the formula ϕ is existential, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' quantifier-free, then we say that C existentially transduces D, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' quantifier-free transduces D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The r-subdivision of a graph G, denoted G(r) below, is obtained by replacing each edge of G by a path of length r + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' A (1, r)-subdivision of a graph G is a graph obtained from G by replacing each edge of G by a path of length at least 2, and at most r + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Say that a graph class C is pattern-free if for every r ⩾ 1, unary expansion � C of C , and quantifier-free formula ϕ(x, y) in the signature of � C , there is some n ⩾ 1 such that ϕ( � C ) avoids the r-subdivision of Kn as an induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Say that a graph M is pattern-free if the class {M} is pattern-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Note that C is not pattern-free if and only if C quantifier-free transduces the class of r- subdivisions of all cliques, for some fixed r ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In this section we study pattern-free classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 we prove that every simply existentially monadically dependent class is pattern-free, proving implication (2)→(3) in Theo- rem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='5 we prove that if C is pattern-free then every M ∈ C is pattern-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In 27 Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='10 we prove that if M is a pattern-free graph and M′ is obtained from M by applying a finite set of flips, then M′ is also pattern-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='9 we describe certain obstructions that are forbidden in pattern-free graphs M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 If a graph class C is simply existentially monadically dependent, then C is pattern-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 immediately follows from the next three lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2 If C is not pattern-free, then for some r ⩾ 1, the class C quantifier-free transduces the class {G(r) | G is a graph} of r-subdivisions of all graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='3 Fix r ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let C be a hereditary graph class that quantifier-free transduces the class of r-subdivisions of all graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then C existentially transduces the class of all graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4 Let C be a graph class that existentially transduces the class of all graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then C is not simply existentially monadically dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Suppose C is not pattern-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then there is a unary expansion � C and a quantifier-free formula ϕ(x, y), such that for every n ⩾ 1 there is some �Gn ∈ � C of G, such that K(r) n is an induced subgraph of ϕ( �G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We argue that C quantifier-free transduces the class of r-subdivisions of all graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let G be a graph with vertices {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let Gn and �Gn be as above, so that K(r) n is an induced subgraph of ϕ( �G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Since G(r) is an induced subgraph of K(r) n , it is also an induced subgraph of ϕ( �G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Therefore, C quantifier-free transduces the class of r-subdivisions of all graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Assume that C quantifier-free transduces the class of r-subdivisions of all graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then there is a unary expansion � C of C and a quantifier-free formula ϕ(x, y) such that for every graph G for some �HG ∈ C such that G(r) is an induced subgraph of ϕ( �HG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Without loss of generality, the class � C is hereditary, since C is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We may assume that V( �HG) = V(G(r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This is because V(G(r)) ⊆ V( �HG), and if �H′ G denotes the substructure of �HG induced by V(G(r)), then ϕ( �H′ G) = ϕ( �HG)[V(G(r))] = G(r), where the first equality holds since ϕ is quantifier-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Moreover, �H′ G ∈ � C since � C is hereditary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We therefore assume that V( �HG) = V(G(r)), for all graphs G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We may also assume that the formula ϕ(x, y) is symmetric and irreflexive, that is, ∀x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='ϕ(x, y) → (x ̸= y) ∧ ϕ(y, x) holds in every k-colored graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' (Otherwise replace ϕ(x, y) by the formula (ϕ(x, y) ∨ ϕ(y, x)) ∧ x ̸= y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=') We show that C existentially transduces the class of all graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Fix any finite graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then for u, v ∈ G(r) we have that �HG |= ϕ(u, v) if and only if G(r) |= E(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let W ⊆ V(G(r)) denote the set of vertices of G(r) that correspond to the original vertices of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In particular, if η(x, y) denotes the formula η(x, y) := ∃x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' xr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='E(x, x1) ∧ E(x1, x2) ∧ · · · ∧ E(xr−1, xr) ∧ E(xr, y), then η(G(r))[W] is isomorphic to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Now consider the formula ψ(x, y) := ∃x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' xr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='ϕ(x, x1) ∧ ϕ(x1, x2) ∧ · · · ∧ ϕ(xr−1, xr) ∧ ϕ(xr, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' It follows that for u, v ∈ W we have that �HG |= ψ(u, v) if and only if G(r) |= η(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In particular, ψ( �HG)[W] is isomorphic to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 28 Therefore, the existential formula ψ(x, y) (which does not depend on G), witnesses that C existentially transduces the class of all graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Suppose that C existentially transduces the class of all graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then there is an existential formula ψ(x, y) such that every finite graph G is an induced subgraph of ψ( �H), for some �H ∈ � C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For n ⩾ 1, let Gn be the bipartite graph with parts {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , n} and 2{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=',n}, and edges iJ such that i ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let �Hn ∈ � C be such that Gn is an induced subgraph of ψ( �Hn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This means that there are vertices v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , vn and wJ, for J ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , n}, such that �Hn |= ψ(vi, wJ) if and only if i ∈ J, for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , n} and J ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Since n is arbitrary, this shows that C is not simply existentially monadically dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We may assume without loss of generality that C is hereditary, since if a class C is simply existentially monadically dependent, then so is the class of all induced subgraphs of C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This is because that taking an induced subgraph may be simulated by using a unary predicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 now follows from Lemmas 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='3, and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We now prove that every model in the elementary closure of a pattern-free class, is pattern- free as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='5 Let C be a hereditary pattern-free graph class and let M ∈ C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then M is pattern-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let M ∈ C , and let �Σ be a unary expansion of the signature of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Fix an integer r ⩾ 1 and a quantifier-free formula ϕ(x, y) in the signature �Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let � C be the class of all monadic lifts of graphs in C in the signature �Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By assumption, there is a number n such that K(r) n is not an induced subgraph of ϕ(�A), for all �A ∈ � C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let � M be a monadic lift of M in the signature of ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We show that ϕ(� M) does not contain K(r) n as an induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Suppose otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Consider the substructure �A of � M induced by the elements of V(K(r) n ) ⊆ V(� M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then �A is a finite �Σ-structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let A be the graph such that �A is a monadic lift of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We claim that A is an induced subgraph of some graph B ∈ C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Write a sentence ϕA that holds in a graph B if and only if A is isomorphic to an induced subgraph of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Since M |= ϕA and M ∈ C , there is some B ∈ C such that B |= ϕA, proving the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Since C is hereditary, we conclude that A ∈ C as well, and hence �A ∈ � C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In particular, ϕ(�A) is isomorphic to K(r) n , a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' It is convenient to use a relaxed form of patterns, where instead of r-subdivisions, we have (1, r)-subdivisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='6 Fix r ⩾ 1, and let C be a graph class that quantifier-free transduces a class D that contains some (1, r)-subdivision of every clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then C is not pattern-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='6 follows easily by Ramsey’s theorem which we recall below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='7 (Ramsey’s theorem) Fix k ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' There is a monotone, unbounded function f : N → N, such that for every n ⩾ 1, if the edges of Kn are colored using k colors, then there is a set W ⊆ V(Kn) with |W| ⩾ f (n) such that W is monochromatic, that is, all edges uv of Kn with u, v ∈ W, have the same color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='6 follows immediately from the next lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='8 Fix r ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let D be a hereditary class that contains some (1, r)-subdivision of every clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then there is some s ∈ [1, r] such that D contains the s-subdivision of every clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 29 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We want to assign to each n ⩾ 1 a subdivision number cn ∈ [1, r] as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let Gn be some (1, r)-subdivision of Kn that belongs to D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For each edge e of Kn, let ℓ(e) ∈ [1, r] be the number of times it was subdivided in Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then ℓ is a coloring of the edges of Kn using r colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By Ramsey’s theorem, there is a subset Wn ⊆ V(Kn) which is monochromatic, that is, there is a color cn ∈ [1, r] such that for every pair u, v of distinct vertices of Wn, we have ℓ(uv) = cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Moreover, the size of Wn is at least f (n), where f : N → N is some fixed unbounded, monotone function (depending only on r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By the pigeonhole principle, in the infinite sequence (c1, c2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ), some element s ∈ [1, r] occurs infinitely many times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We claim that D contains the s-subdivision of every clique Km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Indeed, pick any m ⩾ 1, and let n ∈ N be such that f (n) ⩾ m and cn = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then Wn ⊆ V(Kn) is such that |Wn| ⩾ m and ℓ(u, v) = s for all distinct u, v ∈ Wn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Recall that Gn ∈ D is a (1, r)- subdivision of Kn, and for all distinct u, v ∈ Wn we have that edges uv of Kn are subdivided s many times in Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' As |Wn| ⩾ m, it follows that Gn contains, as an induced subgraph, the s-subdivision of Km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The following lemma describes certain obstructions (depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 3) that are forbidden in pattern-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='9 Fix r ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let M be a graph such that for any k ∈ N, there is an infinite collection of pairwise disjoint finite sets A, B0, B1, B2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' of vertices of M with the following properties: (i) the set A has cardinality k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' (ii) there is a semi-induced matching between A and a subset Ci of Bi for all i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' (iii) there are no edges between A and Bi − Ci for all i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' (iv) there are no edges between Bi − Ci and Bj for i ̸= j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' and (v) for all i and every pair of disjoint vertices u, v ∈ Ci, there is a path of length ⩾ 2 and ⩽ r that connects u and v and whose all internal vertices belong to Bi − Ci Then M is not pattern-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' FIGURE 3: The obstruction pattern, and the setting of the proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We assume that Bab = B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 30 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We construct a quantifier-free formula ϕ(x, y) using three additional unary predicates, denoted A, B, C, and for each k a lift � Mk of M, such that ϕ(� Mk) contains an (1, r + 1)-subdivision of Kk, as an induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='6 implies that {M} is not pattern-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Fix k ⩾ 1 and consider sets A, B0, B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' given by the assumption of the lemma, so that in particular |A| = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Pick a subfamily of the sets B0, B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' containing exactly (|A| 2 ) sets, and reindex those sets as Bab, for ab ∈ (A 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Likewise, denote the distinguished subsets Cab ⊆ Bab, so that each Cab semi-induces a matching with A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We now describe the construction of � Mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Mark the vertices in A with the unary predicate A, the vertices in Bab, for some ab ∈ (A 2), with the unary predicate B, and the vertices in Cab ⊆ Bab, for some ab ∈ (A 2), with the unary predicate C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Define a formula ϕ0(x, y) as follows: ϕ0(x, y) ≡ (A(x) ∧ C(y)) ∨ (A(y) ∧ C(x)) ∨ (B(x) ∧ B(y) ∧ ¬(C(x) ∧ C(y))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let ϕ(x, y) ≡ ϕ0(x, y) ∧ E(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We argue that ϕ(� Mk) contains an (1, r + 1)-subdivision of Kk, as an induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For each ab ∈ (A 2), let a′, b′ be the unique elements in Cab connected to a and b, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Consider a path πab between a′ and b′, of length ⩾ 2 and ⩽ r, that is internally contained in Bab − Cab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let W denote the vertices in A ∪ {V(πab) | ab ∈ (A 2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Now, it is easy to check that ϕ(� Mk)[W] is a (1, r + 1)-subdivision of Kk (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Finally, we state two lemmas that will be useful later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='10 For any pattern-free graph M, any graph M′ obtained from M by applying a finite set of flips, is also pattern-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' It suffices to consider the case when M′ is obtained from M by applying an atomic flip, specified by a pair (A, B) of subsets of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Add the sets A and B as unary predicates UA and UB to the graph M, obtaining a unary expansion � M of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then the quantifier-free formula η(x, y) := E(x, y)△(UA(x) ∧ UB(y)) is such that for any so that for every a, b ∈ M, M′ |= E(a, b) ⇐⇒ � M |= η(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Suppose M′ is not pattern-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then there is r ⩾ 1, a class � C of unary expansions of {M′}, and a quantifier-free formula ϕ(x, y) such that ϕ( � C ) contains the r-subdivision of every clique, as an induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Replace each atom E(z, t) by the formula η(z, t) in ϕ(x, y), yielding a quantifier-free formula ϕ′(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then for every � M′ ∈ � C there is a unary expansion � M′′ of � M, such that � M′ |= ϕ(a, b) ⇐⇒ � M′′ |= ϕ′(a, b) for all a, b ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' It follows that there is some class � C ′ of unary expansions of {� M} such that ϕ′( � C ′) contains the r-subdivision of every clique as an induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Hence, by transitivity of unary expansions, M is not pattern-free, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' A similar statement holds for graphs with a stable edge relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='11 For any graph M with a stable edge relation, any graph M′ obtained from M by applying a finite set of flips, also has a stable edge relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Again it is enough to assume that M′ is obtained from M by performing a single atomic flip F = (A, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We prove the lemma by contraposition – we argue that if M′ contains ladders of arbitrary length, then so does M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 31 Let a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , ak, b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , ak be the vertices of a ladder of length k in M′ (we have aibj ∈ E(M) iff i ⩽ j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Since each ai has four possibilities of being included or non-included in the sets A, B and the same holds for bi, there exists a subset Z of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , k} of size at least k′ := k 16 such that for all i, i′, j, j′ ∈ Z we have (ai, bj) ∈ (A × B) ∪ (B × A) if and only if (ai′, bj′) ∈ (A × B) ∪ (B × A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Without loss of generality assume that Z = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , k′} (this can be achieved by forgetting all ai, bi such that i ̸∈ Z and renaming the indices which are left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then there are two options – either (a1, b1) ̸∈ (A × B) ∪ (B × A) and then a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , ak′, b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , ak′ is a ladder in M′ ⊕ F of length k 16, or (a1, b1) ∈ (A × B) ∪ (B × A) and then the edges between the sides of the ladder semi-induced by a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , ak′, b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , ak′ in M′ get complemented, which results in a ladder of length k 16 − 1 in M′ ⊕ F given by vertices c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , ck′−1, d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , dk′−1, where ci = ak′−i+1 and di = bk′−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Since by our assumption on M′ we can choose k to be arbitrarily large, this means that there are arbitrarily large ladders in M′ ⊕ F, which finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 7 Finite separators in pattern-free stable models Say that a graph M is r-separable if for every elementary extension N of M, and every v ∈ N − M, there is a finite set S ⊆ M such that v and M are r-separated over S in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This section is dedicated to proving the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 Let M be a pattern-free graph with a stable edge relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then M is r-separable, for every r ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='7 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='5, we immediately get the following corollary, proving the implication (3)→(4) in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2 If C is a pattern-free class of graphs with stable edge relation and r ∈ N, then every M ∈ C is r-separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We will prove Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 by induction on r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 Case of finitely many types The following lemma is a generalization of the case r = 1 of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1, where instead of one vertex v we have a set of vertices U with a bounded number of types over M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='3 For any graphs M and N with M ≺ N and such that the edge relation is stable in N, and for any set U ⊆ N − M such that TypesE(U/M) is finite, there exists a finite set S ⊆ M and an S-flip which: – 1-separates U from M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' and – does not flip the S-class T := {v ∈ N : ∀s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ¬E(v, s)} with any other S-class (including itself), as long as T ∩ U is nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We will construct the sought set S in a series of steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ▷ Claim 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4 There is a finite set SU ⊆ M such that any two vertices in M with the same SU-class also have the same U-class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For each set in TypesE(U/M), select one vertex v ∈ U of that type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then apply Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='3 to the formula E(x, y) and the element v ∈ N: there is a positive boolean 32 combination ψ(x) of formulas of the form E(x, m) for m ∈ M, and for every u ∈ M, N |= E(v, u) ⇐⇒ M |= ψ(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' It follows that there is a finite set Sv ⊆ M, namely the set of parameters of ψ(x), so that whether or not a vertex u ∈ M is adjacent to v depends only on its Sv-class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The finite set SU := � v Sv has the desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ◁ ▷ Claim 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='5 There is a finite set SM ⊆ M such that any two vertices in U with the same SM-class also have the same M-class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Notice that since TypesE(U/M) is finite, TypesE(M/U) is also finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Thus we can obtain SM by including one vertex from each set in TypesE(M/U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ◁ Now, for each of the finitely many ordered pairs (A, B) of distinct (SU ∪ SM)-classes, apply- ing Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='5 on A ∩ M and B ∩ M yields a finite set SA,B which is either contained in A ∩ M and dominates B ∩ M, or contained in B ∩ M and antidominates A ∩ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Note that it may be that SA,B and SB,A are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We now set S to be S := SU ∪ SM ∪ � (A,B) SA,B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Note that S is finite and contained in M, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Now we define an S-flip N′ of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let N′ be the graph obtained from N by flipping between S-classes A and B (possibly with A = B) if there exists an edge in N which has one end in M and the other end in U, and at the same time has one end in A and the other end in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We will now show that N′ satisfies the required properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We begin by considering the S-class T := {v ∈ N : ∀s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ¬E(v, s)} of vertices non-adjacent to all of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ▷ Claim 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='6 If T ∩ U is nonempty, then T is not flipped with any other S-class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Assume by contradiction that there is some T′ (possibly with T′ = T) such that T and T′ are flipped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Moreover, let w be some vertex of T ∩ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' First, observe that there is no edge in N between w and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Indeed, as SM contains a rep- resentative of every equivalence class in TypesE(M/U) and yet w is not connected in N to any vertex in SM, then it is not connected to any vertex in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Therefore, there is no edge in N be- tween T ∩ U and M, so in particular there is no edge in N between T ∩ U and T′ ∩ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' However, a flip was made between T and T′, so there must be an edge in N between T ∩ M and T′ ∩ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Denote by v and u two vertices connected by an edge in N such that v ∈ T ∩ M and u ∈ T′ ∩ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We know that the neighborhood of u in M can be defined by a positive boolean combination of the formulas of the form E(x, s) for s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Since v satisfies none of these formulas (and yet is connected to u), we infer that the neighborhood of u must be described by a formula that is always true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Therefore, the neighborhood of u contains all of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' It follows that U contains two vertices u and w such that in N one of them is connected to every vertex in M and the other is disconnected from every vertex in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' However, this cannot happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Indeed, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='5 applied to A = B = M there is a finite R ⊆ M that either dominates every vertex in M or antidominates every vertex in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' As M is an elementary substructure of N, then R also dominates every vertex in N or antidominates every vertex in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This is a contradiction, as R neither dominates w nor antidominates u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Therefore, T is not flipped with any other S-class (including itself).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ◁ 33 To complete the proof, we now show that N′ has no edge connecting a vertex in M with a vertex in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Suppose towards a contradiction that N′ has an edge ab so that a ∈ M and b ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let A and B denote the S-classes of N which contain a and b, respectively (so maybe A = B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' If ab ∈ E(N), then we would have flipped between A and B, removing the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Thus ab /∈ E(N), and we flipped between A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' So there must have been some other edge a′b′ ∈ E(N) such that a′ ∈ A, b′ ∈ B, one of a′, b′ is in M, and the other is in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ▷ Claim 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='7 Let x ∈ M, y ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Assume that a and x have the same SU ∪ SM-class, and that b and y have the same SU ∪ SM-class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then xy /∈ E(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Since, in N, the vertices a and x have the same SU ∪ SM-class, they also have the same SU-class and therefore the same U-class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' As ab /∈ E(N), a ∈ M and b ∈ U, we infer that xb /∈ E(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Likewise, b and y have the same M-class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Therefore, xy /∈ E(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ◁ First suppose that a′ ∈ M and b′ ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then Claim 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='7 applies with (x, y) = (a′, b′), contra- dicting the fact that a′b′ ∈ E(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Thus we may assume that a′ ∈ U and b′ ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let A0 and B0 denote the SU ∪ SM-classes of a′ and b′, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' note that A ⊆ A0 and B ⊆ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' If A0 = B0, then Claim 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='7 applies with (x, y) = (b′, a′), again contradicting that a′b′ ∈ E(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Thus A0 ̸= B0, and a, b, a′, b′ are four different vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' FIGURE 4: Illustration for the end of the proof of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ▷ Claim 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='8 S does not contain a dominating set of B0 ∩ M entirely contained in A0 ∩ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Consider the intersection Ra of S and A0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' see Figure 4 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' If Ra = ∅, then the claim is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Otherwise, let us pick an arbitrary element r of Ra, aiming to show that rb′ is a nonedge in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' As b′ ∈ B0 ∩ M, this immediately finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Since r and a belong to M and have the same SU-class, they have the same U-class and therefore rb is a non-edge in N, just as ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Now since b and b′ have the same S-class (as they both belong to B), they have the same connection to r ∈ S, and therefore rb′ is a nonedge as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ◁ However, we also have that: ▷ Claim 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='9 S does not contain an antidominating set of A0 ∩ M entirely contained in B0 ∩ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Similarly, we let Rb′ be the intersection of S and B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' A symmetric argument shows that there are all possible edges between a and Rb′, and thus S cannot contain an an- tidominating set for A0 ∩ M which is contained in B0 ∩ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ◁ 34 Claims 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='8 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='9 directly contradict the construction of SA0,B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2 Balls have finitely many types over a model Observe that the base case of r = 1 of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1, which follows by Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='3, only assumes that the edge relation in M is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In the inductive argument we will use the assumption that M is pattern-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='10, every graph obtained from M by performing some flips remains pattern-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Using Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='9, we prove the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='10 Fix r ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let M be pattern-free graph with a stable edge relation, let N be its elemen- tary extension, and let v ∈ N be such that the r-ball Br(v) around v in N is disjoint from M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then TypesE(Br(v)/M) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' First note that N is pattern-free and a stable edge relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This is because N ∈ {M}, and we can apply Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='5 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Suppose, going for a contradiction, that TypesE(Br(v)/M) is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='6, the bipartite graph semi-induced in N between Br(v) and M either contains an infinite induced matching, an infinite induced co-matching, or an infinite induced ladder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Since M has a sta- ble edge relation, the last case is excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Moreover, up to performing a flip (over ∅) which exchanges edges and non-edges, we may assume without loss of generality (again thanks to Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='8 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='10 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='11) that we are in the first case: there is an infinite induced match- ing between Br(v) and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We now show that the assumptions of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='9 are satisfied with radius 2r: for any k ∈ N, we will construct A and the Bi’s as in the statement of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='9, this implies that N is not pattern-free, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let A ⊆ M and C ⊆ Br(v) be sets of cardinality k that semi-induce a matching in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Now for each c ∈ C, consider a path of length ⩽ r from v to c, let B ⊆ Br(v) be the union of these paths (as sets of vertices);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' note that |B| ⩽ kr + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let ¯y be a tuple of variables with | ¯y| = |B|, and let ¯n ∈ N ¯y be a ¯y-tuple comprised of all elements of B as its components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Note that since Br(v) avoids M, there can be no edges in N between M and vertices at distance < r from v, therefore C is exactly the set of vertices which are at distance r from v in N[B] (and they are also at distance r from v in N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Observe that for every pair of distinct vertices c1, c2 in C, there exists a simple path of length at least 2 and at most 2r internally contained in B − C, constructed as the concatenation of a nonempty suffix of the path from v to c1 and a nonempty suffix of the path from v to c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We now apply Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4, which yields a sequence of tuples ¯b0, ¯b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' in M such that for each i, ¯bi has the same atomic type as ¯n over A ∪ {¯b0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , ¯bi−1}, and such that the atomic types of (¯bi, ¯bj) are the same for i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For each i, we let Bi ⊆ N be the set of vertices appearing as components of ¯bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Note that in particular, the ¯bi’s have same atomic type as ¯n (in other words, the N[Bi]’s are all isomorphic copies of N[B]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Thus for each i, we let vi ∈ Bi be the vertex corresponding to v ∈ B, and we let Ci ⊆ Bi be the set of vertices at distance r from vi in N[Bi], which are also the vertices corresponding to C ⊆ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Note that conditions (i) and (v) from the statement of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='9 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By definition, there is a semi-induced matching between A ⊆ M and C in N, and moreover there are no edges between B − C and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Now since each ¯bi has the same atomic type as ¯n over A, there is also a semi-induced matching between A and Ci for all i and there are no edges between Bi − Ci and A: conditions (ii) and (iii) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Recall from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4 that for i ̸= j, the atomic type of (¯bi, ¯bj) is the same as the atomic type of (¯b1, ¯b0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Moreover, ¯b1 and ¯n have the same atomic type over A ∪ ¯b0, so the atomic type 35 of (¯b1, ¯b0) is the same as the atomic type of ( ¯n, ¯b0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Now as noticed above, there are no edges in N between M and B − C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' As also ¯b0 is contained in M, there are no edges between the vertices of ¯n (outside of C) and the vertices of ¯b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By the equality of atomic types of (¯bi, ¯bj) and ( ¯n, ¯b0), it follows that there are no edges between the vertices of ¯bi (outside of Ci) and the vertices of ¯bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Hence, there are no edges in N between Bi − Ci and Bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Therefore, (iv) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Therefore Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='9 tells us that N is not pattern-free, a contradiction, which proves that TypesE(Br(v)/M) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='3 Inductive proof An inductive proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 now follows by putting together Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='3 and Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We proceed by induction on r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The base case r = 0 is immediate as we may take S to be ∅ since v /∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In the inductive step, assume that the result is proved for the distance r ∈ N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' that is, there is a finite S ⊆ M such that v |⌣ r S M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Stated differently, there is an S-flip N′ of N in which the r-ball around v is disjoint from M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By working in N′ instead of N, we may assume, thanks to Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='8 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='10 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='11, that the r-ball Br(v) around v in N is disjoint from M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='10, TypesE(Br(v)/M) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Now Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='3 applied to Br(v) finishes the inductive step and the proof (we are using the fact that we obtain a set S and an S-flip which doesn’t flip the S-class which contains Br−1(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 8 Separator wins in monadically stable classes The goal of this section is to prove the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 Fix r ∈ N, and let C be a class of graphs such that every G ∈ C is r-separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then there exists k ∈ N such that Separator wins the Separation Game with radius r in k rounds on every G ∈ C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This proves the implication (4)→(5) in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2 and Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2 we immediately get the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2 Let C be a monadically stable class of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then for any r ∈ N, there exists k ∈ N such that Flipper wins the Flipper game with radius r in k rounds on every G ∈ C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The rest of Section 8 is devoted to the proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Fix an enumeration ϕ1, ϕ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' of all formulas (in the signature of graphs) of the form ϕ(y, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , xℓ), with ℓ ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We define a strategy of Separator in any graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In the kth round, after Connector picks ck ∈ Ak−1, Separator first sets S := Sk−1 ∪ {ck} and marks ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then, for every i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , k, for the formula ϕi(y, ¯x), Separator does the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For each valuation ¯a ∈ S ¯x such that G |= ∃y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='ϕi(y, ¯a), Separator marks any vertex b ∈ V(G) such that G |= ϕi(b, ¯a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We say that any strategy of Separator which has this property is Connector-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The marked vertices form Separator’s response in the kth round, and we set Sk to be the union of Sk−1 and all the marked vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Note that there is a function f : N → N such that |Sk| ⩽ f (k) for all k ∈ N, regardless of which vertices Connector picks or which of the formulas ∃y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='ϕi(y, ¯a) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We prove that there is a number k ∈ N such that when Separator plays according to any Connector-complete strategy on a graph G ∈ C , then he wins in at most k rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Assume 36 that the conclusion of the theorem does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then, there exists a sequence of graphs G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ∈ C , where in Gn Connector has a strategy ensuring that Separator does not win for at least n rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We shall now prove that there is some graph G in the elementary closure of C and a vertex in the graph that survives in the arena indefinitely, when Separator plays ac- cording to a Connector-complete strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We will then show that this contradicts r-separability of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ▷ Claim 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='3 There exists a graph G ∈ C , a strategy of Connector, and a Connector-complete strategy of Separator for which the Separation Game on G lasts indefinitely and � n<ω An is nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For every graph Gn ∈ C , choose any Connector-complete strategy of Sepa- rator, and any strategy of Connector ensuring the game continues for more than n rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We define a new class C ′ of structures by adding constants to graphs G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' – Add constant symbols c1, c2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , cω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For each graph Gn, interpret ck as: – Connector’s move in the kth round if k ⩽ n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' – any vertex remaining in the arena after n rounds if k = ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' – an arbitrary vertex of Gn otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' – Add constant symbols sn,i for 1 ⩽ n < ω and 1 ⩽ i ⩽ f (n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For each graph Gn ∈ C , interpret sk,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , sk,f (k) as: – the vertices marked by Separator in the kth round if k ⩽ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We allow these symbols to be interpreted as the same vertex if Separator plays fewer than f (k) vertices, and we ensure that sk,1 is interpreted in the same way as ck;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' – arbitrary vertices of Gn otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Now, for convenience, for each 1 ⩽ n < ω, we write Sn−1 := {sk,i | k < n, 1 ⩽ i ⩽ f (k)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Finally, let T be the theory which is obtained by including: – the sentences ϕ which hold in all structures G ∈ C ′ (that is, the theory of C ′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' – sentences which express that cn is a valid move in the nth round, namely, that cn ̸ |⌣ r Sn−1 ck for 1 ⩽ k < n (we remark that these sentences are existential);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' – analogous sentences which express that cω is a valid move in each round n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' – sentences which express that Separator plays according to a Connector-complete strategy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' that is, sn,1 = cn, and for each 1 ⩽ i ⩽ n, for the formula ϕi(y, ¯x) and for each valuation ¯c ∈ (Sn−1 ∪ {sn,1}) ¯x, the sentence ∃y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='ϕi(y, ¯c) =⇒ ϕi(sn,2, ¯c) ∨ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ∨ ϕi(sn,f (n), ¯c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We now show that T is consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' To this end, pick a finite subset T′ of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then there is a number n0 ∈ N such that no constants cn or sn,i, with n0 < n < ω, occur in a sentence in T′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Since Connector avoids losing in Gn0+1 for at least n0 + 1 rounds in our fixed strategies, the structure corresponding to Gn0+1 in C ′ models T′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' It follows by compactness (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1) that T is consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Since T is consistent, there exists a model G′ of T, which is a graph equipped with constant symbols c1, c2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , cω and sk,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let G be the graph obtained from G′ by forgetting these constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' As T contains the theory of C ′, which in turn contains the theory of C , we infer that G ∈ C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Now consider the instance of the Separation Game where in the nth round, Connector picks the 37 vertex cn and Separator picks the vertices sn,i with 1 ⩽ i ⩽ f (n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This is a valid strategy for Connector by the fact that G′ |= T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For the same reason, the vertex cω remains in the arena after each round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Finally, in G, Separator’s strategy as defined above is Connector-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ◁ Let G ∈ C be the graph produced by Claim 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='3, along with the strategies of Connector and Separator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By assumption, G is r-separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Recall that A0 ⊇ A1 ⊇ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' is the sequence of arenas in the play, c1, c2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' is the sequence of moves of Connector, and S0 ⊆ S1 ⊆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' is the sequence of sets of vertices marked by Separator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Denote Aω := � n<ω An, and Sω := � n<ω Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We will get a contradiction with the previous claim by proving the following claim: ▷ Claim 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4 Aω is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Observe that for each k ∈ N, we have ck /∈ Sk−1: as soon as Connector plays ck in Sk−1, the arena Ak shrinks to a single vertex and Separator wins in the following round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then, Ak is disjoint from Sk−1: since Connector plays ck outside of Sk−1, each vertex of Sk−1 becomes separated from ck and thus is removed from the arena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' It follows that Aω ∩ Sω = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Since Separator follows a Connector-complete strategy, Sω induces an elementary substruc- ture of G by the Tarski-Vaught test (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We also have that c1, c2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ∈ Sω by con- struction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Now suppose for a contradiction that there exists some cω ∈ Aω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We remark that cω /∈ Sω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1, there exists a finite set S ⊆ Sω such that cω |⌣ r S Sω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' As S is finite, there is some n < ω such that S ⊆ Sn, so in particular, cω |⌣ r Sn Sω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' On the other hand, cω ̸ |⌣ r Sn cn+1, as cω ∈ An+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This is a contradiction since cn+1 ∈ Sω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ◁ However, this means that there exists a graph G ∈ C and strategies of Connector and Sep- arator, for which Aω is simultaneously nonempty (Claim 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='3) and empty (Claim 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This contradicts the existence of the graphs G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ∈ C and completes the proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Part III Algorithmic Flipper game 9 Outline In this part we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='5, recalled below for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='5 Let C be a monadically stable class of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then for every radius r ∈ N there exist k ∈ N and a Flipper strategy flip⋆ such that the following holds: – When playing according to flip⋆ in the Flipper game of radius r on any graph G ∈ C , Flipper wins within at most k rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' – The moves of flip⋆ on an n-vertex graph G ∈ C can be computed in time OC ,r(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Recall here that notation OC ,r(·) hides multiplicative factors that depend only on the class C and the radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let us first sketch a natural approach to use the flip-wideness characterization of monadic stability (see Definition 1) to derive a winning strategy for Flipper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Consider the radius-r Flipper game on a graph G from a monadically stable class C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For convenience we may assume for now 38 that we work with an extended version of the game where at each round Flipper can apply a bounded (in term of the round’s index) number of flips, instead of just one (see the discussion in Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' As making a vertex isolated requires one flip — between the vertex in question and its neighborhood — we can always assume that the flips applied by Flipper in round i make all the i vertices previously played by Connector isolated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Hence, Connector needs to play a new vertex in each round, thus building a growing set X of her moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Fix some constant m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' According to flip-wideness, there exists some number N := N2r(m) with the property that once X has grown to the size N, we find a set of flips F — whose size is bounded independently of m — and a set Y of m vertices in X that are pairwise at distance greater than 2r in G ⊕ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' It now looks reasonable that Flipper applies the flips from F within his next move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Indeed, since after applying F the vertices of Y are at distance more than 2r from each other, the intuition is that F robustly “disconnects” the graph so that the subsequent move of the Connector will necessarily localize the game to a simpler setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This intuition is, however, difficult to capture: flip-wideness a priori does not provide any guarantees on the disconnectedness of G ⊕ F other than that the vertices of Y are far from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The main idea for circumventing this issue is to revisit the notion of flip-wideness and strengthen it with an additional predictability property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Intuitively, predictability says that being given any set of 5 vertices in Y as above is sufficient to uniquely reconstruct the set of flips F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Formally, in Section 10 we prove the following strengthening of the results of [DMST22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Here and later on, O(G) denotes the set of linear orders on the vertices of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 (Predictable flip-wideness) Fix radius r ∈ N and a monadically stable class of graphs C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then there exist the following: – An unbounded non-decreasing function αr : N → N and a bound λr ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' – A function FWr that maps each triple (G ∈ C , ≼ ∈ O(G), X ⊆ V(G)) to a pair (Y, F) such that: – F is a set of at most λr flips in G, and – Y is a set of αr(|X|) vertices of X that is distance-r independent in G ⊕ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' – A function Predictr that maps each triple (G ∈ C , ≼ ∈ O(G), Z ⊆ V(G)) with |Z| = 5 to a set F of flips in G such that the following holds: – For every X ⊆ V(G), if (Y, F) = FWr(G, ≼, X) and Z ⊆ Y, then F = Predictr(G, ≼, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Moreover, given G, ≼, and Z, Predictr(G, ≼, Z) can be computed in time OC ,r(|V(G)|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let us explain the intuition behind the mappings FWr and Predictr provided by Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The existence of bounds αr and λr and of the function FWr with the properties as above is guaranteed by the standard flip-wideness, see Definition 1 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' However, in the proof we pick the function FWr in a very specific way, so that the flip set F is defined in a somewhat minimal way with respect to a given vertex ordering ≼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This enables us to predict what the flip set F should be given any set of 5 vertices from Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This condition is captured by the function Predictr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We remark that the predictability property implies the following condition, which we call canonicity, and which may be easier to think about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' (We assume the notation from Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=') – For every G ∈ C , ≼∈ O(G), and X, X′ ⊆ V(G), if we denote (Y, F) = FWr(G, ≼, X) and (Y′, F′) = FWr(G, ≼, X′), then |Y ∩ Y′| ⩾ 5 entails F = F′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 39 Indeed, to derive canonicity from predictability note that F = Predictr(G, ≼, Z) = F′, where Z is any 5-element subset of Y ∩ Y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Predictability strengthens canonicity by requiring that the mapping from 5-element subsets to flip sets is governed by a single function Predictr, which is moreover efficiently computable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We now outline how Flipper can use predictable flip-wideness for radius 2r to win the radius-r Flipper game in a bounded number of rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Suppose the game is played on a graph G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' we also fix an arbitrary ordering ≼ of vertices of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Flipper will keep track of a growing set X of vertices played by the Connector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The game proceeds in a number of eras, where at the end of each era X will be augmented by one vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In an era, Flipper will spend 2 · (|X| 5 ) rounds trying to robustly disconnect the current set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' To this end, for every 5-element subset Z of X Flipper performs a pair of rounds: – In the first round, Flipper computes F := Predict2r(G, ≼, Z) and applies the flips from F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Subsequently, Connector needs to localize the game to a ball of radius r in the F-flip of the current graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' – In the second round, Flipper reverses the flips by applying F again, and Connector again localizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Thus, after performing a pair of rounds as above, we end with an induced subgraph of the original graph, which moreover is contained in a ball of radius r in the F-flip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Having performed all the (|X| 5 ) pairs of rounds as above, Flipper makes the last round of this era: he applies flips that isolates all vertices of X, thus forcing Connector to play any vertex outside of X that is still available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This adds a new vertex to X and a new era begins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let us sketch why this strategy leads to a victory of Flipper within a bounded number of rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Suppose the game proceeds for N eras, where N is such that α2r(N) ⩾ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then we can apply predictable flip-wideness to the set X built within those eras, thus obtaining a pair (Y, F) := FW2r(G, ≼, X) such that |Y| = 7 and F is a set of flips such that Y is distance-2r inde- pendent in G ⊕ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Enumerate Y as {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , v7}, according to the order in which they were added to X during the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let Z := {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , v5} and note that F = Predict2r(G, ≼, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Observe that in the era following the addition of v5 to X, Flipper considered Z as one of the 5-element subsets of the (current) set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Consequently, within one of the pairs of rounds in this era, he applied flips from F and forced Connector to localize the game subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Since v6 and v7 are at distance larger than 2r in G ⊕ F, this necessarily resulted in removing v6 or v7 from the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This is a contradiction with the assumption that both v6 and v7 were played later in the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In Section 10 we prove Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In Section 11 we formalize the strategy outline pre- sented above and analyze the time complexity needed to compute the moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This will amount to proving Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 10 Predictable flip-wideness This section is devoted to proving Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1: monadic stability implies predictable flip- wideness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We will first collect some facts from the work of Dreier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' [DMST22] and shape them to our convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 40 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 Classifiers The main idea behind the proof of [DMST22] is to perform gradual classification of vertices while showing that this classification needs to satisfy very rigid conditions, imposed by monadic stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' To formalize this we will rely on the notion of a classifier, presented below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For a vertex u, N(u) denotes the (open) neighborhood of u, that is, the set comprising all the neighbors of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The neighborhood of u in a set of vertices B is N(u) ∩ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Further, s is adjacent to B if N(u) ∩ B ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Usually, the graph, in which neighborhoods and adjacencies are evaluated, will be clear from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Otherwise, we specify it in the subscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' A classifier in a graph G is a quadruple B = (B, S, exc, rep), where B is a family of pairwise disjoint vertex subsets of G, called further blobs, S is a non-empty subset of vertices of G, and exc: V(G) → B ∪ {⊥} and rep: V(G) → S are mappings satisfying the following properties: (a) S ∩ � B = ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' that is, no vertex of S belongs to any blob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' (b) Every s ∈ S is adjacent either to all the blobs in B or to no blob in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' (c) For all distinct s, s′ ∈ S and each blob B ∈ B, N(s) ∩ B ̸= N(s′) ∩ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' (d) For each v ∈ � B, we have exc(v) ̸= ⊥ and v ∈ exc(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' (e) For all v ∈ V(G) and B ∈ B − {exc(v)}, we have N(v) ∩ B = N(rep(v)) ∩ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The size of a classifier (B, S, exc, rep) is |B|, and its order is |S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let us give some intuition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In a classifier we have a family of disjoint blobs B and a set of representative vertices S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Further, with every vertex v we can associate its exceptional blob exc(v) ∈ B and its representative rep(v) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The key condition (e) says the following: every vertex v behaves in the same way as its representative rep(v) with respect to all the blobs in B, except for its (single) exceptional blob exc(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We allow exc(v) to be equal to ⊥, which indicates that v has no exceptional blob (this will be convenient in notation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Condition (d) says that if v is contained in some blob B ∈ B, then in fact B must be the exceptional blob of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Conditions (a), (b), and (c) are technical assertions that expresses that the representative set S is reasonably chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' A classifier naturally partitions the vertex set of the graph, as formalized below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For a classifier B = (B, S, exc, rep), the partition raised by B is the partition ΠB of the vertex set of G defined as follows: ΠB := {rep−1(s): s ∈ S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For s ∈ S, we write ΠB(s) := rep−1(s) to indicate the part of ΠB associated with s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The following observation is easy, but will be the key to our use of classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Observation 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 Let B = (B, S, exc, rep) be classifier of size at least five in a graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then for every pair of vertices u, v of G, the following conditions are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' (i) u and v are in the same part of ΠB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' (ii) u and v have the same neighborhood in at least three blobs from B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' (iii) u and v have different neighborhoods in at most two blobs from B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 41 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Implication (i)⇒(iii) follows by observing that since rep(u) = rep(v), u and v must have exactly the same neighborhood in every blob, possibly except for exc(u) and exc(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Implication (iii)⇒(ii) is immediate due to |B| ⩾ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Finally, for implication (ii)⇒(i), observe that u and rep(u) have the same neighborhood in all but at most one blob from B, and similarly for v and rep(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Since u and v have the same neighborhood in at least three blobs from B, it follows that rep(u) and rep(v) have the same neighborhood in at least one blob from B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By condition (c) of Definition 3, this means that rep(u) = rep(v), so u and v belong to the same part of ΠB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' From Observation 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 we can derive a canonicity property for classifiers: whenever two classifiers share at least five blobs in common, the associated partitions are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In the next lemma we show an even stronger property: (efficient) predictability for classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2 There exists an algorithm that given a graph G and family B◦ consisting of five pairwise disjoint subsets of V(G), computes a partition Π◦ of V(G) with the following property: for every classi- fier B = (B, S, exc, rep) in G with B◦ ⊆ B, we have Π◦ = ΠB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The running time of the algorithm is O(|Π◦| · n2), where n = |V(G)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We first present the construction of Π◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Along the way we also construct a set of repre- sentatives S, and at each point vertices s ∈ S are in one-to-one correspondence with parts Π◦(s) of Π◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We start with Π◦ = ∅ and S = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then we iterate through the vertices of G in any order, and when considering the next vertex v we include it in the partition as follows: – If there exists s ∈ S such that v and s have the same neighborhood in at least three of the blobs of B◦, select such s that was added the earliest to S and add v to Π◦(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' – Otherwise, if no s as above exists, add v to S and associate with v a new part Π◦(v) = {v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' It is straightforward to implement the algorithm to work in time O(|Π◦| · n2), where |Π◦| = |S| is the size of the output partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We now verify that Π◦ constructed in this manner satisfies the requested properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let then B = (B, S, exc, rep) be any classifier with B◦ ⊆ B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' we need to argue that Π◦ = ΠB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Consider any pair u, v of vertices of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We need to prove that u, v are in the same part of Π◦ if and only if they are in the same part of ΠB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For the forward implication, suppose u and v belong to the same part of Π◦, say Π◦(s) for some s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By construction, u and s have the same neighborhood in at least three of the blobs of B◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By Observation 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1, this implies that u and s are in the same part of ΠB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Similarly, v and s are in the same part of ΠB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By transitivity, u and v are in the same part of ΠB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For the other direction, suppose u ∈ Π◦(s) and v ∈ Π◦(s′) for some s ̸= s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By symmetry, we may assume that s′ was added to S later than s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Since v was included in Π◦(s′) instead of Π◦(s), by construction it follows that v and s must have different neighborhoods in at least three different blobs of B◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' So by Observation 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1, v and s belong to different parts of ΠB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Since u and s belong to the same part of Π◦, by the forward implication they also belong to the same part of ΠB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Hence u and v belong to different parts of ΠB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We next observe that, at the cost of a moderate loss on the size of a classifier, we may choose the representatives quite freely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='3 Let B = (B, S, exc, rep) be a classifier in a graph G and let S′ be any set such that rep is a bijection from S′ to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then there is a classifier B′ = (B′, S′, exc′, rep′) in G such that B′ ⊆ B and |B′| ⩾ |B| − |S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 42 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let B′ be obtained from B by removing exc(s′) for each s′ ∈ S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Note that by condition (d) of Definition 3, S′ is disjoint from � B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Next, for each vertex u set exc′(u) := exc(u), except for the case when exc(u) ∈ B − B′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' then set exc′(u) := ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Since rep is a bijection from S′ to S, for every vertex u there exists exactly one vertex s′ ∈ S′ satisfying rep(u) = rep(s′), and we set rep′(u) := s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We claim that B′ := (B′, S′, exc′, rep′) is a classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For this, observe that for every s′ ∈ S′, since exc(s′) has been removed when constructing B′, we in fact have N(rep(s′)) ∩ B = N(s′) ∩ B for every B ∈ B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' With this observation in mind, all conditions of Definition 3 for B′ follow directly from those for B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let G be a graph and ≼ be any linear order on V(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We shall say that a classifier B = (B, S, exc, rep) is canonical with respect to ≼ if the following condition holds: each s ∈ S is the ≼-minimum element of ΠB(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We note the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4 Let G be a graph, ≼ be a linear order on G, and B = (B, S, exc, rep) be a classifier in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then there is also a classifier B′ = (B′, S′, exc′, rep′) such that |S′| = |S|, B′ ⊆ B, |B′| ⩾ |B| − |S|, and B′ is canonical with respect to ≼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' It suffices to apply Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='3 to S′ := {min≼ ΠB(s): s ∈ S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The following lemma is the main outcome of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Here, an r-ball in a graph G is a set of the form {w ∈ V(G) | distG(v, w) ⩽ r} for some vertex v (the center of the ball).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='5 Fix a monadically stable class of graphs C and r ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then there exist a constant κ ∈ N and an unbounded non-decreasing function β: N → N such that the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For every G ∈ C , linear order ≼ on V(G), and a non-empty family A of pairwise disjoint r-balls in G, there exists a classifier B = (B, S, exc, rep) in G such that: – B has size at least β(|A|) and order at most κ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' – B ⊆ A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' and – B is canonical with respect to ≼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' To prove Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='5 we need the following lemma, which follows from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2 of [DMST22] by setting ϕ(x, y) = adj(x, y) and α(x, y) = dist⩽r(x, y), where adj(x, y) is the adja- cency predicate and dist⩽r(x, y) is the first-order formula expressing that the distance between x and y is at most r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='6 (follows from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2 of [DMST22]) Fix a monadically stable class of graphs C and r ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then there exist k ∈ N and a function M: N → N such that for every m ⩾ 3 and family A of size at least M(m) of pairwise disjoint r-balls in a graph G ∈ C , the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' There exists a subfamily B ⊆ A of size at least m and a set S ⊆ V(G) of at most k vertices such that for every v ∈ V(G) there exists a single exceptional ball Bv ∈ B and an element sv ∈ S such that for every ball B ∈ B − {Bv} we have N(v) ∩ B = N(sv) ∩ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Furthermore, if v ∈ B for some B ∈ B, then Bv = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In essence, Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='6 already gives us the needed classifier, except that we need to mas- sage it using Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4 and a pigeonhole argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 43 Proof of Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let k and M be the constant and the function provided by Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='6 for the class C and the radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We may assume that M is non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We set κ := k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Further, for every n ∈ N, set β(n) to be the largest positive integer m such that n ⩾ M(kk · (k + 1) · (m + k) + k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' if there is no such m, set β(n) := 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Clearly, β defined in this way is non-decreasing and un- bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Consider any family A of pairwise disjoint r-balls A in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let m := β(|A|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' If m = 0, then we set B to be the unique canonical classifier of size 0 and order 1, that is B := (∅, {v0}, v �→ ⊥, v �→ v0) where v0 = min≼ V(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' So from now on we may assume that m ⩾ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' thus we have |A| ⩾ M(kk · (k + 1) · (m + k) + k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Apply Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='6 to A, yielding suitable B1, S1, and mappings v �→ Bv and v �→ sv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Note that we have |S1| ⩽ k and |B1| ⩾ kk · (k + 1) · (m + k) + k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We would like to claim that these objects constitute a classifier, but for this we have to make sure that conditions (a), (b), and (c) of Definition 3 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This will be done using a pigeonhole argument as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Construct B′ 1 from B1 by removing the exceptional ball Bs for each s ∈ S1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' thus |B′ 1| ⩾ kk · (k + 1) · (m + k) and S1 is disjoint from � B′ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For a ball B ∈ B′ 1, let the profile of B be the pair consisting of: – the following equivalence relation on S1: s, s′ ∈ S1 are equivalent if they have the same neighborhood in B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' and – the unique equivalence class of the relation above whose members have empty neighbor- hood in B, or ⊥ if there is no such equivalence class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Note that the total number of profiles is at most |S1||S1| · (|S1| + 1) ⩽ kk · (k + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Hence, there exists B2 ⊆ B′ 1 with |B2| ⩾ m + k such that all balls from B2 have the same profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Say this profile is (≡, C), where ≡ is an equivalence relation on S1 and C is either an equivalence class of ≡ or ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let S2 ⊆ S1 be any set consisting of one member of each equivalence class of ≡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Note that for all distinct s, s′ ∈ S2 and B ∈ B2, the neighborhoods of s and s′ in B are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Furthermore, every member of S2 is adjacent to all the balls from B2, except possibly for the vertex chosen from C (if existent), which is non-adjacent to all the balls in B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We now define B2 := (B2, S2, exc2, rep2) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For each vertex u of G, set exc2(u) = Bu, unless Bu /∈ B2, in which case set exc2(u) := ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Finally, set rep2(u) = η(su), where η : S1 → S2 maps every s ∈ S1 to the unique s′ ∈ S2 such that s ≡ s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' It is now straightforward to verify that B2 is a classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' It now remains to apply Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='4 to the classifier B2, yielding a classifier B = (B, S, exc, rep) that is canonical with respect to ≼ and satisfies |S| = |S2| ⩽ k, B ⊆ B2 ⊆ A, and |B| ⩾ m = β(|A|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2 Proof of the result We are ready to prove Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The proof follows closely the reasoning from [DMST22], except that we define the flip set somewhat more carefully in order to ensure the predictability property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 44 Proof of Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Throughout the proof we fix the monadically stable class C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For t ∈ N, by C [t] we denote the class consisting of all ⩽ t-flips of graphs from C , that is, C [t] := {G ⊕ F | G ∈ C and F is a flip set of size at most t in G}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Note that since flips can be simulated by unary predicates, monadic stability of C entails monadic stability of C [t], for every fixed t ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Since we will be using Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='5 for different radii and different classes, in notation we follow the convention that κD s and βD s denote the constant κ and the function β obtained from applying Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='5 to a monadically stable class D and radius s ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The proof proceeds by induction on r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Recall that our goal is to define suitable functions FWr and Predictr, along with bounds λr and αr, for all r ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For this, we may use functions FWr′ and Predictr′ and bounds λr′ and αr′ for r′ < r, obtained from the induction assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Case 1: Base Case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For r = 0, we may simply set FW0(G, ≼, X) := (X, ∅) and Predict0(G, ≼, Z) := ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Thus, we can set the bounds λ0 := 0 and α0(n) := n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Case 2: Inductive Case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We first define the function FWr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For this, let us consider any G ∈ C , ≼ ∈ O(G), and X ⊆ V(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We would like to find a pair (Y, F) satisfying the requirements for the value FWr(G, ≼, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let (Yr−1, Fr−1) := FWr−1(G, ≼, X) and H := G ⊕ Fr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By induction, Fr−1 consists of at most λr−1 flips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Thus H ∈ D, where D := C [λr−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Moreover, Yr−1 has size at least αr−1(|X|) and is (r − 1)-independent in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For convenience, we denote r′ := ⌈r/2⌉ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let A be the family of r′-balls in H whose centers are the vertices of Yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Note that the balls of A are pairwise disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Apply Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='5 to radius r′, graph H ∈ D, order ≼, and family of r′-balls A, thus obtain- ing a classifier B = (B, S, exc, rep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Hence we have |B| ⩾ βD r′ (|A|) and |S| ⩽ κD r′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By Ramsey’s Theorem, we may select a subfamily B⋆ ⊆ B of size at least log |B| satisfying the following property: either the centers of the balls in B⋆ are pairwise at distance greater than r in H, or they are pairwise at distance exactly r in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We set αr(n) := � log βD r′ (αr−1(n)) � 45 and note that by the construction, we have |B⋆| ⩾ αr(|X|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We define Y to be the set of centers of the balls in B⋆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' thus |Y| = |B⋆| ⩾ αr(|X|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' It remains to construct a set of flips F such that Y is distance-r independent in G ⊕ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We proceed by cases, each time exposing a flip set F′ such that we may set F := Fr−1△F′ and |F′| ⩽ max(4, k2 · 4k), where k := κD r′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Note that thus we will have |F| ⩽ λr, assuming we set λr := λr−1 + max(4, k2 · 4k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Also, we will have G ⊕ F = H ⊕ F′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1: The vertices of Y are pairwise at distance greater than r in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In this case Y is already distance-r independent, hence we may simply set F′ := ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2: The vertices of Y are pairwise at distance exactly r in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We may assume that |Y| ⩾ 5, for otherwise we can choose F′ to be a set of at most 4 flips that isolate every vertex of Y in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In what follows, whenever speaking about adjacencies or distances, we mean adjacencies and distances in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Recall that B is canonical with respect to ≼, hence s = min ≼ ΠB(s) for every s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By definition, every vertex of S is adjacent either to all the balls in B⋆, or to none.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Further, since vertices of S have pairwise different neighborhoods in every ball B ∈ B⋆, there is at most one vertex of S that is not adjacent to any ball of B⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let S′ ⊆ S consist of those vertices of S that are adjacent to every ball in B⋆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' thus either S′ = S or |S − S′| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let W := � s∈S′ ΠB(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We observe that the vertices of W are the ones that keep the vertices of Y at close distance, in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ▷ Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='7 For every vertex v ∈ V(G), the following conditions are equivalent: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' v belongs to W;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' v is at distance exactly r′ + 1 from all the vertices of Y, possibly except for one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' v is at distance at most r′ + 1 from at least two vertices of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Implication (2)→(3) is trivial due to |Y| ⩾ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For implication (1)→(2) we use that B is a classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let s ∈ S′ be such that v ∈ ΠB(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By definition, s is adjacent to all the balls in B⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Therefore, v is adjacent to all the balls in B⋆, possibly except for exc(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Recalling that v ∈ exc(v) in case v ∈ � B⋆, we conclude that v is 46 at distance exactly r′ + 1 from the centers of all the balls in B⋆, that is, vertices of Y, possibly except for one — the center of exc(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We are left with implication (3)→(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let s ∈ S be such that v ∈ ΠB(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' As v is at distance at most r′ + 1 from the centers of two balls in B⋆, at least one of them, say B, is different from exc(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In particular v /∈ B, so v being at distance r′ + 1 from the center of B means that v has to be adjacent to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' As B ̸= exc(v), we infer that s is also adjacent to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' It follows that s ∈ S′, implying that v ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ◁ Next, we make a case distinction depending on whether r is odd or even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In both cases, we use the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For s ∈ S and U ⊆ S, we write Qs,U := {v ∈ ΠB(s) | NH(v) ∩ S = U}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Further, we let Q := {Qs,U : s ∈ S, U ⊆ S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Note that Q is a partition of the vertex set of H into at most k · 2k parts, and the definition of Q only depends on the graph H, partition ΠB, and set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In order to later prove the predictability property, it will be crucial that, in both of the following two cases, the definition of the exposed set of flips F′ only depends on the partition Q (and therefore on H, ΠB, and S), the set S′, and the order ≼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1: r is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We define F′ as the set of all pairs (Qs1,U1, Qs2,U2) ∈ Q2 satisfying the following conditions: – s1, s2 ∈ S′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' – Qs1,U1 ̸= ∅ and Qs2,U2 ̸= ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' – s1 ∈ U2 or s2 ∈ U1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' and – min≼ Qs1,U1 ≼ min≼ Qs2,U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Thus |F′| ⩽ (|Q| 2 ) ⩽ k2 · 4k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' As desired, F′ depends only on Q, S′, and ≼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The following claim explains the flip set F′ in more friendly terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ▷ Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='8 For any u1, u2 ∈ V(G), applying F′ flips the adjacency between u1 and u2 if and only if u1, u2 ∈ W and (u2 ∈ NH(rep(u1)) or u1 ∈ NH(rep(u2))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let s1, U1, s2, U2 be such that u1 ∈ Qs1,U1 and u2 ∈ Qs2,U2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' in particular Qs1,U1 ̸= ∅ and Qs2,U2 ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By symmetry, we may assume that min≼ Qs1,U1 ≼ min≼ Qs2,U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By definition, the adjacency between u1 and u2 is flipped when applying F′ if and only if (Qs1,U1, Qs2,U2) ∈ F′, which in turn is equivalent to the conjunction of conditions s1, s2 ∈ S′ and (s1 ∈ U2 or s2 ∈ U1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' It now remains to note that condition s1, s2 ∈ S′ is equivalent to u1, u2 ∈ W, and condition (s1 ∈ U2 or s2 ∈ U1) is equivalent to (u2 ∈ NH(rep(u1)) or u1 ∈ NH(rep(u2))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ◁ Further, we note that the vertices of W may only lie outside the balls of B⋆ or on their boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ▷ Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='9 If v ∈ W, then for every y ∈ Y we have distH(v, y) ⩾ r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 47 B1 B2 y1 y2 v1 v2 s1 = rep(v1) FIGURE 5: Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 in a nutshell: Up to symmetry, the (depicted in red) adjacency between v1 and v2 is the same as between s1 and v2, hence the edge v1v2 is flipped away when applying F′ if and only if it was present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Suppose distH(v, y) ⩽ r′ − 1 for some y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' As v ∈ W, by Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='7 there exists some other y′ ∈ Y, y′ ̸= y, such that distH(v, y′) = r′ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Hence distH(y, y′) ⩽ 2r′ = r − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This is a contradiction with the assumption that Y is (r − 1)-independent in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ◁ We are now ready to argue the following: Y is distance-r independent in H ⊕ F′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' See Figure 6 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For contradiction, suppose in H ⊕ F′ there exists a path P of length at most r connecting some distinct y1, y2 ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let B1, B2 ∈ B⋆ be the r′-balls with centers y1, y2, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Since the flips of F′ only affect the adjacency between the vertices of W, and these vertices have to be at distance at least r′ = r−1 2 from y1, y2 due to Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='9, we infer the following: P can be written as P = (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , y2), where (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , v1) and (v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , y2) are paths of length r′ in H that are entirely contained in B1 in B2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In particular, P has length exactly 2r′ + 1 = r and v1v2 is the only edge on P that might have been flipped when applying F′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Observe that if the edge v1v2 appeared when applying the flip F′, then we necessarily have v1, v2 ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Otherwise, if v1v2 was present in H, then path P witnesses that already in H, both v1 and v2 are at distance at most r′ + 1 from both y1 and y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='7, this implies that v1, v2 ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' So in any case, we have v1, v2 ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let s1 := rep(v1) and s2 := rep(v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Since v1 ∈ B1 and v2 ∈ B2, we have exc(v1) = B1 and exc(v2) = B2, hence NH(s1) ∩ B2 = NH(v1) ∩ B2 and NH(s2) ∩ B1 = NH(v2) ∩ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In particular, v1, v2 are adjacent in H ⇔ v1, s2 are adjacent in H ⇔ v1 ∈ NH(s2), and similarly v1, v2 are adjacent in H ⇔ s1, v2 are adjacent in H ⇔ v2 ∈ NH(s1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Therefore, v1, v2 are adjacent in H ⇔ (v1 ∈ NH(s2) or v2 ∈ NH(s1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' As v1, v2 ∈ W, by Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='8 we conclude that v1 and v2 are adjacent in H if and only if their ad- 48 B1 B2 y1 y2 v1 v2 s = rep(u) u FIGURE 6: Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2 in a nutshell: Up to symmetry, the (depicted in red) adjacency between v1 and u is the same as between v1 and s, hence the edge uv1 is flipped away when applying F′ if and only if it was present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' jacency gets flipped when applying F′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' So v1 and v2 are non-adjacent in H ⊕ F′, a contradiction with the existence of the edge v1v2 on P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2: r is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This time, F′ is defined as the set of all pairs (Qs1,U1, Qs2,U2) ∈ Q2 satis- fying the following conditions: – Qs1,U1 ̸= ∅ and Qs2,U2 ̸= ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' – (s1 ∈ S′ and s1 ∈ U2) or (s2 ∈ S′ and s2 ∈ U1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' and – min≼ Qs1,U1 ≼ min≼ Qs2,U2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Again, |F′| ⩽ |Q|2 ⩽ k2 · 4k and the definition of F′ depends only on Q, S′, and ≼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Also, we may similarly explain flipping according to F′ as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ▷ Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='10 For any u1, u2 ∈ V(G), applying F′ flips the adjacency between u1 and u2 if and only if (u1 ∈ W and u2 ∈ NH(rep(u1))) or (u2 ∈ W and u1 ∈ NH(rep(u2))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Analogous to the proof of Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='8, we leave the details to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ◁ Note that Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='8 implies in particular that whenever the adjacency between two vertices is flipped when applying F′, at least one of them belongs to W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' (However, contrary to the odd case, there might be flips in F′ that affect vertices outside of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=') In this vein, the following observation will be convenient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ▷ Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='11 W ∩ � B⋆ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For contradiction, suppose there exists B ∈ B⋆ and v ∈ B such that v ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Letting y be the center of B, we have distH(v, y) ⩽ r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='7, there exists another y′ ∈ Y, y′ ̸= y, such that distH(v, y′) ⩽ r′ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Hence distH(y, y′) ⩽ 2r′ + 1 = r − 1, contradicting the distance-(r − 1) independence of Y in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ◁ As in the odd case, we are left with arguing that Y is distance-r independent in H ⊕ F′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For contradiction, suppose that there exist distinct y1, y2 ∈ Y and a path P of length at most r that connects y1 and y2 in H ⊕ F′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' As before, let B1, B2 ∈ B⋆ be the balls with centers y1, y2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='10, the flips of F′ affect only the vertices of W ∪ � s∈S′ NH(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='11 and as S′ is disjoint with � B⋆, all vertices of W ∪ � s∈S′ NH(s) are at distance (in H) at least r′ 49 from all the vertices of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Since r = 2r′ + 2, similarly as in Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 it follows that P has length 2r′ + 1 = r − 1 or 2r′ + 2 = r and can be written as P = (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , y2) or P = (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , v1, u, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , y2), where (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , v1) and (v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , y2) are paths of length r′ in H entirely contained in B1 and B2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Consider the first case: P has length r − 1 and is of the form (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Ob- serve that edge v1v2 cannot be present in H, because then P would be entirely contained in H, a contradiction with distance-(r − 1) independence of Y in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' On the other hand, note that v1, v2 /∈ W due to Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='11, so by Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='10 the adjacency between v1 and v2 is not flipped when applying F′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We conclude that v1 and v2 remain non-adjacent in H ⊕ F′, a contradiction with the presence of the edge v1v2 on P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We are left with the second case: P has length r and is of the form (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , v1, u, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let us first argue that u ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' If u is adjacent both to v1 and to v2 in H, then u is at distance at most r′ + 1 from both y1 and y2 in H, hence that u ∈ W follows directly from Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' On the other hand, if u is non-adjacent in H to one of v1 or v2, say to v1, then the adjacency between u and v1 must get flipped when applying F′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='10 this means that at least one of u and v1 belongs to W, but it cannot be v1 due to Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' So u ∈ W in this case as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let s := rep(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By symmetry, we may assume that B1 ̸= exc(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This means that v1, u are adjacent in H ⇔ v1, s are adjacent in H ⇔ v1 ∈ NH(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Since u ∈ W and v1 /∈ W (due to Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='11), by Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='10 we conclude that u and v1 are adjacent in H if and only if their adjacency gets flipped when applying F′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' So in any case, u and v1 are non-adjacent in H ⊕ F′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This is a contradiction with the presence of the edge uv1 on P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This concludes the construction of the pair (Y, F) that we set for FWr(G, ≼, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We are left with defining a suitable inductive case for the function Predictr and showing that it can be computed efficiently, in time OC ,r(|V(G)|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For G ∈ C and Z ⊆ V(G) with |Z| = 5, Predictr(G, ≼, Z) is defined as the flip set F◦ output by the following procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' – Let F◦ r−1 := Predictr−1(G, ≼, Z), where the function Predictr−1 is provided by the induction assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let H◦ := G ⊕ F◦ r−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' – If Z is not distance-(r − 1) independent in H◦, output F◦ := ∅ and terminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' – Similarly, if Z is distance-r independent in H◦, output F◦ := F◦ r−1 and terminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' – Otherwise, let B◦ consist of the five r′-balls in H◦ with centers in vertices of Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' note that the balls of B◦ are pairwise disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Apply the algorithm of Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2 to the family B◦, thus obtaining a partition Π◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' – Let S◦ := {min≼ A : A ∈ Π◦} and let S′◦ be the subset of vertices of S◦ that are adjacent to every ball of B◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' – Output the flip set F◦ := F◦ r−1△F′◦, where F′◦ is defined from H◦, Π◦, S◦, and S′◦ exactly in the way described in Cases 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 50 We now argue that provided FWr(G, ≼, X) = (Y, F) and Z ⊆ Y is a set of size 5, we have Predictr(G, ≼, Z) = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Adopt the notation from the definition of FWr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We revisit the case study presented above and show that in each case, the set F◦ output by the procedure defining Predictr(G, ≼, Z) coincides with the set of flips F constructed in the definition of FWr(G, ≼, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By the induction assumption, we have F◦ r−1 = Fr−1, which implies that H◦ = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In partic- ular, as Z ⊆ Y is distance-(r − 1) independent in H, the termination in the second point above cannot happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Also, if the vertices of Y are pairwise at distance more than r in H, then so is the case for Z, and we have F◦ = F◦ r−1 (termination in the third point above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In the definition of FWr(G, ≼, X) Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 applies here, yielding F = Fr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' So F = Fr−1 = F◦ r−1 = F◦, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We are left with Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2: the vertices of Y are pairwise at distance exactly r in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2 we have Π◦ = ΠB, where B = (B, S, exc, rep) is the classifier provided by Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='5 that is used in the construction of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Since B is canonical with respect to ≼, we have S = {min ≼ A : A ∈ ΠB} = {min ≼ A : A ∈ Π◦} = S◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Similarly, as B is a classifier in H = H◦, we have that a vertex from S = S◦ is adjacent to every ball of B⋆ ⊆ B if and only if it is adjacent to every ball of B◦, and we conclude that S′ = S′◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' As now H◦ = H, Π◦ = ΠB, S◦ = S, and S′◦ = S′, the construction presented in Cases 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2 provides the same flip set for H◦, Π◦, S◦, and S′◦, as for H, ΠB, S, and S′: we have F◦ = F, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Finally, we need to argue that Predictr(G, ≼, Z) can be computed in time OC ,r(|V(G)|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For this, observe that the procedure presented above executes r inductive calls, each of which consists of internal computation that is easy to implement in time OC ,r(|V(G)|2), and one call to the algorithm of Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Since we a priori know that the partition Π◦ returned by this call should be of size OC ,r(1), we may terminate this call once the elapsed running time exceeds OC ,r(|V(G)|2), and if so, return ∅ as Predictr(G, ≼, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Therefore, each of the r inductive calls runs in time OC ,r(|V(G)|2), giving a total time complexity of OC ,r(|V(G)|2) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 11 Winning strategy We are almost ready to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Before commencing to the proof, we will first clarify the notion of a strategy for Flipper, and what we mean by a running time of a strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Furthermore, in this section, we will work with an extension of the Flipper game, which we call Induced-Subgraph-Flipper game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In this variant of the game, we allow Connector to localize the graph to an induced subgraph of an r-ball in the current arena, instead of requiring her to pick an entire r-ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Intuitively, “losing” additional vertices on purpose yields no benefit to Connector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' However, the ability to work with induced subgraphs is useful for the design of algorithms, as exhibited in [DMS].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For this reason we will explicitly prove the algorithmic winning strategy for Flipper for this variant of the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' As the modification only expands the move pool of Connector, the proven strategy then also works for the unmodified Flipper game and implies Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 Strategies and runtimes Strategies and runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Strategies are commonly represented by functions mapping the history of the game to a new (played) position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In our context, it will be convenient to use the following 51 equivalent abstraction, which will fit better to our algorithmic perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Fix radius r ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Graphs considered in consecutive rounds of the Induced-Subgraph-Flipper game will be often called arenas, for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' A radius-r Connector strategy is a function con: (Gi) �→ (Gloc i ) that maps the arena Gi at round i to a graph Gloc i that is an induced subgraph of the ball of radius r around some vertex v in Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' A radius-r Flipper strategy is a function flip: (Gloc i , Ii) �→ (F, Ii+1) that maps the graph Gloc i obtained from Connector’s move to the atomic flip F chosen by Flip- per;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' the next arena will be Gi+1 := Gloc i ⊕ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Additionally, we allow Flipper to keep an auxil- iary memory: the strategy takes, as the second argument, an internal state Ii from the previous round, and outputs an updated internal state Ii+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The initial state I0 = I0(flip, G) will be computed from the initial graph at the beginning of the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The internal states will be used as memory and to precompute flips for future turns, which makes them convenient from an algorithmic point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Strategies operating with game histories instead of internal states can simulate the latter in the following sense: knowing the game history, Flipper can compute the current internal state by replaying the entire game up to the current round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Note that since we are interested in Flipper’s strategies that work against any behavior of Connector, it is not necessary to equip Connector’s strategies with memory as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Given radius-r Connector and Flipper strategies con and flip, and a graph G, we define the run R(con, flip, G) to be the infinite sequence of positions R(con, flip, G) := (G0, I0), (G1, I1), (G2, I2), (G3, I3), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' such that G0 = G, I0 = I0(flip, G), and for all i ⩾ 0 we have Gi+1 = con(Gi) ⊕ F, where (Ii+1, F) = flip(con(Gi), Ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' A winning position is a tuple (Gi, Ii) where Gi contains only a single vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' A radius-r Flipper strategy flip is ℓ-winning on a class of graphs C , if for every G ∈ C and for every radius-r Connector strategy con, the ℓth position of R(con, flip, G) is a winning position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Note that while R(con, flip, G) is an infinite sequence, once a winning position is reached, it is only followed by winning positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let r ∈ N and let flip be a radius-r Flipper strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For a function f : N → N, we say that flip has runtime f if the following holds: – given a graph G, the internal state I0(flip, G) can be computed in time f (|G|);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' and – given a graph H and an internal state I, the value flip(H, I) can be computed in time f (|G|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Note that in the second item above, the time complexity is allowed to depend on the original graph G, which is possibly much larger than the current arena H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' On the other hand, we do not require a dependence on the size of the encoding of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Namely, it will always be the case that in positions that may appear in runs of flip on graphs from the considered class of graphs, 52 the encoding size of I will be linear in the encoding size of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Hence, positions with larger encoding size of I can be just ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' (Formally, the algorithm outputs anything on them while not reading the whole internal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=') We will often say that a strategy has runtime F for a class of functions F to indicate that it has runtime f ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' (For instance, we may say that a strategy has runtime OC ,r(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=') Playing multiple flips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' As discussed in Section 3, we may also consider the variant of Flipper game where in every round, Flipper can apply not a single atomic flip F, but a set of flips F of size at most g(i), where i is the index of the round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Here, g: N → N is a function and we call this variant of the game g-bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We may also speak about the k-bounded variant of the game where k ∈ N, and by this we mean the g-bounded game for g being the constant function g(i) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Thus, the standard game is 1-bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The notions of strategies, runs, and runtimes translate to the setting of g-bounded Flipper game naturally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The following simple lemma shows that when designing a strategy for Flipper on a graph class, it suffices to consider the setting where playing multiple flips in a single move is allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 Let C be a class of graphs and r ∈ N be a fixed radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Suppose that for some ℓ ∈ N and functions f, g: N → N, Flipper has a strategy in the g-bounded radius-r Induced-Subgraph-Flipper game that is ℓ-winning on C , and moreover this strategy has runtime f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Then Flipper also has a strategy in the standard (1-bounded) radius-r Induced-Subgraph-Flipper game that is ℓ′-winning on C , where ℓ′ = ∑ℓ i=1 g(i), and this strategy has runtime Og,ℓ( f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let flip be the assumed strategy in the g-bounded game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We define a strategy flip′ in the 1-bounded game as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' When playing flip′, Flipper simulates flip by replacing the ith move in the g-bounded game by g(i) consecutive moves in the 1-bounded game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' More precisely, supposing that flip proposes to play a flip set F of size at most g(i), in flip′ Flipper plays the atomic flips of F one by one, in |F| consecutive rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Within the formal framework of strategies, these atomic flips are saved in a queue within the internal state, and popped from the queue one by one until the queue is empty — and the next move of flip in the simulated game needs to be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' The moves of Connector along the way are ignored, except for the last one, which is considered the next Connector’s move in the simulated g-bounded game for the purpose of computing the next Flipper’s move proposed by flip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' A straightforward induction argument shows that for every j ∈ N, the arena after ∑ j i=1 g(i) rounds in the 1-bounded game played according to flip′ is an induced subgraph of the arena after j rounds in the simulated g-bounded game played according to flip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Consequently, flip′ is ℓ′-winning on C for ℓ′ = ∑ℓ i=1 g(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' As for the runtime, the algorithm computing the next move of flip′ either pops the next atomic flip from the queue stored in the internal state, or, in case the queue is empty, invokes the algorithm to compute the next move of flip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' It is straightforward to see that this can be done in time Og,ℓ( f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2 Finalizing the argument With the definitions above settled, we can now rephrase and prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='5 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2 For every monadically stable class of graphs C and radius r ∈ N, there exists ℓ ∈ N and a radius-r Flipper strategy for the Induced-Subgraph-Flipper game that is ℓ-winning on C and has runtime OC ,r(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 53 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In notation, we fix the objects provided by Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1 for the class C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let then t := α−1 2r (7), k := λ2r, and ℓ := 2 · ��t 5 � + 1 �2 , where by α−1 2r (7) we mean the least integer N such that α2r(N) ⩾ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We will describe a strategy flip⋆ for Flipper in the g-bounded radius-r game, where g(i) := max(i, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Strategy flip⋆ will be ℓ-winning on C and will have runtime OC ,r(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' By Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1, this suffices to prove Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We explain now flip⋆ in natural language;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' the easy translation to the formal layer of strate- gies with internal states, described in Section 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1, is left to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We fix the graph G ∈ C on which the game is played, together with an arbitrary linear order ≼ on V(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' First, Flipper will always play moves in move pairs: Having constructed some flip set F, Flip- per first applies F to the current arena H, then lets the Connector localize the game to a radius-r ball in H ⊕ F, and finally he applies F again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In this way, the following invariant will be satis- fied: After applying every move pair, the arena is an induced subgraph of G (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='2 and Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We will say that a move pair as described above is defined by F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Second, Flipper proceeds in a sequence of eras, each consisting of a number of consecutive moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Along the way, he keeps track of a growing chain of vertex subsets ∅ = X0 ⊊ X1 ⊊ X2 ⊊ X3 ⊊ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , where Xi is obtained from Xi−1 at the end of era i by adding one vertex that is still contained in the arena, but not contained in Xi−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Up until Flipper wins the game, we will ensure that such a vertex always exists, and therefore |Xi| = i for every i ∈ N until the game concludes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We now describe Flipper’s moves in era i (i = 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For every Z ⊆ Xi−1 with |Z| = 5, we compute the flip set FZ := Predict2r(G, ≼, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Note that, instead of the current arena, the original graph G is used to compute FZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' First, Flipper performs (|Xi−1| 5 ) move pairs, each defined by FZ for a different Z as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Finally, let F be a flip set of size |Xi−1| = i − 1 such that in G ⊕ F, every vertex of Xi−1 is isolated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' (Such F can be obtained by iteratively isolating vertices of Xi−1 by performing a flip between a vertex and its neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=') At the end of the era, Flipper applies the move pair defined by F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' After the first application of F within the move pair, the resulting arena is an induced subgraph of G ⊕ F where all the vertices of Xi−1 are isolated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Therefore, the induced subgraph chosen as Connector’s response must contain a vertex x not belonging to Xi−1, otherwise Connector loses immediately after making her move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Followingly, we may set Xi := Xi−1 ∪ {x} and proceed to the next era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This concludes the description of flip⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Clearly, flip⋆ is a valid strategy in the g-bounded game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We now argue that following flip⋆ leads to a quick victory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ▷ Claim 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='3 If Flipper follows flip⋆, the game concludes within at most t eras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' For contradiction, suppose the game enters era t + 1 without termination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 54 Denote X := Xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' we have |X| = t = α−1 2r (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let (Y, F) := FW2r(G, ≼, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Thus, |Y| ⩾ 7 and Y is distance-2r independent in G ⊕ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , y7 be any seven distinct vertices of Y, where yi was added earlier to X than yj for all i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let Z := {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' , y5} and FZ := Predict2r(G, ≼, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Further, let s be the index of the era that concluded with adding y6 to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' (That is, we have Xs = Xs−1 ∪ {y6} and in particular Xs−1 ∩ {y6, y7} = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=') Note that Z ⊆ Xs−1, hence within era s, Flipper applied the move pair defined by FZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Let H be the arena during that era right before the first application of FZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Note that H is an induced subgraph of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' After the first application of FZ, Connector responded by restricting the arena to an induced subgraph H′ of some radius-r ball in H ⊕ FZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Clearly, the vertex set of H′ is entirely contained in some radius-r ball in G ⊕ FZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Since Y is distance-2r independent in G ⊕ FZ and y6, y7 ∈ Y, we conclude {y6, y7} ⊈ V(H′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' In other words, at least one of the vertices y6, y7 got removed from the arena as a consequence of Connector’s move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' This contradicts the assumption that both y6 and y7 were later added to X, which requires them to both be contained in the arena at the end of era s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' ◁ Note that in era i, Flipper applies exactly (|Xi−1| 5 ) + 1 = (i−1 5 ) + 1 move pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Hence, by Claim 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='3, the game terminates within at most t ∑ i=1 2 · ��i − 1 5 � + 1 � ⩽ ℓ rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' We conclude that flip⋆ is ℓ-winning on C , as promised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Finally, computing Flipper’s moves for an era boils down to at most (t 5) = OC ,r(1) applications of the algorithm provided by Theo- rem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='1, which runs in time OC ,r(n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' It follows that flip⋆ has runtime OC ,r(n2) (in a suitable formalization of the strategy through internal states).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' References [AA14] Hans Adler and Isolde Adler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Interpreting nowhere dense graph classes as a classical notion of model theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' Cambridge Uni- versity Press, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' [war16] Algorithms, Logic and Structure Workshop in Warwick – Open Prob- lem Session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' https://warwick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='uk/fac/sci/maths/people/staff/ daniel_kral/alglogstr/openproblems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content='pdf, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' accessed 23- Jan-2023].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} +page_content=' 57' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNFST4oBgHgl3EQfKDg1/content/2301.13735v1.pdf'} diff --git a/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf b/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..742a978db76ecbd76472f81709027be47673d735 --- /dev/null +++ b/n9FST4oBgHgl3EQfMDgV/content/2301.13742v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9d770053a66fc197bda635aa922aa3953c0daa83ba845b6ec1d51cabae6d153c +size 13183409 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In this paper, we consider the Yang–Mills–Higgs flow for twisted Higgs pairs +over K¨ahler manifolds. We prove that this flow converges to a reflexive twisted Higgs +sheaf outside a closed subset of codimension 4, and the limiting twisted Higgs sheaf is +isomorphic to the double dual of the graded twisted Higgs sheaves associated to the +Harder–Narasimhan–Seshadri filtration of the initial twisted Higgs bundle. +1. +Introduction +Let (X, ω) be a compact K¨ahler manifold, (V, hV ) be a Hermitian holomorphic vector +bundle over X. A V -twisted Higgs bundle is a pair (E, φ), where E is a holomorphic +vector bundle and φ : E → V ⊗ E is a holomorphic morphism satisfying 0 = φ ∧ φ ∈ +End(E) ⊗ ∧2V . For untwisted Higgs bundle (i.e. V = T ∗X), it was first studied by +Hitchin ([23]) on Riemann surface, and by Simpson ([39, 40]) on K¨ahler manifold. There +are many interesting research about twisted Higgs bundles (see [7, 18, 19, 20, 36], etc.). +Let H be a Hermitian metric on E. Then we can define a dual morphism φ∗H : E⊗V → E +by using the Hermitian metrics H and hV . Let [φ, φ∗H] = φ ◦ φ∗H − φ∗H ◦ φ ∈ End(E). A +Hermitian metric H is said to be Higgs–Hermitian–Einstein if it satisfies +√ +−1ΛωF¯∂E,H + [φ, φ∗H] = λ · IdE, +(1.1) +where F¯∂E,H is the curvature form of the Chern connection D¯∂E,H and λ = +2πµω(E) +Vol(X,ω) is +a constant. The Donaldson–Uhlenbeck–Yau theorem for twisted Higgs bundles ([2, 8, +23, 39]) guarantees the existence of Higgs–Hermitian–Einstein metrics for the polystable +case. It was originally proved by Narasimhan–Seshadri ([34]), Donaldson ([15, 16]) and +Uhlenbeck–Yau ([42]) for holomorphic bundles. +There are also many interesting and +important generalized Donaldson–Uhlenbeck–Yau theorems (see [4, 5, 6, 9, 25, 31, 32, 33, +37, 44, 45] and references therein). +Given a fixed Hermitian metric H on E. Let A1,1 +H be the space of integrable unitary con- +nections, and BH be the space of V -twisted Higgs pairs. The Yang–Mills–Higgs functional +on BH is defined by +YMH(A, φ) = +� +X +(|FA|2 + 2|∂A,V φ|2 + |[φ, φ∗]|2 − 2⟨φ, φ⟩V )dvg, +(1.2) +2020 Mathematics Subject Classification. 53C07, 58E15. +Key words and phrases. Yang–Mills–Higgs flow, twisted Higgs pair, Harder–Narasimhan–Seshadri +filtration. +1 + +2 +C. Pan et al. +where dvg = ωn +n! , ⟨φ, φ⟩V = tr(φ√−1ΛωFhV φ∗H). The critical point of the Yang–Mills– +Higgs functional satisfies +� +D∗ +AFA + (∂A − ¯∂A)[φ, φ∗H] = 0, +[ +√ +−1ΛωFA + [φ, φ∗H], φ] = 0. +(1.3) +We say that (A, φ) ∈ BH is a Yang–Mills–Higgs pair if it is a critical point of the +Yang–Mills–Higgs functional. By the K¨ahler identities, we know that if (A, φ) satisfies +√−1ΛωFA + [φ, φ∗H] = λ · IdE, then it is a Yang–Mills–Higgs pair and H is the Higgs– +Hermitian–Einstein metric of V -twisted Higgs bundle (E, ¯∂A, φ). The gradient flow of the +Yang–Mills–Higgs functional is + + + + + +∂A(t) +∂t += −D∗ +A(t)FA(t) − (∂A(t) − ¯∂A(t))[φ(t), φ∗H(t)], +∂φ(t) +∂t += −[ +√ +−1ΛωFA(t) + [φ(t), φ∗H(t)], φ(t)]. +(1.4) +The existence of long time solution for the above gradient flow will be discussed in Section +2. If the initial data (A0, φ0) ∈ BH is stable, the heat flow converges to a V -twisted Higgs +pair and the limit must lie in the same orbit of the initial data. In this article, we are +interested in the convergence of this flow in the general case. +Let (E, ¯∂A, φ) be a V -twisted Higgs bundle. There is a filtration of (E, ¯∂A) given by φ- +invariant subsheaves which is called Harder–Narasimhan–Seshadri (abbr. HNS) filtration. +Let GrHNS(E, ¯∂A, φ) be the associated graded object (the direct sum of the stable quo- +tients) of the Harder–Narasimhan–Seshadri filtration. For the holomorphic vector bundle, +Atiyah–Bott ([1]) and Bando–Siu ([3]) conjectured that there should be a correspondence +between the limit of the Yang–Mills flow and the double dual of GrHNS(E, ¯∂A). It was +proved by Daskalopoulos ([13]), Daskalopoulos–Wentworth ([14]) Jacob ([26]) and Sibley +([38]) in different cases. For the untwisted Higgs bundle, Wilkin ([43]) and Li–Zhang +([28, 29]) also proved the similar correspondence between Yang–Mills–Higgs flow and the +double dual of GrHNS(E, ¯∂A, φ). In [30], the authors proved this conjecture for reflexive +sheaves. In the meanwhile, Zhang ([46]) considered this problem for T ∗X ⊗ L-twisted +Higgs bundles over Riemann surface, and he proved the related correspondence in that +case. In the present paper, we extend the above results to V -twisted Higgs bundles over +K¨ahler manifolds. In fact, we prove the following theorem. +Theorem 1.1. Let (A(t), φ(t)) be a solution of Yang–Mills–Higgs flow (1.4) with initial +data (A0, φ0) ∈ BH. Then we have: +(1) for every sequence tk → +∞, there is a subsequence tkj such that as tkj → ++∞, (A(tkj), φ(tkj)) converges modulo gauge transformations to a pair (A∞, φ∞) +satisfying (1.3) on the Hermitian vector bundle (E∞, H∞) in C∞ +loc topology outside +Σ, where Σ is a closed set of Hausdorff codimension at least 4. The limiting +(E∞, ¯∂A∞, φ∞) can be extended to the whole X as a reflexive V -twisted Higgs +sheaf with a holomorphic orthogonal splitting +(E∞, H∞, ¯∂A∞, φ∞) = ⊕l +i=1(Ei +∞, Hi +∞, ¯∂Ai∞, φi +∞), +(1.5) + +Yang–Mills–Higgs Flow for Twisted Higgs Pairs +3 +where Hi +∞ is an admissible Higgs–Hermitian–Einstein metric on the reflexive V - +twisted Higgs sheaf (Ei +∞, ¯∂Ai∞, φi +∞). +(2) let {Ei,j} be the HNS filtration of the V -twisted Higgs bundle (E, ¯∂A0, φ0), the +associated graded object GrHNS(E, A0, φ0) = ⊕l +i=1 ⊕li +j=1 Qi,j be uniquely deter- +mined by the isomorphism class of (A0, φ0). +We have GrHNS(E, A0, φ0)∗∗ ≃ +(E∞, ¯∂A∞, φ∞). +In order to prove the convergence of the flow, we proved the energy inequality, the +monotonicity formula of certain quantities and the ǫ-regularity. Following Hong–Tian’s +arguments ([24]) and using Bando–Siu’s extension technique ([3]), we obtain the first part +of Theorem 1.1. The proof of the second part of Theorem 1.1 can be divided into two +steps. The first step is to prove that the Harder–Narasimhan (abbr. HN) type of the +limiting V -twisted Higgs sheaf is in fact equal to that of (E, A0, φ0). The second step is +to construct a non-zero φ-invariant holomorphic map from Qi,j to the limiting sheaf. The +idea of the proof is the same as for untwisted case ([28, 29]), but there are some differences +in the treatment of certain details. +This paper is organized as follows. In Section 2, we build some basic estimates for +Donaldson heat flow and the Yang–Mills–Higgs flow for twisted Higgs pairs. In Section 3, +we prove the monotonicity inequality and the ǫ-regularity estimate for the Yang–Mills– +Higgs flow and complete the first part of Theorem 1.1. In Section 4, we first prove that +the HN type of the limiting twisted Higgs sheaf is in fact equal to the type of the initial +twisted Higgs bundle, and then complete the proof of the second part of Theorem 1.1. +2. Preliminary +Let (X, ω) be an n-dimensional compact K¨ahler manifold, and (E, φ) be a V -twisted +Higgs vector bundle over X. We consider the following Donaldson heat flow + + + +H−1(t)∂H(t) +∂t += −2( +√ +−1ΛωFH(t) + [φ, φ∗H(t)] − λ · IdE), +H(0) = H0. +(2.1) +It is a strictly parabolic equation, so the standard parabolic theory gives the short time +existence. And the long time existence can be proved by the same method of Simpson’s +article ([39]). The following lemma can be obtained by direct calculation. +Lemma 2.1. Let H(t) be the solution of the flow (2.1) and set Φ(H(t)) = √−1ΛωFH(t) + +[φ, φ∗H(t)] − λ · IdE, then +� ∂ +∂t − ∆ +� +tr(Φ(H(t))) = 0, +(2.2) +and +� ∂ +∂t − ∆ +� +|Φ(H(t))|2 +H(t) = −4|¯∂EΦ(H(t))|2 +ω,H(t) − 4|[φ, Φ(H(t))]|2 +hV ,H(t). +(2.3) +Let (E, H0) be a Hermitian vector bundle, and BH0 be the space of V -twisted Higgs +pairs. Let GC (resp. G) be the complex gauge group (resp. unitary gauge group) of + +4 +C. Pan et al. +(E, H0). The complex gauge group GC acts on BH0 as follows +σ · (¯∂A, φ) = (σ ◦ ¯∂A ◦ σ−1, σ ◦ φ ◦ σ−1), +∀σ ∈ GC, (A, φ) ∈ BH0. +(2.4) +Following the methods in [15, 28], we have the following proposition. +Proposition 2.2. There is a family of complex gauge transformations σ(t) ∈ GC such +that (A(t), φ(t)) = σ(t) · (A0, φ0) is a long time solution of the Yang–Mills–Higgs flow +(1.4) with the initial data (A0, φ0), where σ∗H0(t)σ(t) = H−1 +0 H(t) and H(t) is the long +time solution of Donaldson heat flow (2.1) for (E, ¯∂A0, φ0). +Along the Yang–Mills–Higgs flow, we have the following energy identity. +Proposition 2.3. Let (A(t), φ(t)) be a solution of the Yang–Mills–Higgs flow (1.4) with +initial twisted Higgs pair (A0, φ0). Then +YMH(A(t), φ(t)) + 2 +� t +0 +� +X +����� +∂A +∂t +���� +2 ++ +���� +∂φ +∂t +���� +2 ++ +���� +∂φ∗ +∂t +���� +2� +dvgdt = YMH(A0, φ0). +(2.5) +Proof. Let (At, φt) be a family of twisted Higgs pairs and +d +dt +��� +t=0(At, φt) = ( ˙A, ˙φ), then +d +dt +��� +t=0YMH(At, φt) =2ℜ +� +X +(⟨DA ˙A, FA⟩ + 2⟨[ ˙A1,0, φ] + ∂A,V ˙φ, ∂A,V φ⟩ ++ ⟨[ ˙φ, φ∗] + [φ, ˙φ∗], [φ, φ∗]⟩ − 2⟨ ˙φ, φ⟩V )dvg, +where +⟨[ ˙A1,0, φ], ∂A,V φ⟩ = ⟨ ˙A1,0, [∂A,V φ, φ∗]⟩ = −⟨ ˙A0,1, [φ, ¯∂A,V ∗φ∗]⟩, +⟨ ˙φ, ∂∗ +A,V ∂A,V φ⟩ − ⟨ ˙φ, φ⟩V = ⟨ ˙φ, [ +√ +−1ΛωFA, φ]⟩ = −⟨ ˙φ∗, [ +√ +−1ΛωFA, φ∗]⟩, +and +⟨[ ˙φ, φ∗], [φ, φ∗]⟩ = ⟨ ˙φ, [[φ, φ∗], φ]⟩, +⟨[φ, ˙φ∗], [φ, φ∗]⟩ = −⟨ ˙φ∗, [[φ, φ∗], φ∗]⟩. +Since ¯∂A,V φ = 0 and ∂A,V ∗φ∗ = 0, we have +d +dt +��� +t=0YMH(At, φt) =2 +� +X +(⟨ ˙A, D∗ +AFA⟩ + ⟨ ˙A, (∂A − ¯∂A)[φ, φ∗]⟩ ++ ⟨ ˙φ, [ +√ +−1ΛωFA + [φ, φ∗], φ]⟩ − ⟨ ˙φ∗, [ +√ +−1ΛωFA + [φ, φ∗], φ∗]⟩)dvg. +Using the Yang–Mills–Higgs flow equation (1.4), we have +d +dtYMH(A(t), φ(t)) = −2 +� +X +����� +∂A +∂t +���� +2 ++ +���� +∂φ +∂t +���� +2 ++ +���� +∂φ∗ +∂t +���� +2� +dvg. +(2.6) +Integrating the above equality (2.6) from 0 to t gives (2.5). +□ +Proposition 2.4. Let (A(t), φ(t)) be a solution of the heat flow (1.4) with initial twisted +Higgs pair (A0, φ0). Then +� ∂ +∂t − ∆ +� +|φ|2 +H0,hV ≤ − 2|∂A,V φ|2 − C3(|φ|2 + 1)2 + C4(|φ|2 + 1), +(2.7) + +Yang–Mills–Higgs Flow for Twisted Higgs Pairs +5 +where C3, C4 are constants depending on supX |√−1ΛωFhV |. Moreover, we have +sup +X +|φ|2 ≤ max{sup +X +|φ0|2, C4/C3}. +(2.8) +Proof. By direct calculation, there holds +� ∂ +∂t − ∆ +� +|φ|2 +H0,hV = − 2|∂A,V φ|2 − 2⟨φ, [[φ, φ∗], φ]⟩ + 2⟨φ, +√ +−1ΛωFhV (φ)⟩ += − 2|∂A,V φ|2 − 2|[φ, φ∗]|2 + 2⟨φ, +√ +−1ΛωFhV (φ)⟩. +Let {vi} be a local orthonormal frame of V and φ = φivi. Since φ ∧ φ = 0, we have +φiφj = φjφi and +|[φ, φ∗]|2 = +� +i,j +tr([φi, φ∗ +i ][φj, φ∗ +j]) = +� +i,j +tr([φi, φ∗ +j][φj, φ∗ +i ]) += +� +i,j +|[φi, φ∗ +j]|2 ≥ +� +i +|[φi, φ∗ +i ]|2. +According to the Lemma 2.7 in [40], we have +|[φi, φi]|2 ≥ C1(|φi|2 + 1)2 − C2(|φi|2 + 1). +The above calculation leads to the proof. +□ +In the following, we derive the local energy monotonic inequality along the flow. Let +e2(A, φ) = |FA|2 + 2|∂A,V φ|2 and f ∈ C∞(X), then +d +dt +� +X +f 2e2(A, φ)dvg =2ℜ +� +X +��∂A +∂t , D∗ +A(f 2FA) +� ++ f 2 +�∂A +∂t , (∂A − ¯∂A)[φ, φ∗] +� ++ 2 +�∂φ +∂t , ∂∗ +A,V (f 2∂A,V φ) +�� +dvg, +where +D∗ +A(f 2FA) = +√ +−1[Λω, ¯∂A − ∂A](f 2FA) += +√ +−1Λω(¯∂ − ∂)(f 2) ∧ FA − f 2D∗ +AFA − (¯∂ − ∂)(f 2) +√ +−1ΛωFA, +and +∂∗ +A,V (f 2∂A,V φ) = +√ +−1Λω ¯∂(f 2) ∧ ∂A,V φ + f 2∂∗ +A,V ∂A,V φ. + +6 +C. Pan et al. +Therefore, +d +dt +� +X +f 2e2(A, φ)dvg += 2 +� +X +f 2 +��∂A +∂t , D∗ +AFA +� ++ +�∂A +∂t , (∂A − ¯∂A)[φ, φ∗] +� ++ +�∂φ +∂t , [ +√ +−1ΛωFA, φ] +� +− +�∂φ∗ +∂t , [ +√ +−1ΛωFA, φ∗] +�� +dvg ++ 2ℜ +� +X +��∂A +∂t , +√ +−1Λω(¯∂ − ∂)(f 2) ∧ FA +� +− +�∂A +∂t , +√ +−1(¯∂ − ∂)(f 2)ΛωFA +� ++ 2 +�∂φ +∂t , +√ +−1Λω ¯∂(f 2) ∧ ∂A,V φ +� ++ 2f 2 +�∂φ +∂t , φ +�� +dvg. +Let f be a cut-off function on B2R(x0), satisfy 0 ≤ f ≤ 1, f ≡ 1 on BR(x0) and |df| ≤ 2 +R. +Then +���� +d +dt +� +X +f 2e2(A, φ)dvg + 2 +� +X +f 2 +����� +∂A +∂t +���� +2 ++ +���� +∂φ +∂t +���� +2 ++ +���� +∂φ∗ +∂t +���� +2� +dvg +���� +≤C1 +R +� � +X +����� +∂A +∂t +���� +2 ++ +���� +∂φ +∂t +���� +2 ++ +���� +∂φ∗ +∂t +���� +2� +dvg +�1/2 ++ C2 +� � +X +����� +∂A +∂t +���� +2 ++ +���� +∂φ +∂t +���� +2 ++ +���� +∂φ∗ +∂t +���� +2� +dvg +�1/2 +, +where C1, C2 are constants depending on supX |√−1ΛωFhV |, supX |φ0|, YMH(A0, φ0) and +the geometry of (X, ω). Then we have the following proposition. +Proposition 2.5. Let (A(t), φ(t)) be a solution of the Yang–Mills–Higgs flow (1.4), then +for any B2R(x0) ⊂ X, s, τ, we have +� +BR(x0) +e2(A, φ)(·, s)dvg +≤ +� +B2R(x0) +e2(A, φ)(·, τ)dvg + 2 +� max{s,τ} +min{s,τ} +� +X +����� +∂A +∂t +���� +2 ++ +���� +∂φ +∂t +���� +2 ++ +���� +∂φ∗ +∂t +���� +2� +dvgdt ++ C1 +�|s − τ| +R2 +� max{s,τ} +min{s,τ} +� +X +����� +∂A +∂t +���� +2 ++ +���� +∂φ +∂t +���� +2 ++ +���� +∂φ∗ +∂t +���� +2� +dvgdt +�1/2 ++ C2 +� +|s − τ| +� max{s,τ} +min{s,τ} +� +X +����� +∂A +∂t +���� +2 ++ +���� +∂φ +∂t +���� +2 ++ +���� +∂φ∗ +∂t +���� +2� +dvgdt +�1/2 +, +where C1, C2 are constants depending on supX |√−1ΛωFhV |, supX |φ0|, YMH(A0, φ0) and +the geometry of (X, ω). +Let θ(t) = √−1ΛωFA(t) + [φ(t), φ∗(t)] and +I(t) = +� +X +(|DA(t)θ(t)|2 +ω,H0 + 2|[φ(t), θ(t)]|2 +hV ,H0)dvg. +Then we have the following proposition: + +Yang–Mills–Higgs Flow for Twisted Higgs Pairs +7 +Proposition 2.6. +I(t) → 0, +as t → +∞. +(2.9) +Proof. The proof is exactly the same as untwisted case ([28]), so we omit here. +□ +Let ∇A,V be the induced connection on Ω∗ +X ⊗End(E) ⊗V induced by DA, DhV and the +Chern connection on TM and ∇A be the covariant derivative corresponding to DA. +Proposition 2.7. Along the Yang–Mills–Higgs flow (1.4), we have +� ∂ +∂t − ∆ +� +|∂A,V φ|2 + 2|∇A,V ∂A,V φ|2 +≤C1(|φ|2 + |FA| + |FhV | + |Ric|)|∂A,V φ|2 + C2|∂A +√ +−1ΛωFhV ||φ||∂A,V φ| +(2.10) +and +� ∂ +∂t − ∆ +� +|FA|2 + |∇AFA|2 +≤C3(|FA| + |φ|2 + |Rm|)|FA|2 + C4|FA||∂A,V φ|2 + C5|φ|2|FhV ||FA|, +(2.11) +where the constants Ci(i = 1, · · · , 5) are depending only on the dimension n. +Proof. In local normal coordinates, we have +∆|∂A,V φ|2 = 2|∇A,V ∂A,V φ|2 + 2⟨∇α∇¯α∂A,V φ, ∂A,V φ⟩ + 2⟨∂A,V φ, ∇¯α∇α∂A,V φ⟩, +where +∇α∇¯α∂A,V φ =∇α∇¯α∇βφdzβ += − ∇α(FA,V ;β ¯αφ)dzβ += − ∇α(FA,V ;β ¯α)φdzβ − FA,V ;β ¯α∇αφdzβ += − ∇β(FA,V ;α¯α)φdzβ − FA,V ;β,¯α∇αφdzβ += − ∂A( +√ +−1ΛωFA,V )φ − FA,V ;β,¯α∇αφdzβ += − [∂A( +√ +−1ΛωFA), φ] − ∂A( +√ +−1ΛωFhV )φ − FA,V ;β,¯α∇αφdzβ, +∇¯α∇α∂A,V φ =∇¯α∇α∇βφdzβ + ∇βφ∇¯α∇αdzβ +=∇¯α∇β∇αφdzβ + ∇βφ∇¯α∇αdzβ +=∇β∇¯α∇αφdzβ − FA,V ;β ¯α∇αφdzβ + ∇βφ∇¯α∇αdzβ += − ∂A( +√ +−1ΛωFA,V φ) − FA,V ;β ¯α∇αφdzβ + ∇βφ∇¯α∇αdzβ += − ∂A([ +√ +−1ΛωFA, φ] + +√ +−1ΛωFhV φ) − FA,V ;β ¯α∇αφdzβ + ∇βφ∇¯α∇αdzβ += − [∂A +√ +−1ΛωFA, φ] + [ +√ +−1ΛωFA, ∂Aφ] + (∂A +√ +−1ΛωFhV )φ ++ +√ +−1ΛωFhV ∂Aφ − FA,V ;β ¯α∇αφdzβ + ∇βφ∇¯α∇αdzβ. + +8 +C. Pan et al. +On the other hand, using the heat flow equations (1.4), we have +∂ +∂t|∂A,V φ|2 +ω,hV ,H0 =2ℜ +���∂A1,0 +∂t , φ +� +, ∂A,V φ +� ++ +� +∂A,V +∂φ +∂t , ∂A,V φ +�� += − 2ℜ +� +⟨[∂A( +√ +−1ΛωFA + [φ, φ∗]), φ], ∂A,V φ⟩ ++ ⟨∂A,V [ +√ +−1ΛωFA + [φ, φ∗], φ], ∂A,V φ⟩ +� += − 2ℜ +� +2⟨[∂A( +√ +−1ΛωFA + [φ, φ∗]), φ], ∂A,V φ⟩ ++ ⟨[ +√ +−1ΛωFA + [φ, φ∗], ∂A,V φ], ∂A,V φ⟩ +� +. +Then (2.10) follows from the above identities. The proof of the other equation (2.11) is +similar and we omit here. +□ +3. Convergence of the Yang–Mills–Higgs flow for twisted Higgs pairs +In this section, we consider the convergence of the Yang–Mills–Higgs flow (1.4) for +twisted Higgs pairs on the Hermitian bundle (E, H0). We first prove the monotonicity in- +equality and the ǫ-regularity theorem for the flow. We will adapt the same arguments used +in studying the Yang–Mills flow ([10, 11]) and the Yang–Mills–Higgs flow for untwisted +Higgs pairs ([28]) to the Yang–Mills–Higgs flow for twisted Higgs pairs. +Let u = (x, t) ∈ X × R. For any u0 = (x0, t0) ∈ X × R+, set +Sr(u0) = X × {t = t0 − r2}, +Tr(u0) = X × [t0 − 4r2, t0 − r2], +Pr(u0) = Br(x0) × [t0 − r2, t0 + r2]. +For simplicity, we denote Sr(0, 0), Tr(0, 0), Pr(0, 0) by Sr, Tr, Pr. +The fundamental solution of (backward) heat equation with singularity at u0 = (x0, t0) +is +Gu0(x, t) = G(x0,t0)(x, t) = +1 +(4π(t0 − t))n exp +� +− |x − x0|2 +4(t0 − t) +� +, +t ≤ t0. +For simplicity, denote G(0,0)(x, t) by G(x, t). +Given 0 < R ≤ iX, we take f ∈ C∞ +0 (BR) satisfying 0 ≤ f ≤ 1, f ≡ 1 on BR/2 and +|∇f| ≤ 2/R on BR \ BR/2. Let (A(t), φ(t)) be a solution of the Yang–Mills–Higgs flow +with initial value (A0, φ0) and set +e2(A, φ) = |FA|2 + 2|∂A,V φ|2, +Φ(r) = r2 +� +Tr(u0) +e2(A, φ)f 2Gu0dvgdt. +Then we have the following theorem. +Theorem 3.1. Let (A(t), φ(t)) be a solution of the Yang–Mills–Higgs flow (1.4) with ini- +tial value (A0, φ0). For any u0 = (x0, t0) ∈ X×[0, T] and 0 < r1 ≤ r2 ≤ min{R/2, √t0/2}, + +Yang–Mills–Higgs Flow for Twisted Higgs Pairs +9 +we have +Φ(r1) ≤C exp(C(r2 − r1))Φ(r2) + C(r2 +2 − r2 +1) ++ CR2−2n +� +PR(u0) +e2(A, φ)dvgdt, +(3.1) +where the constant C depends only on the geometry of (X, ω), supX |√−1ΛωFhV | and the +initial data (A0, φ0). +Proof. Choosing normal geodesic coordinates {xi}2n +i=1 in the geodesic ball BR(x0), then it +follows that +|gij(x) − δij| ≤ C|x|2, +|∂kgij(x)| ≤ C|x|, +∀x ∈ Br(x0), +(3.2) +where C is a positive constant depending only on x0. +Let x = r˜x, t = t0 + r2˜t. There holds that +Φ(r) = r2 +� +Tr(u0) +e2(A, φ)f 2Gu0dvgdt += r2 +� t0−r2 +t0−4r2 +� +R2n e2(A, φ)(x, t)f 2(x)Gu0(x, t) +� +det (gij)(x)dxdt += r4 +� −1 +−4 +� +R2n e2(A, φ)(r˜x, t0 + r2˜t)f 2(r˜x)G(˜x, ˜t) +� +det (gij)(r˜x)d˜xd˜t. +Then one can see that +dΦ(r) +dr += 4r3 +� −1 +−4 +� +R2n e2(A, φ)(r˜x, t0 + r2˜t)f 2(r˜x)G(˜x, ˜t) +� +det (gij)(r˜x)d˜xd˜t ++ r3 +� −1 +−4 +� +R2n{xi∂ie2(A, φ)(r˜x, t0 + r2˜t)}f 2(r˜x)G(˜x, ˜t) +� +det (gij)(r˜x)d˜xd˜t ++ r3 +� −1 +−4 +� +R2n{2(t − t0)∂te2(A, φ)(r˜x, t0 + r2˜t)}f 2(r˜x)G(˜x, ˜t) +� +det (gij)(r˜x)d˜xd˜t ++ r4 +� −1 +−4 +� +R2n e2(A, φ)(r˜x, t0 + r2˜t) d +dr{f 2(r˜x) +� +det (gij)(r˜x)}G(˜x, ˜t)d˜xd˜t +=I1 + I2 + I3 + I4. +For the second term I2, we have +I2 =r +� +Tr(u0) +{xi∂ie2(A, φ)(x, t)}f 2(x)Gu0(x, t) +� +det (gij)(x)dxdt, +where +xi∂ie2(A, φ) =2ℜ(⟨xi∇iFA, FA⟩ + 2⟨xi∇i∂A,V φ, ∂A,V φ⟩). + +10 +C. Pan et al. +By the Bianchi identity, we have +2⟨xi∇iFA, FA⟩ =⟨xi∇iFA(∂j, ∂k)dxj ∧ dxk, FA⟩ +=⟨xi∇jFA(∂i, ∂k)dxj ∧ dxk, FA⟩ + ⟨xi∇kFA(∂j, ∂i)dxj ∧ dzk, FA⟩ +=2⟨xiDA(FA,ikdxk) − xi(FA(∇j∂i, ∂k) + FA(∂i, ∇j∂k))dxj ∧ dxk, FA⟩ +=2⟨DA(xiFA,ikdxk) − xiFA(∇j∂i, ∂k)dxj ∧ dxk, FA⟩ − 4|FA|2. +Set +x ⊙ FA = 1 +2xiFA,ijdxj, +we have +ℜ⟨x ⊙ FA, D∗ +AFA⟩ = −ℜ⟨x ⊙ FA, ∂A +∂t ⟩ + ℜ⟨x ⊙ FA, (¯∂A − ∂A)[φ, φ∗]⟩. +(3.3) +In addition, +⟨xi∇i∇A,V φ, ∇A,V φ⟩ =⟨xi∇i(∇jφdxj), ∇A,V φ⟩ +=⟨xi∇j∇iφdxj, ∇A,V φ⟩ + ⟨xiFA,V ;ijφdxj, ∇A,V φ⟩ ++ ⟨xi∇jφ∇idxj), ∇A,V φ⟩ +=⟨DA,V (xi∇iφ), ∇A,V φ⟩ − ⟨xi∇A,V φ(∇i∂j)dxj), ∇A,V φ⟩ ++ ⟨xiFA,V ;ijφdxj, ∇A,V φ⟩ − |∇A,V φ|2, +and +⟨xi∇iφ, ∂∗ +A,V ∂A,V φ⟩ =⟨xi∇iφ, [ +√ +−1ΛωFA, φ] + +√ +−1ΛωFhV φ⟩ += − ⟨xi∇iφ, dφ +dt ⟩ − ⟨xi∇iφ, [[φ, φ∗], φ] + +√ +−1ΛωFhV φ⟩. +Note that, for any α ∈ Ω1(End(E)), α∗ = −α, we have +ℜ⟨α, (¯∂A − ∂A)[φ, φ∗]]⟩ + 2ℜ⟨[α, φ], ∂A,V φ⟩ = 0. + +Yang–Mills–Higgs Flow for Twisted Higgs Pairs +11 +Set x ⊙ ∇A,V φ = 1 +2xi∇A,V ;iφ. Since Gu0 > 0, we have +I1 + I2 =4r +� +Tr(u0) +|∂A,V φ|2f 2Gu0dvgdt +− 4rℜ +� +Tr(u0) +⟨d(f 2Gu0) ∧ x ⊙ FA, FA⟩dvgdt +− 8rℜ +� +Tr(u0) +⟨d(f 2Gu0)x ⊙ ∇A,V φ, ∇A,V φ⟩dvgdt +− 4rℜ +� +Tr(u0) +⟨x ⊙ FA, dA +dt ⟩f 2Gu0dvgdt +− 8rℜ +� +Tr(u0) +⟨x ⊙ ∇A,V φ, dφ +dt ⟩f 2Gu0dvgdt +− 8rℜ +� +Tr(u0) +⟨x ⊙ ∇A,V φ, [[φ, φ∗], φ] + +√ +−1ΛωFhV φ⟩f 2Gu0dvgdt +− 2rℜ +� +Tr(u0) +⟨xiFA(∇j∂i, ∂k)dxj ∧ dxk, FA⟩f 2Gu0dvgdt +− 4rℜ +� +Tr(u0) +⟨xi∇A,V φ(∇j∂i)dxj, ∇A,V φ⟩f 2Gu0dvgdt ++ 8rℜ +� +Tr(u0) +⟨x ⊙ FV φ, ∇A,V φ⟩f 2Gu0dvgdt. +For the second term I3, we have +I3 = 2r +� +Tr(u0) +(t − t0)∂te2(A, φ)(x, t)f 2(x)Gu0(x, t) +� +det (gij)(x)dxdt, +where +∂te2(A, φ) = 2ℜ +�� +DA +�∂A +∂t +� +, FA +� ++ 2 +��∂A1,0 +∂t , φ +� +, ∂A,V φ +� ++ 2 +� +∂A,V +∂φ +∂t , ∂A,V φ +�� +. +So we obtain that +I3 = − 4rℜ +� +Tr(u0) +(t − t0) +����∂A +∂t +��� +2 ++ 2 +���∂φ +∂t +��� +2� +f 2Gu0dvgdt +− 4rℜ +� +Tr(u0) +(t − t0) +� +d(f 2Gu0) ∧ ∂A +∂t , FA +� +dvgdt +− 8rℜ +� +Tr(u0) +(t − t0) +� +d(f 2Gu0)∂φ +∂t , ∇A,V φ +� +dvgdt +− 8rℜ +� +Tr(u0) +(t − t0) +�∂φ +∂t , [[φ, φ∗], φ] + +√ +−1ΛωFhV φ +� +f 2Gu0dvgdt. + +12 +C. Pan et al. +Note that ∂iGu0 = +xiGu0 +2(t−t0). Set +x · FA = 1 +2gijxjFA,ikdxk, +x · ∇A,V φ = 1 +2xjgij∇A,V ;iφ, +∇f · FA = 2gijf −1∂jfFA;ik, +∇f · ∇A,V φ = 2gijf −1∂jf∇iφ. +For any α ∈ Ω1(End(E)), β ∈ Γ(V ⊗ End(E)), we have +⟨d(f 2Gu0) ∧ α, FA⟩ = ⟨α, ∇f · FA⟩f 2Gu0 + +1 +t − t0 +⟨α, x · FA⟩f 2Gu0, +and +⟨d(f 2Gu0)β, ∇A,V φ⟩ = ⟨β, ∇f · ∇A,V φ⟩f 2Gu0 + +1 +t − t0 +⟨β, x · ∇A,V φ⟩f 2Gu0. +Combining the above inequalities, we have +I1 + I2 + I3 += 4r +� +Tr(u0) +1 +|t − t0| +���|t − t0|∂A +∂t − x ⊙ FA +��� +2 +f 2Gu0dvgdt ++ 4rℜ +� +Tr(u0) +1 +|t − t0| +� +x · FA − x ⊙ FA, x ⊙ FA − |t − t0|∂A +∂t +� +f 2Gu0dvgdt ++ 4rℜ +� +Tr(u0) +� +|t − t0|∂A +∂t − x ⊙ FA, ∇f · FA +� +f 2Gu0dvgdt ++ 8r +� +Tr(u0) +1 +|t − t0| +���|t − t0|∂φ +∂t − x ⊙ ∇A,V φ +��� +2 +f 2Gu0dvgdt ++ 8rℜ +� +Tr(u0) +1 +|t − t0| +� +x · ∇A,V φ − x ⊙ ∇A,V φ, x ⊙ ∇A,V φ − |t − t0|∂φ +∂t +� +f 2Gu0dvgdt ++ 8rℜ +� +Tr(u0) +� +|t − t0|∂φ +∂t − x ⊙ ∇A,V φ, ∇f · ∇A,V φ +� +f 2Gu0dvgdt +− 8rℜ +� +Tr(u0) +⟨x ⊙ ∇A,V φ, [[φ, φ∗], φ] + +√ +−1ΛωFhV φ⟩f 2Gu0dvgdt +− 2rℜ +� +Tr(u0) +⟨xiFA(∇j∂i, ∂k)dxj ∧ dxk, FA⟩f 2Gu0dvgdt +− 4rℜ +� +Tr(u0) +⟨xi∇A,V φ(∇j∂i)dxj, ∇A,V φ⟩f 2Gu0dvgdt ++ 8rℜ +� +Tr(u0) +⟨x ⊙ FV φ, ∇A,V φ⟩f 2Gu0dvgdt +− 8rℜ +� +Tr(u0) +(t − t0) +�∂φ +∂t , [[φ, φ∗], φ] + +√ +−1ΛωFhV φ +� +f 2Gu0dvgdt ++ 4r +� +Tr(u0) +|∂A,V φ|2f 2Gu0dvgdt. + +Yang–Mills–Higgs Flow for Twisted Higgs Pairs +13 +By the Lemma 2.1 and Proposition 2.4, we have +���∂φ +∂t +��� +2 +≤ C, +|φ|2 ≤ C, +where the constant C depends on the initial data (A0, φ0) and supX |√−1ΛωFhV |. Since +r ≤ R ≤ iX. According to the Yang’s inequality, we have +I1 + I2 + I3 ≥ − Cr +� +Tr(u0) +1 +|t − t0||x · FA − x ⊙ FA|2f 2Gu0dvgdt +− Cr +� +Tr(u0) +|t − t0||∇f · FA|2f 2Gu0dvgdt +− Cr +� +Tr(u0) +1 +|t − t0||x · ∇A,V φ − x ⊙ ∇A,V φ|2f 2Gu0dvgdt +− Cr +� +Tr(u0) +|t − t0||∇f · ∇A,V φ|2f 2Gu0dvgdt +− Cr +� +Tr(u0) +|x|2|∇A,V φ|2f 2Gu0dvgdt +− 2rℜ +� +Tr(u0) +⟨xiFA(∇j∂i, ∂k)dxj ∧ dxk, FA⟩f 2Gu0dvgdt +− 4rℜ +� +Tr(u0) +⟨xi∇A,V φ(∇j∂i)dxj, ∇A,V φ⟩f 2Gu0dvgdt +− Cr, +where the constant C depends on the initial data (A0, φ0) and supX |√−1ΛωFhV |. +For the last term I4, we have +I4 =r +� +Tr(u0) +e2(A, φ)xi∂i(f 2� +det (gij))Gu0dxdt +=r +� +Tr(u0) +e2(A, φ)2xif∂i(f)Gu0dvgdt + r +� +Tr(u0) +e2(A, φ)f 2xi∂i( +� +det (gij))Gu0dxdt +=r +� +Tr(u0) +e2(A, φ)2xif∂i(f)Gu0dvgdt + r +2 +� +Tr(u0) +e2(A, φ)xitr(g−1∂ig)f 2Gu0dvgdt. +Since +|gij − δij| ≤ C|x|2, +|∂igjk| ≤ C|x|, +|Γk +ij| ≤ C|x|, +then +|x · FA − x ⊙ FA|2 ≤ C|x|6|FA|2, +|x · ∇A,V φ − x ⊙ ∇A,V φ|2 ≤ C|x|6|∇A,V φ|2, +⟨xiFA(∇k∂i, ∂j)dxj ∧ dxk, FA⟩ ≤ C|x|2|FA|2, +tr(g−1∂ig) ≤ C|x|. + +14 +C. Pan et al. +Hence, we have +dΦ(r) +dr +≥ − Cr +� +Tr(u0) +|x|6 +|t − t0|e2(A, φ)f 2Gu0dvgdt +− Cr +� +Tr(u0) +|t − t0||∇f · FA|2f 2Gu0dvgdt +− Cr +� +Tr(u0) +|t − t0||∇f · ∇A,V φ|2f 2Gu0dvgdt +− Cr +� +Tr(u0) +|x|2e2(A, φ)f 2Gu0dvgdt +− Cr +� +Tr(u0) +f|∇f||x|e2(A, φ)Gu0dvgdt +− Cr. +According to the Chen–Struwe’s arguments in [12], we know there exists a constant ˜C4 > 0 +such that +r−1|t − t0| · |x|6Gu0 ≤ ˜C4(1 + Gu0), +r−1|x|2Gu0 ≤ ˜C4(1 + Gu0) +on Tr(u0). Then it follows that +− Cr +� +Tr(u0) +� |x|6 +|t − t0| + |t − t0| + |x|2� +e2(A, φ)f 2Gu0dvgdt +≥ −CΦ(r) − CrYMH(A0, φ0), +According to the arguments in [35, P. 1384], we have +−r +� +Tr(u0) +|t − t0| · |∇f · FA|2f 2Gu0dvgdt ≥ −C(n)r +R2n +� +PR(u0) +|FA|2dvgdt, +−r +� +Tr(u0) +|t − t0| · |∇f · ∇A,V φ|2f 2Gu0dvgdt ≥ −C(n)r +R2n +� +PR(u0) +|∇A,V φ|2dvgdt, +−2r +� +Tr(u0) +|x| · |∇f| · |f| · e2(A, φ)Gu0dvgdt ≥ −C(n)r +R2n +� +PR(u0) +e2(A, φ)dvgdt. +Combining the above inequalities, we have +dΦ(r) +dr +≥ −CΦ(r) − Cr − Cr +R2n +� +PR(u0) +e2(A, φ)dvgdt, +(3.4) +where the constant C depends on the geometry of (X, ω), supX |√−1ΛωFhV | and the +initial data (A0, φ0). By integrating the above inequality (3.4) over r, we complete the +proof. +□ + +Yang–Mills–Higgs Flow for Twisted Higgs Pairs +15 +Theorem 3.2. Let (A(t), φ(t)) be a solution of the Yang–Mills–Higgs flow (1.4). There +exist positive constants ǫ0, δ0 < 1/4, such that if +R2−2n +� +PR(u0) +e2(A, φ)dvgdt < ǫ0 +(3.5) +holds for some 0 < R ≤ min{iX/2, √t0/2}, then for any δ ∈ (0, δ0), we have +sup +PδR(u0) +e2(A, φ) ≤ +16 +(δR)4. +(3.6) +Proof. For any δ ∈ (0, 1/4], we define the function +f(r) = (2δR − r)4 +sup +Pr(x0,t0) +e2(A, φ). +Since f(r) is continuous and f(2δR) = 0, we know that f(r) attains its maximum at some +point r0 ∈ [0, 2δR). Suppose (x1, t1) ∈ ¯Pr0(x0, t0) is a point such that +e2(A, φ)(x1, t1) = +sup +Pr(x0,t0) +e2(A, φ). +We claim that f(r0) ≤ 16 when ǫ0, δ0 are small enough. Otherwise, we have +ρ0 := e2(A, φ)(x1, t1)−1/4 = (2δR − r0)f(r0)−1/4 < δR − r0 +2 . +Rescaling the Riemannian metric ˜g = ρ−2 +0 g, ˜hV = ρ2 +0hV and t = t1 + ρ2 +0˜t, we get +|FA|2 +˜g = ρ4 +0|FA|2 +g, +|∂A,V φ|2 +˜g,˜hV = ρ4 +0|∂A,V φ|2 +g,hV . +Set +eρ0(x, ˜t) = |FA|2 +˜g + 2|∂A,V φ|2 +˜g,˜hV = ρ4 +0e2(A, φ)(x, t1 + ρ2 +0˜t), +˜P˜r(x1, 0) = Bρ0˜r(x1) × [−˜r2, ˜r2]. +Then we have eρ0(x1, 0) = ρ4 +0e(A, φ)(x1, t1) = 1, and +sup +˜P1(x1,0) +eρ0 = ρ4 +0 +sup +Pρ0(x1,t1) +e(A, φ) ≤ ρ4 +0 +sup +PδR+r0/2(x0,t0) +e(A, φ) +≤ ρ4 +0f(δR + r0/2)(δR − r0/2)−4 ≤ 16. +Thus +|FA|2 +˜g + 2|∂A,V φ|2 +˜g,˜hV ≤ 16, +on +˜P1(x1, 0). +(3.7) +Combining above inequalities together with the Proposition 2.7 yields that +� ∂ +∂˜t − ∆˜g +� +eρ0 = ρ6 +0 +� ∂ +∂t − ∆g +� +e2(A, φ) +≤C1ρ6 +0(|φ|2 + |FA| + |FhV | + |Ric|)|∂A,V φ|2 + C2ρ6 +0|∂A +√ +−1ΛωFhV ||φ||∂A,V φ| ++ C3ρ6 +0(|FA| + |φ|2 + |Rm|)|FA|2 + C4ρ6 +0|FA||∂A,V φ|2 + C5ρ6 +0|φ|2|FhV ||FA| +≤C6(eρ0 + ρ8 +0) + +16 +C. Pan et al. +on ˜P1(x1, 0), where the constant C6 depends only on the geometry of (X, ω), FhV and +supX |φ0|H0. Then by the parabolic mean value inequality, we observe +1 < +sup +˜P1/2(x1,0) +(eρ0 + ρ8 +0) ≤ C +� +˜P1(x1,0) +(eρ0 + ρ8 +0)dv˜gd˜t += C7ρ2−2n +0 +� +Pρ0(x1,t1) +e2(A, φ)dvgdt + C7ρ8 +0, +(3.8) +where the constant C7 depends only on the geometry of (X, ω), FhV and supX |φ0|H0. +We choose normal geodesic coordinates centred at x1, and let f ∈ C∞ +0 (BR/2(x1)) be +a smooth cut-off function such that 0 ≤ f ≤ 1, f ≡ 1 on BR/4(x1), |df| ≤ 8/R on +BR/2(x1) \ BR/4(x1). +Taking r1 = ρ0 and r2 = δ0R, and applying the monotonicity +inequality, we obtain +ρ2−2n +0 +� +Pρ0(x1,t1) +e2(A, φ)dvgdt +≤Cρ2 +0 +� +Pρ0(x1,t1) +e2(A, φ)G(x1,t1+2ρ2 +0)f 2dvgdt +≤Cρ2 +0 +� +Tρ0(x1,t1+2ρ2 +0) +e2(A, φ)G(x1,t1+2ρ2 +0)f 2dvgdt +≤C∗r2 +2 +� +Tr2(x1,t1+2ρ2 +0) +e2(A, φ)G(x1,t1+2ρ2 +0)f 2dvgdt + C∗C10δ2 +0R2 ++ C∗(R/2)2−2n +� +PR/2(x1,t1) +e2(A, φ)dvgdt +≤C∗δ2−2n +0 +R2−2n +� +PR(x0,t0) +e2(A, φ)dvgdt + C∗C10δ2 +0R2 +≤C8(δ2−2n +0 +ǫ0 + δ2 +0R2), +where the constant C8 depends only on the geometry of (X, ω), FhV and supX |φ0|H0. +Choosing ǫ0, δ0 small enough such that ˜C4 ˜C5(δ2−2n +0 +ǫ0 + δ2 +0R2) + ˜C4δ8 +0R8 < 1, then a con- +tradiction occurs. So we have f(r0) ≤ 16, which implies +sup +PδR(u0) +e2(A, φ) ≤ 16/(δR)4. +□ +Using the above ǫ-regularity theorem, and following the arguments of Hong and Tian +([24]) for the Yang–Mills flow case, we give the proof of the first part of Theorem 1.1. +Proof of Theorem 1.1 (1). By Proposition 2.3, for any tk → +∞ and a > 0, we have +� tk+a +tk−a +� +X +����∂A +∂t +��� +2 ++ +���∂φ +∂t +��� +2 ++ +���∂φ∗ +∂t +��� +2� +dvgdt → 0, +tk → +∞. + +Yang–Mills–Higgs Flow for Twisted Higgs Pairs +17 +Thus for any ǫ > 0, there is a constant K, when k > K there holds that +� tk+a +tk−a +� +X +����∂A +∂t +��� +2 ++ +���∂φ +∂t +��� +2 ++ +���∂φ∗ +∂t +��� +2 +)dvgdt ≤ ǫ. +Let +Σ = +� +0 0, there exist a finite number of geodesic +balls {Bri(xi)}, ri < δ, such that {Bri(xi)} is a cover of Σ, where xi ∈ Σ and Bri/2(xi) ∩ +Brj/2(xj) = ∅ for any i ̸= j. Since xi ∈ Σ, therefore +r4−2n +i +� +Bri/2(xi) +e2(A, φ)(·, tk)dvg > 24−2nǫ1 +for sufficiently large k. That is, +r2n−4 +i +< 22n−4ǫ−1 +1 +� +Bri/2(xi) +e2(A, φ)(·, tk)dvg, +� +i +r2n−4 +i +< 22n−4ǫ−1 +1 +� +∪iBri/2(xi) +e2(A, φ)(·, tk)dvg < +∞. +This implies that H2n−4(Σ) < +∞. +Convergence: From the previous arguments, for any x0 ∈ X \ Σ, there exist r0 and +{tk} such that +sup +Pr0(x0,tk) +e2(A, φ) ≤ C. +By Uhlenbeck’s weak compactness theorem, there exist a subsequence {tk′} and gauge +transformation {σ(k +′)} such that σ(k +′) · (A(tk′), φ(t +′ +k)) converges to a V -twisted Higgs +pair (A∞, φ∞) on (E∞, H∞) weakly in W 1,2 +loc (X \ Σ) and (A∞, φ∞) is a solution of the +equation (1.3) outside Σ. By the standard parabolic estimates and using Hong–Tian’s +argument (Proposition 6 in [23]), we know that σ(k +′)·(A(tk′), φ(t +′ +k)) converges to (A∞, φ∞) +in C∞ +loc-topology outside Σ. +Holomorphic Orthogonal split: By the equation (1.3), we have +DA∞θ∞ = 0, +[θ∞, φ∞] = 0, +where θ∞ = √−1ΛωFA∞ + [φ∞, φ∗H∞ +∞ +]. +Since θ∞ is parallel and θ∗H∞ +∞ += θ∞, we can +decompose E∞ and φ∞ according to the eigenvalues of θ∞. So we can obtain a holomorphic +orthonogal decomposition +E∞ = ⊕l +i=1Ei +∞, +and +φi +∞ : Ei +∞ → V ⊗ Ei +∞. +Let Hi +∞, φi +∞be the restrict of H∞, φ∞ to Ei +∞, then (Ai +∞, φi +∞) is a V -twisted Higgs pair on +(Ei +∞, Hi +∞) and satisfies +√ +−1ΛωFAi∞ + [φi +∞, (φi +∞)∗Hi +∞] = λiIdEi∞. + +Yang–Mills–Higgs Flow for Twisted Higgs Pairs +19 +Extend to reflexive sheaf: Because φ(t) is bounded, so we have +� +X\Σ +|FA∞|2 +H∞dvg ≤ C. +Since H2n−4(Σ) < +∞, φ∞ is holomorphic and C0 bounded, and every metric Hi +∞ satisfies +the Higgs–Hermitian–Einstein equation, from Theorem 2 in Bando and Siu’s article ([3]), +we know that every (Ei +∞, ¯∂Ai∞) can be extended to the whole X as a reflexive sheaf (which +is also denoted by (Ei +∞, ¯∂Ai∞) for simplicity), φi +∞ and Hi +∞ can be smoothly extended over +the place where the sheaf (Ei +∞, ¯∂Ai∞) is locally free. +□ +Corollary 3.3. Let (A(tk), φ(tk)) be a sequence of V -twisted Higgs pairs along the Yang– +Mills–Higgs flow with the limit (A∞, φ∞). Then +(1) θ(A(tk), φ(tk)) → θ(A∞, φ∞) strongly in Lp as k → +∞ for all 1 ≤ p < +∞ and +limt→+∞ ∥θ(A(t), φ(t))∥2 +L2 = ∥θ(A∞, φ∞)∥2 +L2. +(2) ∥θ(A∞, φ∞)∥L∞ ≤ ∥θ(A(tk), φ(tk))∥L∞ ≤ ∥θ(A(t0), φ(t0))∥L∞ for 0 ≤ t0 ≤ tk. +4. Isomorphism of the limit object and GrHNS(E, ¯∂A0, φ0) +4.1. HNS filtration and Lp-δ-approximate critical Hermitian metric of twisted +Higgs bundles. In this subsection, we show that the HN type of the limit V -twisted +Higgs sheaf is consistent with the initial V -twisted Higgs bundle. The key to the proof is to +obtain the existence of Lp-δ-approximate critical Hermitian metric. First, let’s recall the +Harder–Narasimhan–Seshadri filtration of V -twisted Higgs bundle. The proof is almost +the same as the one used in holomorphic bundles case ([27, Sections 7.15, 7.17, 7.18]). +Lemma 4.1. Let (E, ¯∂A, φ) be a V -twisted Higgs bundle on K¨ahler manifold (X, ω). Then +there is a filtration of E by φ-invariant coherent subsheaves +0 = E0 ⊂ E1 ⊂ · · · ⊂ El = E, +(4.1) +called the Harder–Narasimhan filtration of V -twisted Higgs bundle (E, ¯∂A, φ), such that +Qi = Ei/Ei−1 is torsion-free and semistable. Moreover, µω(Qi) > µω(Qi+1). +Lemma 4.2. Let (F, φF) be a semistable V -twisted Higgs sheaf on K¨ahler manifold (X, ω). +Then there is a filtration of F by φF-invariant coherent subsheaves +0 = F0 ⊂ F1 ⊂ · · · ⊂ Fl = F, +(4.2) +called the Seshadri filtration of V -twisted Higgs bundle (F, φF), such that Qi = Fi/Fi−1 is +torsion-free and stable. Moreover, µω(Qi) = µω(F). +Proposition 4.3. Let (E, ¯∂A, φ) be a V -twisted Higgs bundle on K¨ahler manifold (X, ω). +Then there is a double filtration {Ei,j} of E called φ-invariant Harder–Narasimhan– +Seshadri filtration, such that {Ei}l +i=1 is the HN filtration, and {Ei,j}li +j=1 is a Seshadri fil- +tration of Ei/Ei−1. Let Qi,j = Ei,j/Ei,j−1, the associated graded object GrHNS(E, ¯∂A, φ) = +⊕l +i ⊕li +j=1 Qi,j is uniquely determined by the isomorphism class of (E, ¯∂A, φ). +Definition 4.4. For a V -twisted Higgs bundle (E, ¯∂A, φ) of rank r, construct a nonin- +creasing r-tuple of numbers +⃗µ(E, ¯∂A, φ) = (µ1, · · · , µr) +(4.3) + +20 +C. Pan et al. +from the HN filtration by setting: µi = µω(Qj), for rank(Ej−1) + 1 ≤ i ≤ rank(Ej). We +call ⃗µ(E, ¯∂A, φ) the Harder–Narasimhan type of (E, ¯∂A, φ). +Notice that ⃗µ(E∞, ¯∂A∞, φ∞) = Vol(X,ω) +2π +⃗λ∞ = Vol(X,ω) +2π +(λ1, · · · , λr). In the following, we +assume Vol(X, ω) = 2π. +For any φ-invariant subsheaf S of (E, ¯∂A, φ), let H be a metric on E, πH +S : E → E be +the induced projection map of E to S. It is not hard to get +degω(S) = 1 +2π +� +X +tr(( +√ +−1ΛωFH + [φ, φ∗H])πH +S )dvg − 1 +2π +� +X +(|¯∂AπH +S |2 + |[φ, πH +S ]|2)dvg. +(4.4) +Let G be the unitary gauge group of E, and its Lie algebra is denoted by u(E). The +following proposition can be derived directly from equation (4.4). +Proposition 4.5. Let gj ∈ GC and (Aj, φj) = gj · (A0, φ0) be a sequence of complex gauge +equivalent V -twisted Higgs structure on complex vector bundle E of rank r, S be a φ0- +invariant subsheaf of (E, ¯∂A0, φ0). Suppose √−1ΛωFAj + [φj, φ∗ +j] → a in L1 as j → +∞, +where a ∈ L1(√−1u(E)), and that the eigenvalues λ1 ≥ · · · ≥ λrof +1 +2πa are constant. +Then degω(S) ≤ � +i≤rank(S) λi. +According to (2.2) in Lemma 2.1, we have +r +� +i=1 +µi = degω(E, ¯∂A) = degω(E∞, ¯∂A∞) = +r +� +i=1 +λi. +(4.5) +Let {Ei}l +i=1 be the HN filtration of (E, ¯∂A0, φ0). According to Corollary 3.3 and Proposi- +tion 4.5, we have +� +α≤rank(Ei) +µα = degω(Ei) ≤ +� +α≤rank(Ei) +λα. +(4.6) +That is, +⃗µ(E, ¯∂A0, φ0) ≤ ⃗λ∞. +(4.7) +Let u(r) be the Lie algebra of unitary group U(r). Fixing a real number α ≥ 1, for any +a ∈ u(r), we define the function ϕα : u(r) → R by +ϕα(a) = +r +� +j=1 +|λj|α, +where √−1λj are the eigenvalues of a. It is easy to show that there is a family of smooth +convex ad-invariant functions ϕα,ρ, 0 < ρ ≤ 1, such that ϕα,ρ → ϕα uniformly on compact +subsets of u(r) as ρ → 0. Therefore, from [1, Prop. 12.16], we know that ϕα is convex +function. For any real number N, we define +HYMα,N(A, φ) = +� +X +ϕα( +√ +−1(θ(A, φ) + N · IdE))dvg. +(4.8) + +Yang–Mills–Higgs Flow for Twisted Higgs Pairs +21 +For any ⃗µ = (µ1, · · · , µr), set HYMα,N(µ) = 2πϕα(√−1(⃗µ + N)). Then for any smooth +convex ad-invariant function ϕ, we have +� ∂ +∂t − ∆ +� +ϕ( +√ +−1(θ(A(t), φ(t)) + N · IdE)) ≥ 0, +(4.9) +whose proof can be found in [14, 28]. Since we can approximate ϕα by smooth convex +ad-invariant functions ϕα,ρ → ϕα, by (4.9) we know that HYMα,N(A(t), φ(t)) is nonin- +creasing along the flow. By Corollary 3.3, we can choose a sequence tj → +∞, such that +HYMα,N(A(tj), φ(tj)) → HYMα,N(A∞, φ∞). Then we have +lim +t→+∞ HYMα,N(A(t), φ(t)) = HYMα,N(A∞, φ∞) +(4.10) +for any α ≥ 1 and any N. +Lemma 4.6 ([14]). The functional a → ( +� +X ϕα(a)dvg)1/α defines a norm on Lα(u(E)) +which is equivalent to the Lα norm. +Lemma 4.7 ([14]). +(1) If ⃗µ ≤ ⃗λ, then ϕα(√−1⃗µ) ≤ ϕα(√−1⃗λ) for all α ≥ 1. +(2) Assume µr ≥ 0 and λr ≥ 0. If ϕα(√−1⃗µ) = ϕα(√−1⃗λ) for all α in some set +S ⊂ [1, +∞) possessing a limit point, then ⃗µ = ⃗λ. +Let {Ei} be the HN filtration of E. Given a Hermitian metric H on E, we can define +an L2 +1-Hermitian endomorphism +ΨHN(E, φ, H) = +l +� +i=1 +µi(πH +i − πH +i−1), +where πH +i +is the projection map of E to Ei. +More generally, given a filtration F = +{Fi} of E and real numbers {µi}l +i=1, we can also define an L2 +1-Hermitian endomorphism +Ψ(F, (µ1, · · · , µl), H) = �l +i=1 µi(πH +i − πH +i−1). +Definition 4.8. Fix 0 < p < +∞ and δ > 0. An Lp-δ-approximate critical Hermitian +metric on a V -twisted Higgs bundle (E, φ) is a smooth metric H such that +∥ +√ +−1ΛωFAH + [φ, φ∗H] − ΨHN(E, φ, H)∥Lp ≤ δ, +(4.11) +where AH is the Chern connection determined by (¯∂E, H). +Let {Ei,j} be the HNS filtration of the V -twisted Higgs bundle (E, ¯∂A, φ). Set +Σalg = ∪i,j{Sing(Ei,j) ∪ Sing(Qi,j)}. +(4.12) +It is well known that Σalg is a complex analytic subset of complex codimension at least +two. We call it the singular set of the HNS filtration. Since the HNS filtration fails to be +given by subbundles on the singular set Σalg, it makes difficult to do analysis. +When Σalg = ∅, it is easy to show the existence of L∞-δ-approximate critical Hermitian +metric for any δ > 0. In general case, Sibley use Hironaka’s desingularisation theorem +([21, 22]) to resolve the singularities Σalg and obtain a filtration by subbundles. + +22 +C. Pan et al. +Proposition 4.9 ([38]). Let 0 = E0 ⊂ E1 ⊂ · · · ⊂ El−1 ⊂ El = E be a filtration of a +holomorphic vector bundle E on a complex manifold X by saturated subsheaves and let +Qi = Ei/Ei−1. Then there is a finite sequence of blow-ups along complex submanifolds of +X whose composition π : ˜X → X enjoys the following properties. There is a filtration +0 = ˜E0 ⊂ ˜E1 ⊂ · · · ⊂ ˜El−1 ⊂ ˜El = ˜E +(4.13) +by subbundles such that ˜Ei is the saturation of π∗Ei. If ˜Qi = ˜Ei/ ˜Ei−1, then we have exact +sequences: +0 → Ei → π∗ ˜Ei → Ti → 0 +(4.14) +and +0 → Qi → π∗ ˜Qi → T +′ +i → 0, +(4.15) +where Ti and T +′ +i are torsion sheaves supported on the singular sets of Ei and Qi, respec- +tively, and furthermore π∗ ˜Ei = Ei and Q∗∗ +i = (π∗ ˜Qi)∗∗. +Let φ ∈ Γ(End(E) ⊗ V ) be a twisted Higgs field on holomorphic bundle (E, ¯∂A), ˜V = +π∗V and ˜φ = π∗φ ∈ Γ(End( ˜E) ⊗ ˜V ) be the pullback twisted Higgs field on ˜E. If the +filtration {Ei}l +i=1 is by φ-invariant subsheaves, then the filtration { ˜Ei}l +i=1 in the above +proposition is by ˜φ-invariant subbundles. So, we have the following proposition. +Proposition 4.10. Let {Ei,j} be the HNS filtration of a V -twisted Higgs bundle (E, ¯∂A, φ) +on complex manifold X and let Qi,j = Ei,j/Ei,j−1. Then there is a finite sequence of blow- +ups along complex submanifolds of X whose composition π : ˜X → X enjoys the following +properties. There is a filtration { ˜Ei,j} by ˜φ-subbundles such that ˜Ei,j is the saturation of +π∗Ei,j, π∗ ˜Ei,j = Ei,j and Q∗∗ +i,j = (π∗ ˜Qi,j)∗∗, where ˜φ = π∗φ. +Since the blow-up ˜X is also K¨ahler, we have a family of K¨ahler metrics given by +ωǫ = π∗ω + ǫη on it, where η is a certain K¨ahler metric. +Theorem 4.11 ([38]). Let ˜S be a subsheaf (with torsion free quotient ˜Q) of a holomorphic +vector bundle ˜E on ˜X, where π : ˜X → X is given by a sequence of blow-ups along complex +submanifolds of codim ≥ 2. Then there is a uniform constant C independent of ˜S such +that the degrees of ˜S and ˜Q with respect to ωǫ satisfy: +degωǫ( ˜S) ≤ degω(π∗ ˜S) + ǫC, +and +degωǫ( ˜Q) ≥ degω(π∗ ˜Q) − ǫC. +(4.16) +Corollary 4.12. Let ( ˜E, ˜φ) be a ˜V -twisted Higgs bundle over ˜X, (E, φ) be a V -twisted +Higgs bundle on X satisfy E = π∗ ˜E and ˜φ = π∗φ. If (E, φ) is ω-stable, then there is an +ǫ1 such that ( ˜E, ˜φ) is ωǫ-stable for all 0 < ǫ ≤ ǫ1. Let ⃗µǫ( ˜E, ˜φ) be the HN type of ( ˜E, ˜φ) +with respect to ωǫ, then ⃗µǫ( ˜E, ˜φ) → ⃗µ(E, φ) as ǫ → 0. +Lemma 4.13 ([38]). Let (X, ω) be a compact K¨ahler manifold of complex dimension +n, and π : ˜X → X be a blow-up along a smooth complex submanifold Σ of complex +codimension k ≥ 2. Let η be a K¨ahler metric on X, and consider the family of K¨ahler +metrics ωǫ = π∗ω + ǫη, 0 < ǫ < ǫ1. Then for any α and ˜α such that 1 < α < 1 + +1 +2(k−1) +and +α +1−2(k−1)(α−1) < ˜α < +∞. +Let s = +˜α +˜α−α, we have +ηn +ωn +ǫ +∈ L2(α−1)s( ˜X, η), and the +L2(α−1)s-norm of ηn +ωn +ǫ is uniformly bounded in ǫ. + +Yang–Mills–Higgs Flow for Twisted Higgs Pairs +23 +Lemma 4.14 ([38]). Let π : ˜X → X be a blow-up along a smooth complex submanifold +Σ of complex codimension k and the family of metrics ωǫ be the same as in the previous +lemma. Let F be a (1, 1)-form with values in End( ˜E). Let 1 < α < 1 + +1 +4k(k−1) and +α +1−2(k−1)(α−1) < ˜α < 1 + +1 +2(k−1). Then there is a number κ0 such that for any 0 < κ ≤ κ0, +there exists a constant C independent of ǫ, ǫ1, and κ, and a constant C(κ) such that: +∥ΛωǫF∥Lα( ˜ +X,ωǫ) ≤ C(∥Λωǫ1F∥L˜α( ˜ +X,ωǫ1) + κ∥F∥L2( ˜ +X,ωǫ1)) + ǫ1C(κ)∥F∥L2( ˜ +X,ωǫ1). +(4.17) +Using Donaldson’s argument, we can obtain the following proposition. +Proposition 4.15. Let (E, φ) be a V -twisted Higgs bundle on a smooth K¨ahler manifold +(X, ω), and let F = {Ei}l +i=1 be a filtration of (E, φ) by saturated subsheaves. Let π : ˜X → +X be a blow-up along smooth complex submanifold and the family of metrics ωǫ be the +same as in the previous lemma. Let ( ˜E, ˜φ) be the pullback ˜V -twisted Higgs bundle and +˜F = { ˜Ei}l +i=1 = {Sat ˜E(π∗Ei)}l +i=1 be the filtration of ( ˜E, ˜φ). Suppose ˜Ei are subbundles, +and ˜Qi = ˜Ei/ ˜Ei−1 are ωǫ-stable for all 0 < ǫ ≤ ǫ∗. Then for any δ > 0, 1 ≤ p ≤ +∞ and +0 < ǫ ≤ ǫ∗, there is a smooth Hermitian metric ˜H on ˜E such that +∥ +√ +−1ΛωǫF¯∂ ˜ +E, ˜ +H + [˜φ, ˜φ∗ ˜ +H] − Ψ( ˜F, (µ1,ǫ, · · · , µl,ǫ), ˜H)∥Lp ≤ δ, +(4.18) +where µi,ǫ is the slope of quotient ˜Qi with respect to the metric ωǫ. +We also need the following additional propositions in next proof. +Proposition 4.16. Let (E, φ) be a V -twisted Higgs bundle on a smooth K¨ahler manifold +(X, ω), and let F = {Ei}l +i=1 be a filtration of (E, φ) by saturated subsheaves. Let π : +˜X → X be a blow-up along smooth complex submanifold of complex codimension k and +the family of metrics ωǫ be the same as in the previous lemma. Let ( ˜E, ˜φ) be the pullback +˜V -twisted Higgs bundle and ˜F = { ˜Ei}l +i=1 = {Sat ˜E(π∗Ei)}l +i=1 be the filtration of ( ˜E, ˜φ). +Suppose for any ˜δ > 0 and any 0 < ǫ ≤ ǫ∗, there is a smooth Hermitian metric ˜H on ˜E +such that +∥ +√ +−1ΛωǫF¯∂ ˜ +E, ˜H + [˜φ, ˜φ∗ ˜H] − Ψ( ˜F, (µ1,ǫ, · · · , µl,ǫ), ˜H)∥L2(ωǫ) ≤ ˜δ, +(4.19) +Then for any δ +′ > 0 and any 1 < p < 1 + +1 +4k(k−1), there is a smooth Hermitian metric H +on E such that +∥ +√ +−1ΛωF¯∂E,H + [φ, φ∗H] − Ψ(F, (µ1, · · · , µl), H)∥Lp(ω) ≤ δ +′, +(4.20) +where µi is the ω-slope of sheaf Qi. +Proof. Step 1: Let ǫ1 ∈ (0, ǫ∗), by the condition, we can choose a smooth metric ˜H1 +satisfies (4.19) for ǫ1 and ˜δ which will be chosen small enough later. For simplicity, we + +24 +C. Pan et al. +denote Θ1 = √−1F¯∂ ˜ +E, ˜ +H1. Then +∥ +√ +−1ΛωǫF¯∂ ˜ +E, ˜ +H1 + [˜φ, ˜φ∗ ˜H1] − Ψ( ˜F, (µ1, · · · , µl), ˜H1)∥Lp(ωǫ) +≤ +���Λωǫ{Θ1 + ωǫ1 +n [˜φ, ˜φ∗ ˜H1] − ωǫ1 +n Ψ( ˜F, (µ1,ǫ1, · · · , µl,ǫ1), ˜H1)} +��� +Lp(ωǫ) ++ +���1 +nΛωǫ(ωǫ − ωǫ1){[˜φ, ˜φ∗ ˜H1] − Ψ( ˜F, (µ1,ǫ1, · · · , µl,ǫ1), ˜H1)} +��� +Lp(ωǫ) ++ ∥Ψ( ˜F, (µ1, · · · , µl), ˜H1) − Ψ( ˜F, (µ1,ǫ1, · · · , µl,ǫ1), ˜H1)∥Lp(ωǫ) +(4.21) +Set +˜Ψǫ1 =[˜φ, ˜φ∗ ˜H1] − Ψ( ˜F, (µ1,ǫ1, · · · , µl,ǫ1), ˜H1), +Θ2 =Θ1 + ωǫ1 +n +˜Ψǫ1, +Θ3 = (ωǫ − ωǫ1)˜Ψǫ1, +where ∥˜Ψǫ1∥L2(ωǫ1) is uniformly bounded in ǫ1. Applying Lemma 4.14 to Θi, i = 2, 3, we +have +∥ΛωǫΘi∥Lp(ωǫ) ≤ C(∥Λωǫ1Θi∥L˜p(ωǫ1) + κ∥Θi∥L2(ωǫ1)) + ǫ1C(κ)∥Θi∥L2(ωǫ1). +(4.22) +Following the arguments in Sibley’s article ([38, Page 35]), if we choose κ and ǫ1 small +enough, we have +∥ΛωǫΘi∥Lp(ωǫ) ≤ δ +3. +(4.23) +On the other hand, since µi,ǫ → µi as ǫ → 0, we may choose ǫ1 small enough so that the +third term in (4.21) is also smaller than δ/3. Then for any δ > 0 and 1 < p < 1 + +1 +4k(k−1), +we have +∥ +√ +−1ΛωǫF¯∂ ˜ +E, ˜ +H1 + [˜φ, ˜φ∗ ˜H1] − Ψ( ˜F, (µ1, · · · , µl), ˜H1)∥Lp(ωǫ) ≤ δ +(4.24) +for any 0 < ǫ ≤ ǫ1, where µi is the slope of Qi with respect to the metric ω. +Step 2: In order to obtain a smooth metric on E, we need to use a cut-off argument. +Since Σ is a smooth complex submanifold, the open set {(x, v) ∈ NΣ| |v| < R} in the +normal bundle NΣ of Σ, is diffeomorphic to an open neighborhood UR of Σ for R suffi- +ciently small. For any small R, we may choose a smooth cut-off function ψR satisfying +suppψR ⊂ UR, ψR = 1 on UR/2, 0 ≤ ψR ≤ 1, and furthermore |∂ψR|2 +ω + |∂ ¯∂ψR|ω ≤ CR−2, +where C is a positive constant independent of R. +Let HD be a smooth Hermitian metric on bundle E, and ˜H1 be the metric on ˜E such +that (4.24) holds for all 0 < ǫ ≤ ǫ1 where δ ≤ δ +′ +4 . Note that E is isomorphic to ˜E outsides +Σ, we can define +HR = (1 − ψR) ˜H1 + ψRHD +(4.25) +on bundle E, ˜HR = π∗HR and ˜HD = π∗HD on bundle ˜E. + +Yang–Mills–Higgs Flow for Twisted Higgs Pairs +25 +Set Θ( ˜HR) = √−1F¯∂ ˜ +E, ˜HR, then we have +� +˜ +X +|ΛωǫΘ( ˜HR) + [˜φ, ˜φ∗ ˜ +HR] − Ψ( ˜F, (µ1, · · · , µl), ˜HR)|p +˜ +HR +ωn +ǫ +n! +≤ +� +π−1(UR/2) +|ΛωǫΘ( ˜HD) + [˜φ, ˜φ∗ ˜ +HD] − Ψ( ˜F, (µ1, · · · , µl), ˜HD)|p +˜HD +ωn +ǫ +n! ++ +� +˜ +X\π−1(UR) +|ΛωǫΘ( ˜H1) + [˜φ, ˜φ∗ ˜H1] − Ψ( ˜F, (µ1, · · · , µl), ˜H1)|p +˜H1 +ωn +ǫ +n! ++ C(p) +� +π−1(UR\UR/2) +|Λωǫ(Θ( ˜HR) − Θ( ˜H1))|p +˜ +HR +ωn +ǫ +n! ++ C(p) +� +π−1(UR\UR/2) +|ΛωǫΘ( ˜H1) + [˜φ, ˜φ∗ ˜ +HR] − Ψ( ˜F, (µ1, · · · , µl), ˜HR)|p +˜ +HR +ωn +ǫ +n! . +(4.26) +For the first term, we have +� +π−1(UR/2) +|ΛωǫΘ( ˜HD) + [˜φ, ˜φ∗ ˜ +HD] − Ψ( ˜F, (µ1, · · · , µl), ˜HD)|p +˜ +HD +ωn +ǫ +n! +≤C1 +� +π−1(UR/2) +� ηn +ωnǫ +�p−1ηn +n! ++ C2 +� +π−1(UR/2) +|[˜φ, ˜φ∗ ˜HD] − Ψ( ˜F, (µ1, · · · , µl), ˜HD)|p +˜ +HD +ωn +ǫ +n! , +(4.27) +where C1 and C2 are constants independent of ǫ and R. +For the third term, we have +|Λωǫ(F ˜ +HR − F ˜H1)| ˜ +HR ≤ +�C3 +R2 + C4 +� ηn +ωnǫ +, +(4.28) +where C3 and C4 are independent of ǫ and R. Then by H¨older’s inequality, we have +� +π−1(UR\UR/2) +|Λωǫ(Θ( ˜HR) − Θ( ˜H1))|p +˜ +HR +ωn +ǫ +n! +≤ +� � +π−1(UR\UR/2) +� ηn +ωn +ǫ +�(1−p)s ηn +n! +� 1 +s� � +π−1(UR\UR/2) +� C3 +R2˜p + C4 +�ηn +n! +� p +˜p, +(4.29) +where s and ˜p are constants as in Lemma 4.13. Hence +� +π−1(UR\UR/2) +� C3 +R2˜p + C4 +�ηn +n! ≤ C3R2n−2˜p + C4R2n. +(4.30) +Since p < 1 + +1 +4k(k−1), +p +1−2(k−1)(p−1) < +2k +2k−1(1 + +1 +4k(k−1)) ≤ 3 +2, we may choose ˜p < 2. We also +have +� +π−1(UR) +|[˜φ, ˜φ∗ ˜ +HD] − Ψ( ˜F, (µ1, · · · , µl), ˜HD)|p +˜HD +ωn +ǫ +n! → 0, +� +π−1(UR) +|[˜φ, ˜φ∗ ˜HR] − Ψ( ˜F, (µ1, · · · , µl), ˜HR)|p +˜HR +ωn +ǫ +n! → 0 +(4.31) + +26 +C. Pan et al. +as R → 0, uniformly in ǫ. +By above formulas and Lemma 4.13, choosing R small enough, we have +� +˜ +X +|ΛωǫΘ( ˜HR) + [˜φ, ˜φ∗ ˜ +HR] − Ψ( ˜F, (µ1, · · · , µl), ˜HR)|p +˜ +HR +ωn +ǫ +n! ≤ δ +′ +(4.32) +for all 0 < ǫ ≤ ǫ1. Now let ǫ → 0, we have +� +X +| +√ +−1ΛωF¯∂E,HR + [φ, φ∗HR] − Ψ(F, (µ1, · · · , µl), HR)|p +HR +ωn +n! ≤ δ +′. +(4.33) +□ +Theorem 4.17. Let (E, ¯∂A0, φ0) be a V -twisted Higgs bundle on a smooth K¨ahler manifold +(X, ω), and (A(t), φ(t)) be the smooth solution of the Yang–Mills–Higgs flow on Hermitian +vector bundle (E, H0) with initial data (A0, φ0). Suppose that for any δ +′ > 0 and any +1 < p < p0 there is a smooth metric H on (E, ¯∂A0, φ0) such that (4.20) holds, where ⃗µ0 is +the HN type of (E, ¯∂A0, φ0). Let (A∞, φ∞) be an Uhlenbeck limit of (At, φt), and (E∞, H∞) +be the corresponding Hermitian vector bundle defined away from Σan. Then +HYMα,N(A∞, φ∞) = lim +t→∞ HYMα,N(A(t), φ(t)) = HYMα,N(⃗µ0) +(4.34) +for all 1 < α < p0 and all N ∈ R; Moreover, the HN type of (E, ¯∂A0, φ0) is equal to the +HN type of (E∞, ¯∂A∞, φ∞). +Proof. The proof is the same as [28, 29], and we write the main part of the proof here for +the convenience of the reader. First, by triangle inequality and Lemma 4.6, we have +|(HYMα,N(A0, φ0, H))1/α − (HYMα,N(⃗µ0))1/α| +≤ +� � +X +|(ϕα( +√ +−1(θ(A0, φ0, H) + N · IdE)))1/α − (ϕα( +√ +−1(⃗µ0 + N)))1/α|αdvg +�1/α +≤ +� � +X +ϕα( +√ +−1(θ(A0, φ0, H) − Ψ(F, (µ1, · · · , µl), H)))dvg +�1/α +≤C(α)∥ +√ +−1ΛωF¯∂A0,H + [φ0, φ∗H +0 ] − Ψ(F, (µ1, · · · , µl), H)∥Lα. +(4.35) +Combining this with condition (4.20), we know that for any δ > 0 and any 1 < α < p0 +there is a metric H such that +HYMα,N(A0, φ0, H) ≤ HYMα,N(⃗µ0) + δ. +(4.36) +Since the image of the degree map on line bundles is discrete, for fixed α (1 < α ≤ p0) +and fixed N, we can define δ0 > 0 such that +2δ0 + HYMα,N(⃗µ0) = min{HYMα,N(⃗µ) : HYMα,N(⃗µ) > HYMα,N(⃗µ0)}, +(4.37) +where ⃗µ runs over all possible HN types of V -twisted Higgs sheaves with rank r. +Let H be a Hermitian metric on the complex bundle E, and (AH(t), φH(t)) be the +solution of the Yang–Mills–Higgs flow on Hermitian vector bundle (E, H) with initial +pair (AH +0 , φ0) ∈ BH where AH +0 is the Chern connection associated with (¯∂A0, H). Let +(AH +∞, φ∞) be an Uhlenbeck limit along the flow. We can choose the metric H such that +HYMα,N(A0, φ0, H) ≤ HYMα,N(⃗µ0) + δ0. +(4.38) + +Yang–Mills–Higgs Flow for Twisted Higgs Pairs +27 +From (4.7), (4.10) and Lemma 4.6, it follows that +HYMα,N(⃗µ0) ≤ HYMα,N(AH +∞, φ∞) ≤ HYMα,N(⃗µ0) + δ0. +(4.39) +Thus by the definition of δ0, we must have HYMα,N(AH +∞, φ∞) = HYMα,N(⃗µ0). This shows +that the result holds if the metric H0 satisfies (4.38). +Let H (E) be the space of all smooth metrics on E. For any fixed δ, we define a subset +of H (E) denoted by Hδ as follows: H ∈ Hδ if there exists some T ≥ 0 such that +HYMα,N(AH(t), φH(t)) < HYMα,N(⃗µ0) + δ +(4.40) +for all t ≥ T. In our proof, we may assume that 0 < δ ≤ δ0 +2 . It is easy to see that Hδ is +not empty. Following the argument in [14, Lemma 4.3] (or [28, Thoerem 5.13]), we can +show that Hδ is both closed and open in H (E). Since H (E) is connected, we obtain +that H (E) = Hδ. From the previous discussion, we have +HYMα,N(A∞, φ∞) = lim +t→∞ HYMα,N(A(t), φ(t)) = HYMα,N(⃗µ0). +(4.41) +Let ⃗λ∞ be the HN type of (E∞, ¯∂A∞, φ∞), by (4.41), we have ϕα(√−1(⃗µ0 + N)) = +ϕα(√−1(⃗λ∞ + N)) for all 1 < α < p0 and all N. Choosing N large enough, by Lemma +4.7, we get ⃗µ0 = ⃗λ∞. +□ +Let {πi}l +i=1 be a HN filtration of (E, A0, φ0), the action gj produces HN filtration {π(j) +i } +of (E, Aj, φj). Then +Lemma 4.18. Let (E, ¯∂A0, φ0) be a V -twisted Higgs bundle on a smooth K¨ahler manifold +(X, ω), and satisfy the same assumptions as that in Theorem 4.17. +(1) Let {π∞ +i } be the HN filtration of (E∞, ¯∂∞, φ∞), then there is a subsequence of HN +filtration {π(j) +i } converges to a filtration {π∞ +i } strongly in Lp ∩ L2 +1,loc off Σan for +all 1 ≤ p < +∞. +(2) Assume the V -twisted Higgs bundle (E, ¯∂A0, φ0) is semistable and {Es,i} is the +Seshadri filtration of (E, ¯∂A0, φ0), then, after passing to a subsequence, {π(j) +s,i } +converges to a filtration {π∞ +s,i} strongly in Lp ∩ L2 +1,loc off Σan for all 1 ≤ p < +∞, +the rank and degree of π∞ +i +is equal to the rank and degree of π(j) +s,i for all i and j. +Proof. See ([14, 29]) for details. +□ +Combining the previous Lemma 4.18, Corollary 3.3 and the fact that +√ +−1ΛωFA∞ + [φ∞, (φ∞)∗H∞] = ΨHN(A∞, φ∞, H∞), +(4.42) +we have the following proposition. +Proposition 4.19. Let (E, ¯∂A0, φ0) be a V -twisted Higgs bundle on compact K¨ahler man- +ifold (X, ω), and satisfy the same assumptions as that in Theorem 4.17. Then for any +δ > 0 and 1 ≤ p < ∞, (E, A0, φ0) has an Lp-δ-approximate critical Hermitian metric. +According to Proposition 4.10, we can resolve the singularities Σalg by blowing up +finitely many times along complex submanifolds, that is, we have a sequence of blow-ups +Xm +πm +−→ Xm−1 +πm−1 +−→ · · · +π1 +−→ X0 = X. +(4.43) + +28 +C. Pan et al. +Applying Proposition 4.16, Theorem 4.17 and Proposition 4.19 finitely many times, we +end up with the following theorems. +Theorem 4.20. Let (E, ¯∂A0, φ0) be a V -twisted Higgs bundle on compact K¨ahler manifold +(X, ω). Then for any δ > 0 and 1 ≤ p < +∞, (E, ¯∂A0, φ0) has an Lp- δ-approximate +critical Hermitian metric. +Theorem 4.21. Let (A(t), φ(t)) be a smooth solution of the Yang–Mills–Higgs flow on +the Hermitian vector bundle (E, H0) with initial twisted Higgs pair (A0, φ0), and (A∞, φ∞) +be a Uhlenbeck limit. Let E∞ denote the vector bundle obtained from (A∞, φ∞) as that in +Theorem 1.1. Then the Harder–Narasimhan type of the extended reflexive V -twisted Higgs +sheaf (E∞, ¯∂A∞, φ∞) is same as that of the original V -twisted Higgs bundle (E, ¯∂A0, φ0), +that is, ⃗λ∞ = ⃗µ0. +4.2. Construction of non-trivial holomorphic mappings. Let {Ei,j} be the HNS fil- +tration of the V -twisted Higgs bundle (E, ¯∂A0, φ0), the associated graded object GrHNS(E, ¯∂A0, φ0) +is uniquely determined by the isomorphism class of (¯∂A0, φ0). Let (E∞, ¯∂A∞, φ∞) be the +limiting V -twisted Higgs sheaf. In this subsection, we want to show GrHNS(E, ¯∂A0, φ0)∗∗ ≃ +(E∞, ¯∂A∞, φ∞). The key is to construct non-zero holomorphic mappings from the sub- +sheaves in the HNS filtration of (E, ¯∂A0, φ0) to (E∞, ¯∂A∞, φ∞). This result can be proved +by induction on the length of HNS filtration. The inductive hypotheses on a sheaf Q are +following: +(1) There is a sequence of V -twisted Higgs structures (AQ +j , φQ +j ) on Q such that +(AQ +j , φQ +j ) → (AQ +∞, φQ +∞) in C∞ +loc off Σalg ∪ Σan; +(2) (AQ +j , φQ +j ) = gj(AQ +0 , φQ +0 ) for some gj ∈ GC(Q); +(3) (Q, ¯∂AQ +0 , φQ +0 ) and (Q∞, ¯∂AQ +∞, φQ +∞) extended to X as reflexive V -twisted Higgs sheaves +with the same HN type; +(4) ∥φQ +j ∥C0 and ∥√−1Λω(FAQ +j )∥L1(ω) is uniformly bounded in j. +The following proposition is crucial, and its proof is the same as that of Proposition 4.1 +of ([29]). +Proposition 4.22 ([29]). Let (X, ω) be a K¨ahler manifold, (E, ¯∂A0, φ0) be a V -twisted +Higgs sheaf on X with Hermitian metric H0, S be a V -twisted Higgs subsheaf of (E, ¯∂A0, φ0), +and (Aj, φj) = gj(A0, φ0) be a sequence of V -twisted Higgs pairs on E, where gj is a se- +quence of complex gauge transformations. Suppose that there exits a sequence of blow-ups: +πi : Xi → Xi−1, i = 1, · · · , r (where X0 = X, every πi is a blow-up with non-singular +center; denoting π = πr ◦ · · · ◦ π1); such that π∗E and π∗S are bundles, the pulling +back geometric objects π∗(A0, φ0), π∗gj and π∗H0 can be extended smoothly on the whole +Xr. Assume that (Aj, φj) converges to (A∞, φ∞) outside a closed subset Σan of Haus- +dorff complex codimension 2, and |Λω(FAj)|H0 is bounded uniformly in j in L1(ω0). Let +i0 : (S, ¯∂A0) → (E, ¯∂A0) be the holomorphic inclusion, then there is a subsequence of gj ◦i0, +up to rescale, converges to a non-zero holomorphic map f∞ : (S, ¯∂A0) → (E∞, ¯∂A∞) in C∞ +loc +off Σ ∪ Σan, and f∞ ◦ φ0 = φ∞ ◦ f∞, where Σ is the singular set of S and E. +The proof of the following lemma is standard and can be found in [27]. + +Yang–Mills–Higgs Flow for Twisted Higgs Pairs +29 +Lemma 4.23. Let (E1, φ1) and (E2, φ2) be semistable V -twisted Higgs sheaves with rank(E1) = +rank(E2) and deg(E1) = deg(E2), let f : E1 → E2 be a nonzero sheaf homomorphism +satisfying f ◦ φ1 = φ2 ◦ f. If (E1, φ1) is stable, then f is an isomorphism. +Proof of Theorem 1.1 (2). The proof is the same as in [29, Sec. 5], which we write here +for the convenience of the reader. +Let S = E1,1 be the first stable V -twisted Higgs subsheaf corresponding to the HNS +filtration {Ei,j} of (E, ¯∂A0, φ0), π : +˜X → X be the resolution of singularities Σalg. +Then the filtration of ˜E = π∗E is given by subbundles ˜Ei,j, isomorphic to Ei,j off +˜Σ = π−1(Σalg). Setting ( ˜Aj, ˜φj) = π∗(Aj, φj) and ˜gj = π∗gj, then we have ( ˜Aj, ˜φj) = +˜gj( ˜A0, ˜φ0). By Theorem 1.1, we know that ( ˜Aj, ˜φj) converges to ( ˜A∞, ˜φ∞) in C∞ +loc topol- +ogy outside π−1(Σalg ∪ Σan). By Corollary 3.3 and the uniform C0 bound on φ(t), we +have ∥√−1Λω0(F ˜ +Aj)∥L∞, specially ∥√−1Λω0(F ˜ +Aj)∥L1 is uniformly bounded in j, where +ω0 = π∗ω. +Using Proposition 4.22, we have a non-zero smooth ˜φ-invariant holomorphic map ˜f∞ : +˜S → ˜E∞ off π−1(Σalg ∪ Σan). Since ˜S is isomorphic to S off ˜Σ, then we obtain a non- +zero smooth φ-invariant holomorphic map f∞ : S → (E∞, ¯∂A∞) on X \ Σan ∪ Σalg. By +Hartog’s theorem, f∞ extends to a V -twisted Higgs sheaf homomorphism f∞ : (S, φ0) → +(E∞, ¯∂A∞, φ∞) on X. +Let π(j) +1 +denotes the projection to gj(S). By the Lemma 4.18, we know that π(j) +1 +→ π∞ +1 +in Lp∩L2 +1,loc off Σan and π∞ +1 determines a V -twisted Higgs subsheaf E∞ +1,1 of (E∞, ¯∂A∞, φ∞), +with rank(E∞ +1,1) = rank(S) and µ(E∞ +1,1) = µ(S). Since (E∞, ¯∂A∞, φ∞) and (E0, ¯∂A0, φ0) +have the same HN type, thus we have the V -twisted Higgs subsheaf (E∞ +1,1, φ∞) is semistable +and +f∞ : S → E∞ +1,1. +(4.44) +Recall that S = E1,1 is V -twisted Higgs stable. By the Lemma 4.23, we see that the +non-zero holomorphic map f∞ must be injective on X \ Σan ∪ Σalg and E∞ +1,1 must be a +stable V -twisted Higgs subsheaf of (E∞, ¯∂A∞, φ∞). +Let {eα} be a local frame of S, and Hj,α¯β = ⟨gj(eα), gj(eβ)⟩H0. +We can write the +orthogonal projection π(j) +1 +as +π(j) +1 (X) = ⟨X, gj(eβ)⟩H0Hα¯β +j gj(eα) +(4.45) +for any X ∈ Γ(E), where (Hα¯β +j ) is the inverse of the matrix (Hj,α¯β). Because gj ◦i0 → f∞ +in C∞(Ω), and f∞ is injective on X \ Σan ∪ Σalg, then we can prove that π(j) +1 +→ π∞ +1 +in +C∞ +loc off Σan ∪ Σalg. +Let Q = E/S, then we have GrHNS(E, ¯∂A0, φ0) = S ⊕ GrHNS(Q, ¯∂AQ +0 , φQ +0 ). Write the +orthogonal holomorphic decomposition (E∞, ¯∂A∞, φ∞) = E∞ +1 ⊕ Q∞, where Q∞ = (E∞ +1 )⊥. +Using Lemma 5.12 in [13], we can choose a sequence of unitary gauge transformation +uj such that π(j) +1 += uj˜πju−1 +j +where ˜πj(E) = π∞ +1 (E) = E∞ +1 +and uj → IdE in C∞ +loc on +X \ (Σalg ∪ Σan). It is easy to check that uj(Q∞) = (π(j) +1 (E))⊥. Let p : Q → S⊥ be the +C∞ bundle isomorphism outside singularity set. Noting the unitary gauge transformation + +30 +C. Pan et al. +u0 : Q∞ → S⊥, and considering the induced V -twisted Higgs pair on Q, defined by +DAQ +j =p−1 ◦ u0 ◦ u−1 +j +◦ π⊥ +j ◦ DAj ◦ π⊥ +j ◦ uj ◦ u−1 +0 +◦ p, +φQ +j = p−1 ◦ u0 ◦ u−1 +j +◦ π⊥ +j ◦ φj ◦ π⊥ +j ◦ uj ◦ u−1 +0 +◦ p ∈ Γ(V ⊗ End(Q)). +(4.46) +Let +hj = p−1 ◦ u0 ◦ u−1 +j +◦ π⊥ +j ◦ gj ◦ π⊥ +0 ∈ GC(Q). +(4.47) +Then, we have +¯∂AQ +j =hj ◦ ¯∂AQ +0 ◦ h−1 +j , +∂AQ +j =(h∗ +j)−1 ◦ ∂AQ +0 ◦ h∗ +j, +(4.48) +φQ +j = hj ◦ φQ +0 ◦ h−1 +j , +(4.49) +and +¯∂AQ +j φQ +j = 0, +φQ +j ∧ φQ +j = 0, +(4.50) +where we have used h−1 +j += π⊥ +0 ◦ g−1 +j +◦ π⊥ +j ◦ uj ◦ u−1 +0 +◦ p. +On the other hand, by the +definition, it is easy to check that u∗ +0p(AQ +j , φQ +j ) → (AQ∞ +∞ , φQ∞ +∞ ) in C∞ +loc. Now we check the +third statement in the inductive hypotheses. Let’s consider the Gauss–Codazzi equation +on (π(j) +1 (E))⊥ ≃ Qj, +FAQj =(π(j) +1 )⊥ ◦ FAj ◦ (π(j) +1 )⊥ + ∂Ajπ(j) +1 +∧ ¯∂Ajπ(j) +1 , +[φQ +j , (φQ +j )∗] =(π(j) +1 )⊥ ◦ [φj, (φj)∗] ◦ (π(j) +1 )⊥ + [π(j) +1 , φ∗ +j] ◦ [φj, π(j) +1 ], +(4.51) +where DAQj = (π(j) +1 )⊥ ◦ DAj ◦ (π(j) +1 )⊥. Setting the V -twisted Higgs field φQj = (π(j) +1 )⊥ ◦ +φj ◦ (π(j) +1 )⊥, we have +� +X +| +√ +−1ΛωFAQ +j + [φQ +j , (φQ +j )∗] − ΨHN(AQ +j , φQ +j , H0)|ωn +n! += +� +X +| +√ +−1ΛωFAQj + [φQj, (φQj)∗] − ΨHN(AQj, φQj, H0)|ωn +n! += +� +X +|(π(j) +1 )⊥{ +√ +−1ΛωFAj + [φj, (φj)∗] − ΨHN(Aj, φj, H0)}(π(j) +1 )⊥ ++ +√ +−1Λω(∂Ajπ(j) +1 +∧ ¯∂Ajπ(j) +1 ) + [π(j) +1 , φ∗ +j] ◦ [φj, π(j) +1 ]|ωn +n! +≤ +� +X +| +√ +−1ΛωFAj + [φj, (φj)∗] − ΨHN(Aj, φj, H0)| + |¯∂Ajπ(j) +1 |2 ++ |[φj, π(j) +1 ]|2ωn +n! +→0. +(4.52) +Since C0 norm of φj is uniformly bounded, then ∥φQ +j ∥C0 and ∥√−1Λω(FAQ +j )∥L1(ω) is +uniformly bounded in j. So, (Q, AQ +j , φQ +j ) satisfy the inductive hypotheses. Since we can +resolve the singularity set Σalg by blowing up finitely many times with non-singular center, +and the pulling back of the HNS filtration is given by subbundles. The sheaf Q and every + +Yang–Mills–Higgs Flow for Twisted Higgs Pairs +31 +geometric objects which we considered are induced by the HNS filtration, so their pulling +back are all smooth. Using Proposition 4.22 again, by induction we have +E∞ ≃ GrHNS(E, ¯∂A0, φ0) = ⊕l +i=1 ⊕li +j=1 Qi,j +(4.53) +on X \(Σalg ∪Σan). By Theorem 1.1, we know that (E∞, ¯∂A∞, φ∞) can be extended to the +whole X as a reflexive V -twisted Higgs sheaf. By the uniqueness of reflexive extension in +[41], we know that there exists a sheaf isomorphism +f : GrHNS(E, ∂A0, φ0)∗∗ → (E∞, ¯∂A∞, φ∞) +(4.54) +on X. This completes the proof. +□ +Acknowledgements: The authors thank Prof. Xi Zhang for his consistent encourage- +ment. +The research was supported by the National Key R and D Program of China +2020YFA0713100. 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Zhang, The limit of the Hermitian–Yang–Mills flow on reflexive sheaves, +Adv. Math. 325 (2018), 165-214. +[31] J. Li, S.-T. Yau, Hermitian–Yang–Mills connection on non-K¨ahler manifolds, in: Mathematical +Aspects of String Theory, World Scientific, 1987, pp. 560-573. +[32] M. L¨ubke and A. Teleman, The Kobayashi–Hitchin correspondence, World Scientific Publishing Co., +Inc., River Edge, NJ, 1995. x+254 pp. +[33] T. Mochizuki, Kobayashi–Hitchin correspondence for analytically stable bundles, Trans. Am. Math. +Soc. 373 (2020), 551-596. +[34] M.S. Narasimhan and C.S. Seshadri, Stable and unitary vector bundles on a compact Riemann +surface, Ann. of Math. 82 (1965), 540-567. +[35] Y.C. Nie and X. Zhang, The limiting behaviour of the Hermitian–Yang–Mills flow over compact +non-K¨ahler manifolds, Sci. China Math. 63 (2020), no. 7, 1369-1390. +[36] N. Nitsure, Moduli space of semistable pairs on a curve, Proc. London Math. Soc. 62 (1991), no. 2, +275-300. +[37] Z. Shen, C. Zhang, X. Zhang, Flat Higgs bundles over non-compact affine Gauduchon manifolds, J. +Geom. Phys. 175 (2022), 104475. +[38] B. Sibley, Asymptotics of the Yang–Mills flow for holomorphic vector bundles over K¨ahler manifolds: +the canonical structure of the limit, J. Reine Angew. Math. 706 (2015), 123-191. +[39] C. Simpson, Constructing variations of Hodge structure using Yang–Mills theory and applications to +uniformization, J. Amer. Math. Soc. 1 (1988), no. 4, 867-918. +[40] C. Simpson, Higgs bundles and local systems, Publ. Math. IHES 75 (1992), 5-95. + +Yang–Mills–Higgs Flow for Twisted Higgs Pairs +33 +[41] Y.T. Siu, A Hartogs type extension theorem for coherent analytic sheaves, Ann. of Math. 93 (1971), +166-188. +[42] K.K. Uhlenbeck and S.T. Yau, On the existence of Hermitian–Yang–Mills connections in stable +vector bundles, Comm. Pure Appl. Math. 39 (1986), S257-S293. +[43] G. Wilkin, Morse theory for the space of Higgs bundles, Comm. Anal. Geom. 16 (2008), 283-332. +[44] D. Wu, X. Zhang, Higgs bundles over foliation manifolds, Sci. China Math. 64 (2021), 399-420. +[45] C. Zhang, P. Zhang, X. Zhang, Higgs bundles over non-compact Gauduchon manifolds, Trans. Am. +Math. Soc. 374 (2021), 3735-3759. +[46] W. Zhang, Convergence of Yang–Mills–Higgs flow for twist Higgs pairs on Riemann surfaces, Sci. +China Math. 57 (2014), 1657-1670. +Changpeng Pan, School of Mathematics and Statistics, Nanjing University of Science +and Technology, Nanjing, 210094,P.R. China, +Email address: mathpcp@njust.edu.cn +Zhenghan Shen, School of Mathematics and Statistics, Nanjing University of Science +and Technology, Nanjing, 210094,P.R. China, +Email address: mathszh@njust.edu.cn +Pan Zhang, School of Mathematical Sciences, Anhui University, Hefei, 230601, P.R. +China, +Email address: panzhang20100@ahu.edu.cn + diff --git a/otE0T4oBgHgl3EQfqgEA/content/tmp_files/load_file.txt b/otE0T4oBgHgl3EQfqgEA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b0bef361c7240f8de0072e5057c8eec3ec4cba00 --- /dev/null +++ b/otE0T4oBgHgl3EQfqgEA/content/tmp_files/load_file.txt @@ -0,0 +1,1177 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf,len=1176 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='02552v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='DG] 4 Jan 2023 THE LIMIT OF THE YANG–MILLS–HIGGS FLOW FOR TWISTED HIGGS PAIRS CHANGPENG PAN, ZHENGHAN SHEN AND PAN ZHANG Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' In this paper, we consider the Yang–Mills–Higgs flow for twisted Higgs pairs over K¨ahler manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' We prove that this flow converges to a reflexive twisted Higgs sheaf outside a closed subset of codimension 4, and the limiting twisted Higgs sheaf is isomorphic to the double dual of the graded twisted Higgs sheaves associated to the Harder–Narasimhan–Seshadri filtration of the initial twisted Higgs bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Introduction Let (X, ω) be a compact K¨ahler manifold, (V, hV ) be a Hermitian holomorphic vector bundle over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' A V -twisted Higgs bundle is a pair (E, φ), where E is a holomorphic vector bundle and φ : E → V ⊗ E is a holomorphic morphism satisfying 0 = φ ∧ φ ∈ End(E) ⊗ ∧2V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' For untwisted Higgs bundle (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' V = T ∗X), it was first studied by Hitchin ([23]) on Riemann surface, and by Simpson ([39, 40]) on K¨ahler manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' There are many interesting research about twisted Higgs bundles (see [7, 18, 19, 20, 36], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Let H be a Hermitian metric on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Then we can define a dual morphism φ∗H : E⊗V → E by using the Hermitian metrics H and hV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Let [φ, φ∗H] = φ ◦ φ∗H − φ∗H ◦ φ ∈ End(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' A Hermitian metric H is said to be Higgs–Hermitian–Einstein if it satisfies √ −1ΛωF¯∂E,H + [φ, φ∗H] = λ · IdE, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='1) where F¯∂E,H is the curvature form of the Chern connection D¯∂E,H and λ = 2πµω(E) Vol(X,ω) is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' The Donaldson–Uhlenbeck–Yau theorem for twisted Higgs bundles ([2, 8, 23, 39]) guarantees the existence of Higgs–Hermitian–Einstein metrics for the polystable case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' It was originally proved by Narasimhan–Seshadri ([34]), Donaldson ([15, 16]) and Uhlenbeck–Yau ([42]) for holomorphic bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' There are also many interesting and important generalized Donaldson–Uhlenbeck–Yau theorems (see [4, 5, 6, 9, 25, 31, 32, 33, 37, 44, 45] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Given a fixed Hermitian metric H on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Let A1,1 H be the space of integrable unitary con- nections, and BH be the space of V -twisted Higgs pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' The Yang–Mills–Higgs functional on BH is defined by YMH(A, φ) = � X (|FA|2 + 2|∂A,V φ|2 + |[φ, φ∗]|2 − 2⟨φ, φ⟩V )dvg, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='2) 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' 53C07, 58E15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Yang–Mills–Higgs flow, twisted Higgs pair, Harder–Narasimhan–Seshadri filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' 1 2 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' where dvg = ωn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' , ⟨φ, φ⟩V = tr(φ√−1ΛωFhV φ∗H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' The critical point of the Yang–Mills– Higgs functional satisfies � D∗ AFA + (∂A − ¯∂A)[φ, φ∗H] = 0, [ √ −1ΛωFA + [φ, φ∗H], φ] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='3) We say that (A, φ) ∈ BH is a Yang–Mills–Higgs pair if it is a critical point of the Yang–Mills–Higgs functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' By the K¨ahler identities, we know that if (A, φ) satisfies √−1ΛωFA + [φ, φ∗H] = λ · IdE, then it is a Yang–Mills–Higgs pair and H is the Higgs– Hermitian–Einstein metric of V -twisted Higgs bundle (E, ¯∂A, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' The gradient flow of the Yang–Mills–Higgs functional is \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 ∂A(t) ∂t = −D∗ A(t)FA(t) − (∂A(t) − ¯∂A(t))[φ(t), φ∗H(t)], ∂φ(t) ∂t = −[ √ −1ΛωFA(t) + [φ(t), φ∗H(t)], φ(t)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='4) The existence of long time solution for the above gradient flow will be discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' If the initial data (A0, φ0) ∈ BH is stable, the heat flow converges to a V -twisted Higgs pair and the limit must lie in the same orbit of the initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' In this article, we are interested in the convergence of this flow in the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Let (E, ¯∂A, φ) be a V -twisted Higgs bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' There is a filtration of (E, ¯∂A) given by φ- invariant subsheaves which is called Harder–Narasimhan–Seshadri (abbr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' HNS) filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Let GrHNS(E, ¯∂A, φ) be the associated graded object (the direct sum of the stable quo- tients) of the Harder–Narasimhan–Seshadri filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' For the holomorphic vector bundle, Atiyah–Bott ([1]) and Bando–Siu ([3]) conjectured that there should be a correspondence between the limit of the Yang–Mills flow and the double dual of GrHNS(E, ¯∂A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' It was proved by Daskalopoulos ([13]), Daskalopoulos–Wentworth ([14]) Jacob ([26]) and Sibley ([38]) in different cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' For the untwisted Higgs bundle, Wilkin ([43]) and Li–Zhang ([28, 29]) also proved the similar correspondence between Yang–Mills–Higgs flow and the double dual of GrHNS(E, ¯∂A, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' In [30], the authors proved this conjecture for reflexive sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' In the meanwhile, Zhang ([46]) considered this problem for T ∗X ⊗ L-twisted Higgs bundles over Riemann surface, and he proved the related correspondence in that case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' In the present paper, we extend the above results to V -twisted Higgs bundles over K¨ahler manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' In fact, we prove the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Let (A(t), φ(t)) be a solution of Yang–Mills–Higgs flow (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='4) with initial data (A0, φ0) ∈ BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Then we have: (1) for every sequence tk → +∞, there is a subsequence tkj such that as tkj → +∞, (A(tkj), φ(tkj)) converges modulo gauge transformations to a pair (A∞, φ∞) satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='3) on the Hermitian vector bundle (E∞, H∞) in C∞ loc topology outside Σ, where Σ is a closed set of Hausdorff codimension at least 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' The limiting (E∞, ¯∂A∞, φ∞) can be extended to the whole X as a reflexive V -twisted Higgs sheaf with a holomorphic orthogonal splitting (E∞, H∞, ¯∂A∞, φ∞) = ⊕l i=1(Ei ∞, Hi ∞, ¯∂Ai∞, φi ∞), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='5) Yang–Mills–Higgs Flow for Twisted Higgs Pairs 3 where Hi ∞ is an admissible Higgs–Hermitian–Einstein metric on the reflexive V - twisted Higgs sheaf (Ei ∞, ¯∂Ai∞, φi ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' (2) let {Ei,j} be the HNS filtration of the V -twisted Higgs bundle (E, ¯∂A0, φ0), the associated graded object GrHNS(E, A0, φ0) = ⊕l i=1 ⊕li j=1 Qi,j be uniquely deter- mined by the isomorphism class of (A0, φ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' We have GrHNS(E, A0, φ0)∗∗ ≃ (E∞, ¯∂A∞, φ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' In order to prove the convergence of the flow, we proved the energy inequality, the monotonicity formula of certain quantities and the ǫ-regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Following Hong–Tian’s arguments ([24]) and using Bando–Siu’s extension technique ([3]), we obtain the first part of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' The proof of the second part of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='1 can be divided into two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' The first step is to prove that the Harder–Narasimhan (abbr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' HN) type of the limiting V -twisted Higgs sheaf is in fact equal to that of (E, A0, φ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' The second step is to construct a non-zero φ-invariant holomorphic map from Qi,j to the limiting sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' The idea of the proof is the same as for untwisted case ([28, 29]), but there are some differences in the treatment of certain details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' In Section 2, we build some basic estimates for Donaldson heat flow and the Yang–Mills–Higgs flow for twisted Higgs pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' In Section 3, we prove the monotonicity inequality and the ǫ-regularity estimate for the Yang–Mills– Higgs flow and complete the first part of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' In Section 4, we first prove that the HN type of the limiting twisted Higgs sheaf is in fact equal to the type of the initial twisted Higgs bundle, and then complete the proof of the second part of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Preliminary Let (X, ω) be an n-dimensional compact K¨ahler manifold, and (E, φ) be a V -twisted Higgs vector bundle over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' We consider the following Donaldson heat flow \uf8f1 \uf8f2 \uf8f3 H−1(t)∂H(t) ∂t = −2( √ −1ΛωFH(t) + [φ, φ∗H(t)] − λ · IdE), H(0) = H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='1) It is a strictly parabolic equation, so the standard parabolic theory gives the short time existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' And the long time existence can be proved by the same method of Simpson’s article ([39]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' The following lemma can be obtained by direct calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Let H(t) be the solution of the flow (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='1) and set Φ(H(t)) = √−1ΛωFH(t) + [φ, φ∗H(t)] − λ · IdE, then � ∂ ∂t − ∆ � tr(Φ(H(t))) = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='2) and � ∂ ∂t − ∆ � |Φ(H(t))|2 H(t) = −4|¯∂EΦ(H(t))|2 ω,H(t) − 4|[φ, Φ(H(t))]|2 hV ,H(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='3) Let (E, H0) be a Hermitian vector bundle, and BH0 be the space of V -twisted Higgs pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Let GC (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' G) be the complex gauge group (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' unitary gauge group) of 4 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' (E, H0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' The complex gauge group GC acts on BH0 as follows σ · (¯∂A, φ) = (σ ◦ ¯∂A ◦ σ−1, σ ◦ φ ◦ σ−1), ∀σ ∈ GC, (A, φ) ∈ BH0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='4) Following the methods in [15, 28], we have the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' There is a family of complex gauge transformations σ(t) ∈ GC such that (A(t), φ(t)) = σ(t) · (A0, φ0) is a long time solution of the Yang–Mills–Higgs flow (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='4) with the initial data (A0, φ0), where σ∗H0(t)σ(t) = H−1 0 H(t) and H(t) is the long time solution of Donaldson heat flow (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='1) for (E, ¯∂A0, φ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Along the Yang–Mills–Higgs flow, we have the following energy identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Let (A(t), φ(t)) be a solution of the Yang–Mills–Higgs flow (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='4) with initial twisted Higgs pair (A0, φ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Then YMH(A(t), φ(t)) + 2 � t 0 � X ����� ∂A ∂t ���� 2 + ���� ∂φ ∂t ���� 2 + ���� ∂φ∗ ∂t ���� 2� dvgdt = YMH(A0, φ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='5) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Let (At,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' φt) be a family of twisted Higgs pairs and d dt ��� t=0(At,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' φt) = ( ˙A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' ˙φ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' then d dt ��� t=0YMH(At,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' φt) =2ℜ � X (⟨DA ˙A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' FA⟩ + 2⟨[ ˙A1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' φ] + ∂A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='V ˙φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' ∂A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='V φ⟩ + ⟨[ ˙φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' φ∗] + [φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' ˙φ∗],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' [φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' φ∗]⟩ − 2⟨ ˙φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' φ⟩V )dvg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' where ⟨[ ˙A1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' φ],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' ∂A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='V φ⟩ = ⟨ ˙A1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' [∂A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='V φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' φ∗]⟩ = −⟨ ˙A0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' [φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' ¯∂A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='V ∗φ∗]⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' ⟨ ˙φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' ∂∗ A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='V ∂A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='V φ⟩ − ⟨ ˙φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' φ⟩V = ⟨ ˙φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' [ √ −1ΛωFA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' φ]⟩ = −⟨ ˙φ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' [ √ −1ΛωFA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' φ∗]⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' and ⟨[ ˙φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' φ∗],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' [φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' φ∗]⟩ = ⟨ ˙φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' [[φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' φ∗],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' φ]⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' ⟨[φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' ˙φ∗],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' [φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' φ∗]⟩ = −⟨ ˙φ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' [[φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' φ∗],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' φ∗]⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Since ¯∂A,V φ = 0 and ∂A,V ∗φ∗ = 0, we have d dt ��� t=0YMH(At, φt) =2 � X (⟨ ˙A, D∗ AFA⟩ + ⟨ ˙A, (∂A − ¯∂A)[φ, φ∗]⟩ + ⟨ ˙φ, [ √ −1ΛωFA + [φ, φ∗], φ]⟩ − ⟨ ˙φ∗, [ √ −1ΛωFA + [φ, φ∗], φ∗]⟩)dvg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Using the Yang–Mills–Higgs flow equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='4), we have d dtYMH(A(t), φ(t)) = −2 � X ����� ∂A ∂t ���� 2 + ���� ∂φ ∂t ���� 2 + ���� ∂φ∗ ∂t ���� 2� dvg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='6) Integrating the above equality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='6) from 0 to t gives (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Let (A(t), φ(t)) be a solution of the heat flow (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='4) with initial twisted Higgs pair (A0, φ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Then � ∂ ∂t − ∆ � |φ|2 H0,hV ≤ − 2|∂A,V φ|2 − C3(|φ|2 + 1)2 + C4(|φ|2 + 1), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='7) Yang–Mills–Higgs Flow for Twisted Higgs Pairs 5 where C3, C4 are constants depending on supX |√−1ΛωFhV |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Moreover, we have sup X |φ|2 ≤ max{sup X |φ0|2, C4/C3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='8) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' By direct calculation, there holds � ∂ ∂t − ∆ � |φ|2 H0,hV = − 2|∂A,V φ|2 − 2⟨φ, [[φ, φ∗], φ]⟩ + 2⟨φ, √ −1ΛωFhV (φ)⟩ = − 2|∂A,V φ|2 − 2|[φ, φ∗]|2 + 2⟨φ, √ −1ΛωFhV (φ)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Let {vi} be a local orthonormal frame of V and φ = φivi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Since φ ∧ φ = 0, we have φiφj = φjφi and |[φ, φ∗]|2 = � i,j tr([φi, φ∗ i ][φj, φ∗ j]) = � i,j tr([φi, φ∗ j][φj, φ∗ i ]) = � i,j |[φi, φ∗ j]|2 ≥ � i |[φi, φ∗ i ]|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' According to the Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='7 in [40], we have |[φi, φi]|2 ≥ C1(|φi|2 + 1)2 − C2(|φi|2 + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' The above calculation leads to the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' □ In the following, we derive the local energy monotonic inequality along the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Let e2(A, φ) = |FA|2 + 2|∂A,V φ|2 and f ∈ C∞(X), then d dt � X f 2e2(A, φ)dvg =2ℜ � X ��∂A ∂t , D∗ A(f 2FA) � + f 2 �∂A ∂t , (∂A − ¯∂A)[φ, φ∗] � + 2 �∂φ ∂t , ∂∗ A,V (f 2∂A,V φ) �� dvg, where D∗ A(f 2FA) = √ −1[Λω, ¯∂A − ∂A](f 2FA) = √ −1Λω(¯∂ − ∂)(f 2) ∧ FA − f 2D∗ AFA − (¯∂ − ∂)(f 2) √ −1ΛωFA, and ∂∗ A,V (f 2∂A,V φ) = √ −1Λω ¯∂(f 2) ∧ ∂A,V φ + f 2∂∗ A,V ∂A,V φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' 6 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Therefore, d dt � X f 2e2(A, φ)dvg = 2 � X f 2 ��∂A ∂t , D∗ AFA � + �∂A ∂t , (∂A − ¯∂A)[φ, φ∗] � + �∂φ ∂t , [ √ −1ΛωFA, φ] � − �∂φ∗ ∂t , [ √ −1ΛωFA, φ∗] �� dvg + 2ℜ � X ��∂A ∂t , √ −1Λω(¯∂ − ∂)(f 2) ∧ FA � − �∂A ∂t , √ −1(¯∂ − ∂)(f 2)ΛωFA � + 2 �∂φ ∂t , √ −1Λω ¯∂(f 2) ∧ ∂A,V φ � + 2f 2 �∂φ ∂t , φ �� dvg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Let f be a cut-off function on B2R(x0), satisfy 0 ≤ f ≤ 1, f ≡ 1 on BR(x0) and |df| ≤ 2 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Then ���� d dt � X f 2e2(A, φ)dvg + 2 � X f 2 ����� ∂A ∂t ���� 2 + ���� ∂φ ∂t ���� 2 + ���� ∂φ∗ ∂t ���� 2� dvg ���� ≤C1 R � � X ����� ∂A ∂t ���� 2 + ���� ∂φ ∂t ���� 2 + ���� ∂φ∗ ∂t ���� 2� dvg �1/2 + C2 � � X ����� ∂A ∂t ���� 2 + ���� ∂φ ∂t ���� 2 + ���� ∂φ∗ ∂t ���� 2� dvg �1/2 , where C1, C2 are constants depending on supX |√−1ΛωFhV |, supX |φ0|, YMH(A0, φ0) and the geometry of (X, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Then we have the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Let (A(t), φ(t)) be a solution of the Yang–Mills–Higgs flow (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' then for any B2R(x0) ⊂ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' we have � BR(x0) e2(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' φ)(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' s)dvg ≤ � B2R(x0) e2(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' φ)(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' τ)dvg + 2 � max{s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='τ} min{s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='τ} � X ����� ∂A ∂t ���� 2 + ���� ∂φ ∂t ���� 2 + ���� ∂φ∗ ∂t ���� 2� dvgdt + C1 �|s − τ| R2 � max{s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='τ} min{s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='τ} � X ����� ∂A ∂t ���� 2 + ���� ∂φ ∂t ���� 2 + ���� ∂φ∗ ∂t ���� 2� dvgdt �1/2 + C2 � |s − τ| � max{s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='τ} min{s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='τ} � X ����� ∂A ∂t ���� 2 + ���� ∂φ ∂t ���� 2 + ���� ∂φ∗ ∂t ���� 2� dvgdt �1/2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' where C1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' C2 are constants depending on supX |√−1ΛωFhV |,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' supX |φ0|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' YMH(A0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' φ0) and the geometry of (X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Let θ(t) = √−1ΛωFA(t) + [φ(t), φ∗(t)] and I(t) = � X (|DA(t)θ(t)|2 ω,H0 + 2|[φ(t), θ(t)]|2 hV ,H0)dvg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Then we have the following proposition: Yang–Mills–Higgs Flow for Twisted Higgs Pairs 7 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' I(t) → 0, as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='9) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' The proof is exactly the same as untwisted case ([28]), so we omit here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' □ Let ∇A,V be the induced connection on Ω∗ X ⊗End(E) ⊗V induced by DA, DhV and the Chern connection on TM and ∇A be the covariant derivative corresponding to DA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Along the Yang–Mills–Higgs flow (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='4), we have � ∂ ∂t − ∆ � |∂A,V φ|2 + 2|∇A,V ∂A,V φ|2 ≤C1(|φ|2 + |FA| + |FhV | + |Ric|)|∂A,V φ|2 + C2|∂A √ −1ΛωFhV ||φ||∂A,V φ| (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='10) and � ∂ ∂t − ∆ � |FA|2 + |∇AFA|2 ≤C3(|FA| + |φ|2 + |Rm|)|FA|2 + C4|FA||∂A,V φ|2 + C5|φ|2|FhV ||FA|, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='11) where the constants Ci(i = 1, · · · , 5) are depending only on the dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' In local normal coordinates, we have ∆|∂A,V φ|2 = 2|∇A,V ∂A,V φ|2 + 2⟨∇α∇¯α∂A,V φ, ∂A,V φ⟩ + 2⟨∂A,V φ, ∇¯α∇α∂A,V φ⟩, where ∇α∇¯α∂A,V φ =∇α∇¯α∇βφdzβ = − ∇α(FA,V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='β ¯αφ)dzβ = − ∇α(FA,V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='β ¯α)φdzβ − FA,V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='β ¯α∇αφdzβ = − ∇β(FA,V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='α¯α)φdzβ − FA,V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='β,¯α∇αφdzβ = − ∂A( √ −1ΛωFA,V )φ − FA,V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='β,¯α∇αφdzβ = − [∂A( √ −1ΛωFA), φ] − ∂A( √ −1ΛωFhV )φ − FA,V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='β,¯α∇αφdzβ, ∇¯α∇α∂A,V φ =∇¯α∇α∇βφdzβ + ∇βφ∇¯α∇αdzβ =∇¯α∇β∇αφdzβ + ∇βφ∇¯α∇αdzβ =∇β∇¯α∇αφdzβ − FA,V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='β ¯α∇αφdzβ + ∇βφ∇¯α∇αdzβ = − ∂A( √ −1ΛωFA,V φ) − FA,V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='β ¯α∇αφdzβ + ∇βφ∇¯α∇αdzβ = − ∂A([ √ −1ΛωFA, φ] + √ −1ΛωFhV φ) − FA,V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='β ¯α∇αφdzβ + ∇βφ∇¯α∇αdzβ = − [∂A √ −1ΛωFA, φ] + [ √ −1ΛωFA, ∂Aφ] + (∂A √ −1ΛωFhV )φ + √ −1ΛωFhV ∂Aφ − FA,V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='β ¯α∇αφdzβ + ∇βφ∇¯α∇αdzβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' 8 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' On the other hand, using the heat flow equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='4), we have ∂ ∂t|∂A,V φ|2 ω,hV ,H0 =2ℜ ���∂A1,0 ∂t , φ � , ∂A,V φ � + � ∂A,V ∂φ ∂t , ∂A,V φ �� = − 2ℜ � ⟨[∂A( √ −1ΛωFA + [φ, φ∗]), φ], ∂A,V φ⟩ + ⟨∂A,V [ √ −1ΛωFA + [φ, φ∗], φ], ∂A,V φ⟩ � = − 2ℜ � 2⟨[∂A( √ −1ΛωFA + [φ, φ∗]), φ], ∂A,V φ⟩ + ⟨[ √ −1ΛωFA + [φ, φ∗], ∂A,V φ], ∂A,V φ⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='10) follows from the above identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' The proof of the other equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='11) is similar and we omit here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Convergence of the Yang–Mills–Higgs flow for twisted Higgs pairs In this section, we consider the convergence of the Yang–Mills–Higgs flow (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='4) for twisted Higgs pairs on the Hermitian bundle (E, H0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' We first prove the monotonicity in- equality and the ǫ-regularity theorem for the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' We will adapt the same arguments used in studying the Yang–Mills flow ([10, 11]) and the Yang–Mills–Higgs flow for untwisted Higgs pairs ([28]) to the Yang–Mills–Higgs flow for twisted Higgs pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Let u = (x, t) ∈ X × R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' For any u0 = (x0, t0) ∈ X × R+, set Sr(u0) = X × {t = t0 − r2}, Tr(u0) = X × [t0 − 4r2, t0 − r2], Pr(u0) = Br(x0) × [t0 − r2, t0 + r2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' For simplicity, we denote Sr(0, 0), Tr(0, 0), Pr(0, 0) by Sr, Tr, Pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' The fundamental solution of (backward) heat equation with singularity at u0 = (x0, t0) is Gu0(x, t) = G(x0,t0)(x, t) = 1 (4π(t0 − t))n exp � − |x − x0|2 4(t0 − t) � , t ≤ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' For simplicity, denote G(0,0)(x, t) by G(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Given 0 < R ≤ iX, we take f ∈ C∞ 0 (BR) satisfying 0 ≤ f ≤ 1, f ≡ 1 on BR/2 and |∇f| ≤ 2/R on BR \\ BR/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Let (A(t), φ(t)) be a solution of the Yang–Mills–Higgs flow with initial value (A0, φ0) and set e2(A, φ) = |FA|2 + 2|∂A,V φ|2, Φ(r) = r2 � Tr(u0) e2(A, φ)f 2Gu0dvgdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Then we have the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Let (A(t), φ(t)) be a solution of the Yang–Mills–Higgs flow (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='4) with ini- tial value (A0, φ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' For any u0 = (x0, t0) ∈ X×[0, T] and 0 < r1 ≤ r2 ≤ min{R/2, √t0/2}, Yang–Mills–Higgs Flow for Twisted Higgs Pairs 9 we have Φ(r1) ≤C exp(C(r2 − r1))Φ(r2) + C(r2 2 − r2 1) + CR2−2n � PR(u0) e2(A, φ)dvgdt, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='1) where the constant C depends only on the geometry of (X, ω), supX |√−1ΛωFhV | and the initial data (A0, φ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Choosing normal geodesic coordinates {xi}2n i=1 in the geodesic ball BR(x0), then it follows that |gij(x) − δij| ≤ C|x|2, |∂kgij(x)| ≤ C|x|, ∀x ∈ Br(x0), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='2) where C is a positive constant depending only on x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Let x = r˜x, t = t0 + r2˜t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' There holds that Φ(r) = r2 � Tr(u0) e2(A, φ)f 2Gu0dvgdt = r2 � t0−r2 t0−4r2 � R2n e2(A, φ)(x, t)f 2(x)Gu0(x, t) � det (gij)(x)dxdt = r4 � −1 −4 � R2n e2(A, φ)(r˜x, t0 + r2˜t)f 2(r˜x)G(˜x, ˜t) � det (gij)(r˜x)d˜xd˜t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Then one can see that dΦ(r) dr = 4r3 � −1 −4 � R2n e2(A, φ)(r˜x, t0 + r2˜t)f 2(r˜x)G(˜x, ˜t) � det (gij)(r˜x)d˜xd˜t + r3 � −1 −4 � R2n{xi∂ie2(A, φ)(r˜x, t0 + r2˜t)}f 2(r˜x)G(˜x, ˜t) � det (gij)(r˜x)d˜xd˜t + r3 � −1 −4 � R2n{2(t − t0)∂te2(A, φ)(r˜x, t0 + r2˜t)}f 2(r˜x)G(˜x, ˜t) � det (gij)(r˜x)d˜xd˜t + r4 � −1 −4 � R2n e2(A, φ)(r˜x, t0 + r2˜t) d dr{f 2(r˜x) � det (gij)(r˜x)}G(˜x, ˜t)d˜xd˜t =I1 + I2 + I3 + I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' For the second term I2, we have I2 =r � Tr(u0) {xi∂ie2(A, φ)(x, t)}f 2(x)Gu0(x, t) � det (gij)(x)dxdt, where xi∂ie2(A, φ) =2ℜ(⟨xi∇iFA, FA⟩ + 2⟨xi∇i∂A,V φ, ∂A,V φ⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' 10 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' By the Bianchi identity, we have 2⟨xi∇iFA, FA⟩ =⟨xi∇iFA(∂j, ∂k)dxj ∧ dxk, FA⟩ =⟨xi∇jFA(∂i, ∂k)dxj ∧ dxk, FA⟩ + ⟨xi∇kFA(∂j, ∂i)dxj ∧ dzk, FA⟩ =2⟨xiDA(FA,ikdxk) − xi(FA(∇j∂i, ∂k) + FA(∂i, ∇j∂k))dxj ∧ dxk, FA⟩ =2⟨DA(xiFA,ikdxk) − xiFA(∇j∂i, ∂k)dxj ∧ dxk, FA⟩ − 4|FA|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Set x ⊙ FA = 1 2xiFA,ijdxj, we have ℜ⟨x ⊙ FA, D∗ AFA⟩ = −ℜ⟨x ⊙ FA, ∂A ∂t ⟩ + ℜ⟨x ⊙ FA, (¯∂A − ∂A)[φ, φ∗]⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='3) In addition, ⟨xi∇i∇A,V φ, ∇A,V φ⟩ =⟨xi∇i(∇jφdxj), ∇A,V φ⟩ =⟨xi∇j∇iφdxj, ∇A,V φ⟩ + ⟨xiFA,V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='ijφdxj, ∇A,V φ⟩ + ⟨xi∇jφ∇idxj), ∇A,V φ⟩ =⟨DA,V (xi∇iφ), ∇A,V φ⟩ − ⟨xi∇A,V φ(∇i∂j)dxj), ∇A,V φ⟩ + ⟨xiFA,V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='ijφdxj, ∇A,V φ⟩ − |∇A,V φ|2, and ⟨xi∇iφ, ∂∗ A,V ∂A,V φ⟩ =⟨xi∇iφ, [ √ −1ΛωFA, φ] + √ −1ΛωFhV φ⟩ = − ⟨xi∇iφ, dφ dt ⟩ − ⟨xi∇iφ, [[φ, φ∗], φ] + √ −1ΛωFhV φ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Note that, for any α ∈ Ω1(End(E)), α∗ = −α, we have ℜ⟨α, (¯∂A − ∂A)[φ, φ∗]]⟩ + 2ℜ⟨[α, φ], ∂A,V φ⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Yang–Mills–Higgs Flow for Twisted Higgs Pairs 11 Set x ⊙ ∇A,V φ = 1 2xi∇A,V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='iφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Since Gu0 > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' we have I1 + I2 =4r � Tr(u0) |∂A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='V φ|2f 2Gu0dvgdt − 4rℜ � Tr(u0) ⟨d(f 2Gu0) ∧ x ⊙ FA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' FA⟩dvgdt − 8rℜ � Tr(u0) ⟨d(f 2Gu0)x ⊙ ∇A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='V φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' ∇A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='V φ⟩dvgdt − 4rℜ � Tr(u0) ⟨x ⊙ FA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' dA dt ⟩f 2Gu0dvgdt − 8rℜ � Tr(u0) ⟨x ⊙ ∇A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='V φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' dφ dt ⟩f 2Gu0dvgdt − 8rℜ � Tr(u0) ⟨x ⊙ ∇A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='V φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' [[φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' φ∗],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' φ] + √ −1ΛωFhV φ⟩f 2Gu0dvgdt − 2rℜ � Tr(u0) ⟨xiFA(∇j∂i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' ∂k)dxj ∧ dxk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' FA⟩f 2Gu0dvgdt − 4rℜ � Tr(u0) ⟨xi∇A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='V φ(∇j∂i)dxj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' ∇A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='V φ⟩f 2Gu0dvgdt + 8rℜ � Tr(u0) ⟨x ⊙ FV φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' ∇A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='V φ⟩f 2Gu0dvgdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' For the second term I3, we have I3 = 2r � Tr(u0) (t − t0)∂te2(A, φ)(x, t)f 2(x)Gu0(x, t) � det (gij)(x)dxdt, where ∂te2(A, φ) = 2ℜ �� DA �∂A ∂t � , FA � + 2 ��∂A1,0 ∂t , φ � , ∂A,V φ � + 2 � ∂A,V ∂φ ∂t , ∂A,V φ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' So we obtain that I3 = − 4rℜ � Tr(u0) (t − t0) ����∂A ∂t ��� 2 + 2 ���∂φ ∂t ��� 2� f 2Gu0dvgdt − 4rℜ � Tr(u0) (t − t0) � d(f 2Gu0) ∧ ∂A ∂t , FA � dvgdt − 8rℜ � Tr(u0) (t − t0) � d(f 2Gu0)∂φ ∂t , ∇A,V φ � dvgdt − 8rℜ � Tr(u0) (t − t0) �∂φ ∂t , [[φ, φ∗], φ] + √ −1ΛωFhV φ � f 2Gu0dvgdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' 12 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Note that ∂iGu0 = xiGu0 2(t−t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Set x · FA = 1 2gijxjFA,ikdxk, x · ∇A,V φ = 1 2xjgij∇A,V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='iφ, ∇f · FA = 2gijf −1∂jfFA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='ik, ∇f · ∇A,V φ = 2gijf −1∂jf∇iφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' For any α ∈ Ω1(End(E)), β ∈ Γ(V ⊗ End(E)), we have ⟨d(f 2Gu0) ∧ α, FA⟩ = ⟨α, ∇f · FA⟩f 2Gu0 + 1 t − t0 ⟨α, x · FA⟩f 2Gu0, and ⟨d(f 2Gu0)β, ∇A,V φ⟩ = ⟨β, ∇f · ∇A,V φ⟩f 2Gu0 + 1 t − t0 ⟨β, x · ∇A,V φ⟩f 2Gu0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Combining the above inequalities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' we have I1 + I2 + I3 = 4r � Tr(u0) 1 |t − t0| ���|t − t0|∂A ∂t − x ⊙ FA ��� 2 f 2Gu0dvgdt + 4rℜ � Tr(u0) 1 |t − t0| � x · FA − x ⊙ FA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' x ⊙ FA − |t − t0|∂A ∂t � f 2Gu0dvgdt + 4rℜ � Tr(u0) � |t − t0|∂A ∂t − x ⊙ FA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' ∇f · FA � f 2Gu0dvgdt + 8r � Tr(u0) 1 |t − t0| ���|t − t0|∂φ ∂t − x ⊙ ∇A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='V φ ��� 2 f 2Gu0dvgdt + 8rℜ � Tr(u0) 1 |t − t0| � x · ∇A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='V φ − x ⊙ ∇A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='V φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' x ⊙ ∇A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='V φ − |t − t0|∂φ ∂t � f 2Gu0dvgdt + 8rℜ � Tr(u0) � |t − t0|∂φ ∂t − x ⊙ ∇A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='V φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' ∇f · ∇A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='V φ � f 2Gu0dvgdt − 8rℜ � Tr(u0) ⟨x ⊙ ∇A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='V φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' [[φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' φ∗],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' φ] + √ −1ΛωFhV φ⟩f 2Gu0dvgdt − 2rℜ � Tr(u0) ⟨xiFA(∇j∂i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' ∂k)dxj ∧ dxk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' FA⟩f 2Gu0dvgdt − 4rℜ � Tr(u0) ⟨xi∇A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='V φ(∇j∂i)dxj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' ∇A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='V φ⟩f 2Gu0dvgdt + 8rℜ � Tr(u0) ⟨x ⊙ FV φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' ∇A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='V φ⟩f 2Gu0dvgdt − 8rℜ � Tr(u0) (t − t0) �∂φ ∂t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' [[φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' φ∗],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' φ] + √ −1ΛωFhV φ � f 2Gu0dvgdt + 4r � Tr(u0) |∂A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='V φ|2f 2Gu0dvgdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Yang–Mills–Higgs Flow for Twisted Higgs Pairs 13 By the Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='1 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='4, we have ���∂φ ∂t ��� 2 ≤ C, |φ|2 ≤ C, where the constant C depends on the initial data (A0, φ0) and supX |√−1ΛωFhV |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Since r ≤ R ≤ iX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' According to the Yang’s inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' we have I1 + I2 + I3 ≥ − Cr � Tr(u0) 1 |t − t0||x · FA − x ⊙ FA|2f 2Gu0dvgdt − Cr � Tr(u0) |t − t0||∇f · FA|2f 2Gu0dvgdt − Cr � Tr(u0) 1 |t − t0||x · ∇A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='V φ − x ⊙ ∇A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='V φ|2f 2Gu0dvgdt − Cr � Tr(u0) |t − t0||∇f · ∇A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='V φ|2f 2Gu0dvgdt − Cr � Tr(u0) |x|2|∇A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='V φ|2f 2Gu0dvgdt − 2rℜ � Tr(u0) ⟨xiFA(∇j∂i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' ∂k)dxj ∧ dxk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' FA⟩f 2Gu0dvgdt − 4rℜ � Tr(u0) ⟨xi∇A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='V φ(∇j∂i)dxj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' ∇A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='V φ⟩f 2Gu0dvgdt − Cr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' where the constant C depends on the initial data (A0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' φ0) and supX |√−1ΛωFhV |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' For the last term I4, we have I4 =r � Tr(u0) e2(A, φ)xi∂i(f 2� det (gij))Gu0dxdt =r � Tr(u0) e2(A, φ)2xif∂i(f)Gu0dvgdt + r � Tr(u0) e2(A, φ)f 2xi∂i( � det (gij))Gu0dxdt =r � Tr(u0) e2(A, φ)2xif∂i(f)Gu0dvgdt + r 2 � Tr(u0) e2(A, φ)xitr(g−1∂ig)f 2Gu0dvgdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Since |gij − δij| ≤ C|x|2, |∂igjk| ≤ C|x|, |Γk ij| ≤ C|x|, then |x · FA − x ⊙ FA|2 ≤ C|x|6|FA|2, |x · ∇A,V φ − x ⊙ ∇A,V φ|2 ≤ C|x|6|∇A,V φ|2, ⟨xiFA(∇k∂i, ∂j)dxj ∧ dxk, FA⟩ ≤ C|x|2|FA|2, tr(g−1∂ig) ≤ C|x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' 14 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Hence, we have dΦ(r) dr ≥ − Cr � Tr(u0) |x|6 |t − t0|e2(A, φ)f 2Gu0dvgdt − Cr � Tr(u0) |t − t0||∇f · FA|2f 2Gu0dvgdt − Cr � Tr(u0) |t − t0||∇f · ∇A,V φ|2f 2Gu0dvgdt − Cr � Tr(u0) |x|2e2(A, φ)f 2Gu0dvgdt − Cr � Tr(u0) f|∇f||x|e2(A, φ)Gu0dvgdt − Cr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' According to the Chen–Struwe’s arguments in [12], we know there exists a constant ˜C4 > 0 such that r−1|t − t0| · |x|6Gu0 ≤ ˜C4(1 + Gu0), r−1|x|2Gu0 ≤ ˜C4(1 + Gu0) on Tr(u0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Then it follows that − Cr � Tr(u0) � |x|6 |t − t0| + |t − t0| + |x|2� e2(A, φ)f 2Gu0dvgdt ≥ −CΦ(r) − CrYMH(A0, φ0), According to the arguments in [35, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' 1384], we have −r � Tr(u0) |t − t0| · |∇f · FA|2f 2Gu0dvgdt ≥ −C(n)r R2n � PR(u0) |FA|2dvgdt, −r � Tr(u0) |t − t0| · |∇f · ∇A,V φ|2f 2Gu0dvgdt ≥ −C(n)r R2n � PR(u0) |∇A,V φ|2dvgdt, −2r � Tr(u0) |x| · |∇f| · |f| · e2(A, φ)Gu0dvgdt ≥ −C(n)r R2n � PR(u0) e2(A, φ)dvgdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Combining the above inequalities, we have dΦ(r) dr ≥ −CΦ(r) − Cr − Cr R2n � PR(u0) e2(A, φ)dvgdt, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='4) where the constant C depends on the geometry of (X, ω), supX |√−1ΛωFhV | and the initial data (A0, φ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' By integrating the above inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='4) over r, we complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' □ Yang–Mills–Higgs Flow for Twisted Higgs Pairs 15 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Let (A(t), φ(t)) be a solution of the Yang–Mills–Higgs flow (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' There exist positive constants ǫ0, δ0 < 1/4, such that if R2−2n � PR(u0) e2(A, φ)dvgdt < ǫ0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='5) holds for some 0 < R ≤ min{iX/2, √t0/2}, then for any δ ∈ (0, δ0), we have sup PδR(u0) e2(A, φ) ≤ 16 (δR)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='6) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' For any δ ∈ (0, 1/4], we define the function f(r) = (2δR − r)4 sup Pr(x0,t0) e2(A, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Since f(r) is continuous and f(2δR) = 0, we know that f(r) attains its maximum at some point r0 ∈ [0, 2δR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Suppose (x1, t1) ∈ ¯Pr0(x0, t0) is a point such that e2(A, φ)(x1, t1) = sup Pr(x0,t0) e2(A, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' We claim that f(r0) ≤ 16 when ǫ0, δ0 are small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Otherwise, we have ρ0 := e2(A, φ)(x1, t1)−1/4 = (2δR − r0)f(r0)−1/4 < δR − r0 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Rescaling the Riemannian metric ˜g = ρ−2 0 g, ˜hV = ρ2 0hV and t = t1 + ρ2 0˜t, we get |FA|2 ˜g = ρ4 0|FA|2 g, |∂A,V φ|2 ˜g,˜hV = ρ4 0|∂A,V φ|2 g,hV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Set eρ0(x, ˜t) = |FA|2 ˜g + 2|∂A,V φ|2 ˜g,˜hV = ρ4 0e2(A, φ)(x, t1 + ρ2 0˜t), ˜P˜r(x1, 0) = Bρ0˜r(x1) × [−˜r2, ˜r2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Then we have eρ0(x1, 0) = ρ4 0e(A, φ)(x1, t1) = 1, and sup ˜P1(x1,0) eρ0 = ρ4 0 sup Pρ0(x1,t1) e(A, φ) ≤ ρ4 0 sup PδR+r0/2(x0,t0) e(A, φ) ≤ ρ4 0f(δR + r0/2)(δR − r0/2)−4 ≤ 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Thus |FA|2 ˜g + 2|∂A,V φ|2 ˜g,˜hV ≤ 16, on ˜P1(x1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='7) Combining above inequalities together with the Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='7 yields that � ∂ ∂˜t − ∆˜g � eρ0 = ρ6 0 � ∂ ∂t − ∆g � e2(A, φ) ≤C1ρ6 0(|φ|2 + |FA| + |FhV | + |Ric|)|∂A,V φ|2 + C2ρ6 0|∂A √ −1ΛωFhV ||φ||∂A,V φ| + C3ρ6 0(|FA| + |φ|2 + |Rm|)|FA|2 + C4ρ6 0|FA||∂A,V φ|2 + C5ρ6 0|φ|2|FhV ||FA| ≤C6(eρ0 + ρ8 0) 16 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' on ˜P1(x1, 0), where the constant C6 depends only on the geometry of (X, ω), FhV and supX |φ0|H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Then by the parabolic mean value inequality, we observe 1 < sup ˜P1/2(x1,0) (eρ0 + ρ8 0) ≤ C � ˜P1(x1,0) (eρ0 + ρ8 0)dv˜gd˜t = C7ρ2−2n 0 � Pρ0(x1,t1) e2(A, φ)dvgdt + C7ρ8 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='8) where the constant C7 depends only on the geometry of (X, ω), FhV and supX |φ0|H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' We choose normal geodesic coordinates centred at x1, and let f ∈ C∞ 0 (BR/2(x1)) be a smooth cut-off function such that 0 ≤ f ≤ 1, f ≡ 1 on BR/4(x1), |df| ≤ 8/R on BR/2(x1) \\ BR/4(x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Taking r1 = ρ0 and r2 = δ0R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' and applying the monotonicity inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' we obtain ρ2−2n 0 � Pρ0(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='t1) e2(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' φ)dvgdt ≤Cρ2 0 � Pρ0(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='t1) e2(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' φ)G(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='t1+2ρ2 0)f 2dvgdt ≤Cρ2 0 � Tρ0(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='t1+2ρ2 0) e2(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' φ)G(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='t1+2ρ2 0)f 2dvgdt ≤C∗r2 2 � Tr2(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='t1+2ρ2 0) e2(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' φ)G(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='t1+2ρ2 0)f 2dvgdt + C∗C10δ2 0R2 + C∗(R/2)2−2n � PR/2(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='t1) e2(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' φ)dvgdt ≤C∗δ2−2n 0 R2−2n � PR(x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='t0) e2(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' φ)dvgdt + C∗C10δ2 0R2 ≤C8(δ2−2n 0 ǫ0 + δ2 0R2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' where the constant C8 depends only on the geometry of (X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' ω),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' FhV and supX |φ0|H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Choosing ǫ0, δ0 small enough such that ˜C4 ˜C5(δ2−2n 0 ǫ0 + δ2 0R2) + ˜C4δ8 0R8 < 1, then a con- tradiction occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' So we have f(r0) ≤ 16, which implies sup PδR(u0) e2(A, φ) ≤ 16/(δR)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' □ Using the above ǫ-regularity theorem, and following the arguments of Hong and Tian ([24]) for the Yang–Mills flow case, we give the proof of the first part of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='1 (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content='3, for any tk → +∞ and a > 0, we have � tk+a tk−a � X ����∂A ∂t ��� 2 + ���∂φ ∂t ��� 2 + ���∂φ∗ ∂t ��� 2� dvgdt → 0, tk → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Yang–Mills–Higgs Flow for Twisted Higgs Pairs 17 Thus for any ǫ > 0, there is a constant K, when k > K there holds that � tk+a tk−a � X ����∂A ∂t ��� 2 + ���∂φ ∂t ��� 2 + ���∂φ∗ ∂t ��� 2 )dvgdt ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfqgEA/content/2301.02552v1.pdf'} +page_content=' Let Σ = � 0